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Contents
New results on modular Golomb rulers, optical orthogonal codes and related
structures
Marco Buratti, Douglas Robert Stinson . . . . . . . . . . . . . . . . . . . . 1
A family of fractal non-contracting weakly branch groups
Marialaura Noce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
From Farey fractions to the Klein quartic and beyond
Ioannis Ivrissimtzis, David Singerman, James Strudwick . . . . . . . . . . 37
On the incidence map of incidence structures
Tim Penttila, Alessandro Siciliano . . . . . . . . . . . . . . . . . . . . . . 51
On plane subgraphs of complete topological drawings
Alfredo García Olaverri, Javier Tejel Altarriba, Alexander Pilz . . . . . . . 69
Graphical Frobenius representations of non-abelian groups
Gábor Korchmáros, Gábor Péter Nagy . . . . . . . . . . . . . . . . . . . . 89
On few-class Q-polynomial association schemes: feasible parameters and
nonexistence results
Alexander L. Gavrilyuk, Janoš Vidali, Jason S. Williford . . . . . . . . . . 103
The enclaveless competition game
Michael A. Henning, Douglas F. Rall . . . . . . . . . . . . . . . . . . . . . 129
Strongly involutive self-dual polyhedra
Javier Bracho, Luis Montejano, Eric Pauli Pérez,
Jorge Luis Ramírez Alfonsín . . . . . . . . . . . . . . . . . . . . . . . . . 143
Geometry of the parallelism in polar spine spaces and their line reducts
Krzysztof Petelczyc, Krzysztof Prażmowski, Mariusz Żynel . . . . . . . . 151
Volume 20, Number 1, Spring/Summer 2021, Pages 1–170
iii

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 1–27
https://doi.org/10.26493/1855-3974.2374.9ff
(Also available at http://amc-journal.eu)
New results on modular Golomb rulers, optical
orthogonal codes and related structures*
Marco Buratti †
Dipartimento di Matematica e Informatica, Università di Perugia, 06123, Perugia, Italy
Douglas Robert Stinson ‡
David R. Cheriton School of Computer Science, University of Waterloo,
Waterloo, Ontario, N2L 3G1, Canada
Received 4 July 2020, accepted 12 October 2020, published online 26 November 2020
Abstract
We prove new existence and nonexistence results for modular Golomb rulers in this
paper. We completely determine which modular Golomb rulers of order k exist, for all
k ≤ 11, and we present a general existence result that holds for all k ≥ 3. We also
derive new nonexistence results for infinite classes of modular Golomb rulers and related
structures such as difference packings, optical orthogonal codes, cyclic Steiner systems and
relative difference families.
Keywords: Golomb ruler, optical orthogonal code, difference family.
Math. Subj. Class. (2020): 05B10
1 Introduction and definitions
A Golomb ruler of order k is a set of k distinct integers, say x1 < x2 < · · · < xk, such
that all the differences xj−xi (i ̸= j) are distinct. To avoid trivial cases, we assume k ≥ 3.
The length of the ruler is xk−x1. For a survey of constructions of Golomb rulers, see [12].
A (v, k)-modular Golomb ruler (or (v, k)-MGR) is a set of k distinct integers,
0 ≤ x1 < x2 < · · · < xk ≤ v − 1,
*We would like to thank Dieter Jungnickel and Hugh Williams for helpful comments and pointers to the
literature. We also thank Shannon Veitch for assistance with programming. Finally, we thank a referee for
pointing out a mistake in Corollary 3.4.
†This work has been performed under the auspices of the G.N.S.A.G.A. of the C.N.R. (National Research
Council) of Italy.
‡D. R. Stinson’s research is supported by NSERC discovery grant RGPIN-03882.
E-mail addresses: buratti@dmi.unipg.it (Marco Buratti), dstinson@uwaterloo.ca (Douglas Robert Stinson)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 20 (2021) 1–27
such that all the differences xj − xi mod v (i ̸= j) are distinct elements of Zv . We define
length and order as before. It is obvious that a modular Golomb ruler is automatically a
Golomb ruler. We can assume without loss of generality that x1 = 0.
Known results on modular Golomb rulers are summarized in [9, §VI.19.3]. We state a
few basic results and standard constructions now.
Theorem 1.1. If there exists a (v, k)-MGR, then v ≥ k2−k+1. Further, a (k2−k+1, k)-
MGR is equivalent to a cyclic (k2 − k + 1, k, 1)-difference set.
Of course a (q2 + q + 1, q + 1, 1)-difference set (i.e., a Singer difference set) is known
to exist if q is a prime power. So we have the following Corollary.
Corollary 1.2. There exists a (k2 − k + 1, k, 1)-MGR if k − 1 is a prime power.
It is widely conjectured that a (q2 + q + 1, q + 1, 1)-difference set exists only if q is a
prime power, and this conjecture has been verified for all q < 2, 000, 000; see [14].
Theorem 1.3 (Bose [3]). For any prime power q, there is a (q2 − 1, q)-MGR.
Theorem 1.4 (Rusza [21]). For any prime p, there is a (p2 − p, p− 1)-MGR.
A (v, k;n)-difference packing is a set of n k-element subsets of Zv , say X1, . . . , Xn,
such that all the differences in the multiset
{x− y : x, y ∈ Xi, x ̸= y, 1 ≤ i ≤ n}
are nonzero and distinct. The following result is obvious.
Theorem 1.5. A (v, k)-MGR is equivalent to a (v, k; 1)-difference packing.
A (v, b, r, k)-configuration is a set system (V,B), where V is a set of v points and B is
a set of b blocks, each of which contains exactly k points, such that the following properties
hold:
1. no pair of points occurs in more than one block, and
2. every point occurs in exactly r blocks.
It is easy to see that the parameters of a (v, b, r, k)-configuration satisfy the equation
bk = vr. For basic results on configurations, see [9, §VI.7]. A (v, b, r, k)-configuration is
symmetric if v = b, which of course implies r = k. In this case we speak of it as a sym-
metric (v, k)-configuration. A symmetric (v, k)-configuration is cyclic if there is a cyclic
permutation of the v points that maps every block to a block.
We state the following easy result without proof.
Theorem 1.6. A (v, k)-MGR is equivalent to a cyclic symmetric (v, k)-configuration.
For additional connections between Golomb rulers and symmetric configurations,
see [7, 10].
A (v, k, λa, λc)-optical orthogonal code of size n is a set C of n (0, 1)-vectors of length
v, which satisfies the following properties:
1. the Hamming weight of x is equal to k, for all x ∈ C,
M. Buratti and D. R. Stinson: New results on modular Golomb rulers, optical orthogonal . . . 3
2. autocorrelation: for all x = (x0, . . . , xv−1) ∈ C, the following holds for all integers
τ such that 0 < τ < v:
v−1∑
i=0
xixi+τ ≤ λa,
where subscripts are reduced modulo v.
3. cross-correlation: for all x = (x0, . . . , xv−1) ∈ C and all y = (y0, . . . , yv−1) ∈ C
with x ̸= y, the following holds for all integers τ such that 0 ≤ τ < v:
v−1∑
i=0
xiyi+τ ≤ λc,
where subscripts are reduced modulo v.
We sometimes abbreviate the phrase “optical orthogonal code” to “OOC.” If λa = λc = λ,
then the optical orthogonal code is denoted as a (v, k, λ)-optical orthogonal code.
Optical orthogonal codes were introduced by Chung, Salehi and Wei [8] in 1989 and
have been studied by numerous authors since then. The following result establishes the
equivalence of OOC and difference packings.
Theorem 1.7 ([8]). A (v, k;n)-difference packing is equivalent to a (v, k, 1)-optical or-
thogonal code of size n.
The following result is proven in [8] by a simple counting argument.
Theorem 1.8. If there exists a (v, k, 1)-optical orthogonal code of size n, then
n ≤
⌊
v − 1
k(k − 1)
⌋
.
A (v, k, 1)-optical orthogonal code is optimal if the relevant inequality in Theorem 1.8
is met with equality.
Relative difference families have been introduced in [5] as a natural generalization of
relative difference sets. We define them now. Let H be a subgroup of a finite additive
group G, and let k, λ be positive integers. A (G,H, k, λ)-relative difference family, or
(G,H, k, λ)-RDF for short, is a collection X of k-subsets of G (called base blocks) whose
list of differences has no element in H and covers all elements of G \H exactly λ times.
If G has order v and H has order w, we say that X is a (v, w, k, λ)-RDF in G relative to
H . If X consists of n base blocks, it is evident that
λ(v − w) = k(k − 1)n. (1.1)
When H = {0} (or, equivalently, if w = 1), one usually speaks of an ordinary (v, k, λ)-
difference family or (v, k, λ)-difference family ((v, k, λ)-DF, for short), in G. If n = 1,
then we refer to a (G,H, k, λ)-relative difference family as a (G,H, k, λ)-relative differ-
ence set. Analogously, a (v, k, λ)-difference family of size n = 1 is a (v, k, λ)-difference
set ((v, k, λ)-DS, for short).
4 Ars Math. Contemp. 20 (2021) 1–27
1.1 Number-theoretic background
In this section, we record some number-theoretic results that we will be using later in the
paper.
Theorem 1.9.
1. A positive integer can be written as a sum of two squares if and only if its prime
decomposition contains no prime p ≡ 3 (mod 4) raised to an odd power.
2. A positive integer can be written as a sum of three squares if and only if it is not of
the form 4a(8b+ 7), where a and b are nonnegative integers.
3. Any positive integer can be written as a sum of four squares.
Proof. Statement 1. is proven in many textbook on elementary number theory, e.g., [20,
Theorem 13.6]. The result 2. is known as Legendre’s Three-square Theorem (for a proof of
it, see, e.g., [18, Chapter 20, Theorem 1]). Finally, 3. is Lagrange’s Four-square Theorem.
Lemma 1.10. For any positive integer t, there exist t consecutive positive integers, none
of which is a sum of two squares.
Proof. Take t distinct primes p1, . . . , pt all of which are ≡ 3 (mod 4) (they exist by the
Dirichlet’s Theorem on primes in an arithmetic progression). By the Chinese Remainder
Theorem, the system of t congruences
x+ i ≡ pi (mod p2i ) (1 ≤ i ≤ t)
has a solution s.
Since s+ i ≡ pi (mod p2i ), it is clear that s+ i is divisible by pi, but not by p2i . Since
pi ≡ 3 (mod 4), it follows from Theorem 1.9 that s+ i is not a sum of two squares. This
holds for 1 ≤ i ≤ t.
Lemma 1.11. Two consecutive integers, say n and n+1, are both not expressible as a sum
of three squares if and only if n = 4a(8b+ 7)− 1, where a ≥ 2 and b ≥ 0.
Proof. This is a consequence of Legendre’s Three-square Theorem (Theorem 1.9). If n is
not expressible as a sum of three squares, then n ≡ 0, 4 or 7 (mod 8). Therefore, if n and
n + 1 are both not expressible as a sum of three squares, then n ≡ 7 (mod 8). It follows
from Legendre’s Three-square Theorem that n and n+1 are both not expressible as a sum
of three squares if and only if n+ 1 = 4a(8b+ 7) where a ≥ 2 and b ≥ 0.
1.2 Our contributions
Section 2 gives existence results for modular Golomb rulers. We summarize exhaustive
searches that we have carried out for all k ≤ 11, and we present a general existence result
that holds for all k ≥ 3. Section 3 proves nonexistence results for various infinite classes
of modular Golomb rulers. Many of our new results are based on counting even and odd
differences and then applying some classical results from number theory which establish
which integers can be expressed as a sum of a two or three squares. Section 4 studies opti-
cal orthogonal codes and provides nonexistence results for certain optimal OOCs. In Sec-
tion 5, we consider cyclic Steiner systems and relative difference families and we present
additional nonexistence results using the techniques we have developed. Finally, Section 6
is a brief summary.
M. Buratti and D. R. Stinson: New results on modular Golomb rulers, optical orthogonal . . . 5
2 Existence results for (v, k)-MGR
In this section, we report the results of exhaustive searches for (v, k)-MGR with k ≤ 11.
We also prove a general existence result that holds for all integers k ≥ 3. First, we discuss
a few preliminary results..
Given a positive integer k ≥ 3, define
MGR(k) = {v : there exists a (v, k)-MGR}.
We are interested in the set MGR(k). In particular, it is natural to try to determine the
minimum integer in MGR(k) as well as the maximum integer not in MGR(k).
Another parameter of interest is the length of a Golomb ruler. There has been consider-
able research done on finding the minimum length of a Golomb ruler of specified order k,
which we denote by L∗(k). In the modular case, we will define L∗m(k) to be the minimum
L such that there exists a (v, k)-MGR of length L for some v.
The following basic lemma is well-known.
Lemma 2.1. Suppose there is a Golomb ruler of order k and length L, and suppose v ≥
2L+ 1. Then there is a (v, k)-MGR.
Proof. We have a Golomb ruler consisting of k integers
0 = x1 < x2 < · · · < xk = L.
Consider these as residues modulo v, where v ≥ 2L+1. Clearly all the “positive residues”
xj − xi mod v (i < j) are nonzero and distinct, as are all the “negative residues” xj −
xi mod v (j < i). The largest positive residue is L and the smallest negative residue is
v − L. Since v > 2L, no positive residue is equal to a negative residue.
The following is an immediate consequence of Lemma 2.1.
Lemma 2.2. For any positive integer k ≥ 2, L∗(k) = L∗m(k).
Given a positive integer k ≥ 3, define
MGR(k) = {v : there exists a (v, k)-MGR}.
We have performed exhaustive backtracking searches in order to determine the sets MGR(k)
for 3 ≤ k ≤ 11. For each value of k, once we have constructed a sufficient number of
“small” (v, k)-MGR, we can apply Lemma 2.1 to conclude that all (v, k)-MGR exist for
larger values of v. To this end, when we compute all the (v, k)-MGR for given values of
v and k, we keep track of the ruler having the smallest possible length. This facilitates the
application of Lemma 2.1
Our computational results are summarized as follows.
Theorem 2.3.
1. MGR(3) = {v : v ≥ 7}.
2. MGR(4) = {v : v ≥ 13}.
3. MGR(5) = {21} ∪ {v : v ≥ 23}.
4. MGR(6) = {31} ∪ {v : v ≥ 35}.
6 Ars Math. Contemp. 20 (2021) 1–27
5. MGR(7) = {v : v ≥ 48}.
6. MGR(8) = {57} ∪ {v : v ≥ 63}.
7. MGR(9) = {73, 80} ∪ {v : v ≥ 85}.
8. MGR(10) = {91} ∪ {v : v ≥ 107}.
9. MGR(11) = {120, 133} ∪ {v : v ≥ 135}.
Proof. Proof details are in Table 1.
Table 1: (v, k)-modular Golomb rulers for 3 ≤ k ≤ 11.
v k ruler
v = 7 3 0, 1, 3
v ≥ 8 3 Lemma 2.1, v = 7, L = 3
v = 13 4 0, 1, 4, 6
v ≥ 14 4 Lemma 2.1, v = 13, L = 6
v = 21 5 0, 2, 7, 8, 11
v = 22 5 does not exist
v ≥ 23 5 Lemma 2.1, v = 21, L = 11
v = 31 6 0, 1, 4, 10, 12, 17
32 ≤ v ≤ 34 6 does not exist
v ≥ 35 6 Lemma 2.1, v = 31, L = 17
43 ≤ v ≤ 47 7 does not exist
v = 48 7 0, 5, 7, 18, 19, 22, 28
v = 49 7 0, 2, 3, 10, 16, 21, 25
v = 50 7 0, 1, 5, 7, 15, 18, 27
v ≥ 51 7 Lemma 2.1, v = 49, L = 25
v = 57, 64, 68 8 0, 4, 5, 17, 19, 25, 28, 35
58 ≤ v ≤ 62 8 does not exist
v = 63, 67 8 0, 1, 8, 20, 22, 25, 31, 35
v = 65 8 0, 2, 10, 11, 16, 28, 31, 35
v = 66 8 0, 2, 10, 21, 24, 25, 30, 37
v = 69 8 0, 1, 4, 9, 15, 22, 32, 34
v ≥ 70 8 Lemma 2.1, v = 69, L = 34
v = 73 9 0, 2, 10, 24, 25, 29, 36, 42, 45
74 ≤ v ≤ 79 9 does not exist
v = 80 9 0, 1, 12, 16, 18, 25, 39, 44, 47
81 ≤ v ≤ 84 9 does not exist
v = 85 9 0, 1, 7, 12, 21, 29, 31, 44, 47
v = 86, 88 9 0, 2, 5, 13, 17, 31, 37, 38, 47
v = 87 9 0, 1, 4, 13, 24, 30, 38, 40, 45
v = 89 9 0, 1, 5, 12, 25, 27, 35, 41, 44
v ≥ 90 9 Lemma 2.1, v = 89, L = 44
v = 91 10 0, 1, 6, 10, 23, 26, 34, 41, 53, 55
92 ≤ v ≤ 106 10 does not exist
v = 107 10 0, 2, 15, 21, 22, 32, 46, 50, 55, 58
M. Buratti and D. R. Stinson: New results on modular Golomb rulers, optical orthogonal . . . 7
Table 1: (v, k)-modular Golomb rulers for 3 ≤ k ≤ 11 (cont.)
v k ruler
v = 108 10 0, 2, 8, 27, 32, 36, 39, 49, 50, 65
v = 109 10 0, 4, 11, 16, 25, 35, 38, 53, 55, 61
v = 110 10 0, 3, 14, 16, 36, 37, 42, 46, 54, 61
v ≥ 111 10 Lemma 2.1, v = 91, L = 55
111 ≤ v ≤ 119 11 does not exist
v = 120 11 0, 1, 4, 9, 23, 30, 41, 43, 58, 68, 74
121 ≤ v ≤ 132 11 does not exist
v = 133 11 0, 1, 9, 19, 24, 31, 52, 56, 58, 69, 72
v = 134 11 does not exist
v = 135 11 0, 5, 7, 11, 31, 41, 49, 50, 63, 66, 78
v = 136 11 0, 2, 11, 27, 37, 42, 45, 59, 65, 66, 78
v = 137 11 0, 1, 16, 21, 24, 33, 43, 61, 68, 72, 74
v = 138 11 0, 4, 5, 23, 25, 37, 52, 59, 65, 68, 76
v = 139 11 0, 1, 3, 11, 25, 41, 45, 54, 60, 72, 77
v = 140 11 0, 4, 10, 24, 25, 27, 36, 43, 65, 73, 78
v = 141 11 0, 2, 3, 7, 20, 29, 41, 52, 60, 66, 76
v = 142 11 0, 1, 13, 16, 22, 33, 47, 51, 70, 75, 77
v = 143, 144 11 0, 3, 7, 22, 27, 43, 56, 57, 66, 68, 74
v ≥ 145 11 Lemma 2.1, v = 133, L = 72
Remark 2.4. Existence of a (110, 10)-MGR also follows from Theorem 1.4, and existence
of a (48, 7)-MGR and a (120, 11)-MGR follow from Theorem 1.3.
The rulers that are presented in Table 1 provide upper bounds on L∗m(k) for 3 ≤ k ≤ 11.
However, it turns out that all these values are in fact exact. This is because the exact values
of L∗(k) are known for small k (see, for example, [11, Table 2.2]) and they match the
minimum lengths of the modular Golomb rulers that we have recorded in Table 1. Thus we
have the following result.
Theorem 2.5. L∗m(3) = 3; L∗m(4) = 6; L∗m(5) = 11; L∗m(6) = 17; L∗m(7) = 25;
L∗m(8) = 34; L
∗
m(9) = 44; L
∗
m(10) = 55; and L
∗
m(11) = 72.
Now we state and prove two general existence results that hold for all k ≥ 3.
Theorem 2.6. For any integer k ≥ 3, there is a (v, k)-MGR for some integer v ≤ 3k2/2.
Proof. For 3 ≤ k ≤ 11, we refer to the results in Table 1. Indeed, for these values of k,
there is a (v, k)-MGR for some integer v ≤ k2 − 1.
For 12 ≤ k ≤ 24, we use Corollary 1.2. There is a (p2 + p+1, p+1, 1)-difference set
in Zp2+p+1 for p = 11, 13, 16, 17, 19 and 23. If we delete δ = p + 1 − k elements from
such a difference set, we obtain a (p2 + p+ 1, k)-MGR. For k = 12, we have p = 11 and
δ = 0; for k = 13, 14, we have p = 13 and δ ≤ 1; for 15 ≤ k ≤ 17, we have p = 16
and δ ≤ 2; for k = 18, we have p = 17 and δ = 0; for k = 19, 20, we have p = 19 and
δ ≤ 1; and for 21 ≤ k ≤ 24, we have p = 23 and δ ≤ 3. So, for 12 ≤ k ≤ 24, there is a
8 Ars Math. Contemp. 20 (2021) 1–27
(v, k)-MGR for some integer
v ≤ (k + δ − 1)2 + (k + δ − 1) + 1
≤ (k + 2)2 + k + 3
= k2 + 5k + 7.
It is easy to verify that k2 + 5k + 7 ≤ 3k2/2 if k ≥ 12.
Finally, suppose k ≥ 25. Let p be the smallest prime such that p ≥ k − 1. By a
result of Nagura [19], we have p ≤ 6(k − 1)/5 < 6k/5. From Corollary 1.2, there exists
a (p2 + p + 1, p + 1, 1)-difference set in Zp2+p+1. Delete p + 1 − k elements from this
difference set to obtain a (p2 + p+ 1, k)-MGR. We have
p2 + p+ 1 <
(
6k
5
)2
+
6k
5
+ 1
<
3k2
2
,
where the last inequality holds for k ≥ 21.
Theorem 2.7. For any integer k ≥ 3 and any integer v ≥ 3k2 − 1, there is a (v, k)-MGR.
Proof. From Theorem 2.6, there exists a (v, k)-MGR for some integer v ≤ 3k2/2. This
ruler has length L ≤ 3k2/2 − 1. Applying Theorem 2.1, there is a (v, k)-MGR for all
v ≥ 2(3k2/2− 1) + 1 = 3k2 − 1.
Remark 2.8. Of course there are stronger results known on gaps between consecutive
primes that hold for larger integers. For example, it was shown by Dusart [13] that, if
k ≥ 89693, then there is at least one prime p such that
k < p ≤
(
1 +
1
ln3 k
)
k.
So improved versions of Theorems 2.6 and 2.7 could be proven that hold for sufficiently
large values of k.
3 Nonexistence results for (v, k)-MGR
We present several nonexistence results for infinite classes of modular Golomb rulers in
this section.
3.1 (k2 − k + 2, k)-MGR
We have noted that v ≥ k2 − k + 1 if a (v, k)-MGR exists, and the (k2 − k + 1, k)-MGR
are equivalent to cyclic difference sets with λ = 1. There has been considerable study of
these difference sets and various nonexistence results are known. We do not discuss this
case further here, but we refer to [16, §8] for a good summary of known results.
The next case is v = k2 − k + 2. First, we note that there are two small examples
of (k2 − k + 2, k)-MGR, namely, an (8, 3)-MGR and a (14, 4)-MGR. These are found in
Table 1. In fact, these are the only examples that are known to exist. We now discuss some
nonexistence results for (k2 − k + 2, k)-MGR.
M. Buratti and D. R. Stinson: New results on modular Golomb rulers, optical orthogonal . . . 9
We next observe that (k2 − k+ 2, k)-MGR are equivalent to certain relative difference
sets in the cyclic group Zk2−k+2. The proof of this easy result is left to the reader.
Theorem 3.1. A (k2 − k + 2, k)-MGR is equivalent to a (Zk2−k+2, H, k, 1)-relative dif-
ference set, where H is the unique subgroup of order 2 in Zk2−k+2, i.e., H = {0, (k2 −
k + 2)/2}.
It is well-known that relative difference sets give rise to certain square divisible designs,
which we define now. A (w, u, k, λ1, λ2)-divisible design is a set system (actually, a type
of group-divisible design) on v = uw points and having blocks of size k, such that the
following conditions are satisfied:
1. the points are partitioned into u groups of size w,
2. two points in the same group occur together in exactly λ1 blocks, and
3. two points in different groups occur together in exactly λ2 blocks.
If the number of blocks is the same as the number of points, then we have a square divisible
design.
The following result is a consequence of Theorem 3.1, since a square divisible design
is obtained by developing a relative difference set through the relevant cyclic group.
Theorem 3.2. If there exists a (k2 − k + 2, k)-MGR, then there exists a square divisible
design with parameters w = 2, u = (k2 − k + 2)/2, λ1 = 0 and λ2 = 1.
We will make use of some results due to Bose and Connor [4], as stated in [15, Propo-
sition 1.8].
Theorem 3.3 (Bose and Connor). Suppose there exists a square divisible design with pa-
rameters w, u, k, λ1 and λ2. Denote v = uw. Then the following hold.
1. If u is even, then k2 − λ2v is a perfect square. If furthermore u ≡ 2 (mod 4), then
k − λ1 is the sum of two squares.
2. If u is odd and w is even, then k − λ1 is a perfect square and the equation
(k2 − λ2v)x2 + (−1)u(u−1)/2λ2wy2 = z2
has a nontrivial solution in integers x, y and z.
We can use Theorem 3.3 to obtain necessary conditions for the existence of (k2 − k +
2, k)-MGR.
Corollary 3.4. Suppose there exists a (k2 − k + 2, k)-MGR. Then the following hold.
1. k ̸≡ 7 (mod 8).
2. If k ≡ 2 (mod 8), then k − 2 is a perfect square and k is the sum of two squares.
3. If k ≡ 3, 6 (mod 8), then k − 2 is a perfect square.
4. If k ≡ 0, 1 (mod 8), then k is a perfect square and the equation
(k − 2)x2 + 2y2 = z2
has a nontrivial solution in integers x, y and z.
10 Ars Math. Contemp. 20 (2021) 1–27
5. If k ≡ 4, 5 (mod 8), then k is a perfect square and the equation
(k − 2)x2 − 2y2 = z2
has a nontrivial solution in integers x, y and z.
Proof. Suppose there exists a (k2 − k + 2, k)-MGR. Then, from Theorem 3.2, there is a
square divisible design with parameters w = 2, u = (k2 − k + 2)/2, v = k2 − k + 2,
λ1 = 0 and λ2 = 1. We apply Theorem 3.3, making use of the fact that k2 − λ2v = k− 2.
First, we observe that u is even if and only if k ≡ 2, 3 (mod 4). Further, u ≡ 2
(mod 4) if and only if k ≡ 2, 7 (mod 8).
If k ≡ 7 (mod 8), then k2 − λ2v = k − 2 ≡ 5 (mod 8), so k2 − λ2v is not a perfect
square. Therefore, from part 1. of Theorem 3.3, a (k2 − k + 2, k)-MGR does not exist if
k ≡ 7 (mod 8).
If k ≡ 2 (mod 8), then part 1. of Theorem 3.3 says that k − 2 is a perfect square and
k is the sum of two squares.
If k ≡ 3, 6 (mod 8), then part 1. of Theorem 3.3 says that k − 2 is a perfect square.
When k ≡ 0, 1 (mod 8), we have u ≡ 1 (mod 4) and hence (−1)u(u−1)/2 = 1.
When k ≡ 4, 5 (mod 8), we have u ≡ 3 (mod 4) and hence (−1)u(u−1)/2 = −1. The
stated results then follow immediately from part 2. of Theorem 3.3.
3.2 (k2 − k + 2ℓ, k)-MGR
For v > k2−k+2, a (v, k)-MGR is not necessarily a relative difference set and it does not
necessarily imply the existence of a square divisible design. So, in general, we cannot apply
the results in Theorem 3.3. However, we can derive some nice necessary conditions for the
existence of certain (v, k)-MGR using elementary counting arguments. These arguments
are in the spirit of techniques introduced in [6, §2]; see also [17]. Before studying MGR,
we present a simple example to illustrate the basic idea.
Example 3.5. Suppose we have a (v, k, λ)-difference set in Zv when v is even. There are
v/2 − 1 nonzero even differences and v/2 odd differences, each of which occurs λ times.
Suppose the difference set consists of a even elements and b odd elements. Then a+ b = k
and 2ab = λv/2. So a and b are the solutions of the quadratic equation
x2 − kx+ λv
4
= 0.
Since a and b are integers, the discriminant must be a perfect square. Therefore, k2 − λv
is a square. However, k(k − 1) = λ(v − 1), so k2 − λv = k − λ must be a perfect square.
(Of course, this condition is the same as in the Bruck-Ryser-Chowla Theorem for v even,
which holds for any symmetric BIBD.)
In the next theorem, we will use this counting technique to obtain necessary conditions
for the existence of a (k2 − k + 2ℓ, k)-MGR for a given integer ℓ ≥ 1. First, we give a
couple of definitions that will be useful in the rest of the paper.
Suppose X is a (v, k)-MGR. Define
∆X = {x− y mod v : x, y ∈ X,x ̸= y}
and
L(X) = Zv \∆X.
M. Buratti and D. R. Stinson: New results on modular Golomb rulers, optical orthogonal . . . 11
Note that ∆X consists of all the differences obtained from pairs of distinct elements in X
and L(X) is the complement of ∆X . The set L(X) is called the leave of X . For i = 0, 1,
define Li(X) to consist of the elements of L(X) that are congruent to i modulo 2.
The following lemma is straightforward but useful.
Lemma 3.6. Suppose X is a (v, k)-MGR where v is even. Then {0, v/2} ⊆ L(X). If v ≡ 0
(mod 4), then |L0(X)| and |L1(X)| are both even. If v ≡ 2 (mod 4), then |L0(X)| and
|L1(X)| are both odd.
Proof. It is evident that 0 ∈ L(X). Also, if we have x − y = v/2 for some pair (x, y) ∈
X × X , then we have y − x = v/2 as well. This would imply that v/2 appears at least
twice as a difference, which is not allowed. Hence {0, v/2} ⊂ L(X).
Now note that if d ∈ ∆X , then v − d ∈ ∆X as well. Consequently, if d ∈ L(X), then
v−d ∈ L(X). Of course d = v−d if and only if d = 0 or d = v/2. The remaining elements
of Zv can be matched into pairs (d, v − d) having the same parity. Thus, considering that
v/2 is even or odd according to whether v ≡ 0 or 2 modulo 4, respectively, it is clear that
|L1(X)| and |L2(X)| are both even in the first case and both odd in the second.
Theorem 3.7. Suppose v = k2− k+2ℓ, where ℓ ≥ 1, and suppose there is a (v, k)-MGR.
Then the following hold.
1. If v ≡ 2 (mod 4), then k − 2ℓ + 2 + 4i is a perfect square for some integer i ∈
{0, . . . , ℓ− 1}.
2. If v ≡ 0 (mod 4), then k − 2ℓ+ 4i is a perfect square for some integer i ∈ {0, . . . ,
ℓ− 1}.
Proof. Let X be a (v, k)-MGR. Since |X| = k, we have
|L(X)| = v − (k2 − k) = 2ℓ.
Suppose X contains a even elements and b odd elements; then a+ b = k.
Suppose first that v ≡ 2 (mod 4), so v/2 is odd. From Lemma 3.6, |L1(X)| is odd,
say |L1(X)| = 2i+ 1, and v/2 ∈ L1(X). Therefore, 0 ≤ i ≤ ℓ− 1.
The quantity 2ab is equal to the number of odd differences in ∆X , so
2ab =
v
2
− (2i+ 1) = v − 2− 4i
2
.
It follows that a and b are the solutions of the quadratic equation
x2 − kx+ v − 2− 4i
4
= 0.
The solutions a and b must be integers, which can happen only if the discriminant is a
perfect square. Hence,
k2 − (v − 2− 4i) = k − 2ℓ+ 2 + 4i
is a perfect square. Hence, k − 2ℓ + 2 + 4i is a perfect square for some integer i ∈
{0, . . . , ℓ− 1}.
12 Ars Math. Contemp. 20 (2021) 1–27
The proof is similar when v ≡ 0 (mod 4). Here, from Lemma 3.6, |L1(X)| is even,
say |L1(X)| = 2i and {0, v/2} ⊆ L0(X). Since {0, v/2} ⊆ L0(X), we have |L1(X)| ≤
2ℓ− 2. Hence i ∈ {0, . . . , ℓ− 1}.
We have
2ab =
v
2
− 2i = v − 4i
2
.
It follows that a and b are the solutions of the quadratic equation
x2 − kx+ v − 4i
4
= 0.
The solutions a and b must be integers, which can happen only if the discriminant is a
perfect square. Hence,
k2 − (v − 4i) = k − 2ℓ+ 4i
is a perfect square. Hence, k − 2ℓ + 4i is a perfect square for some integer i ∈ {0, . . . ,
ℓ− 1}.
Example 3.8. Suppose k = 10 and v = 94 = 10×9+2×2, ℓ = 2. Here v ≡ 2 (mod 4).
Then we compute
10− 2× 2 + 2 + 4i = 8 + 4i
for i = 0, 1, obtaining 8 and 12. Neither of these is a perfect square, so we conclude that a
(94, 10)-MGR does not exist.
It is interesting to see what Theorem 3.7 tells us when ℓ = 1.
Corollary 3.9. Suppose there is a (k2 − k + 2, k)-MGR. Then the following hold.
1. If k ≡ 2, 3 (mod 4), then k − 2 is a perfect square.
2. If k ≡ 0, 1 (mod 4), then k is a perfect square.
Proof. Take ℓ = 1 in Theorem 3.7; then v = k2 − k + 2 and we have i = 0. We note
that v ≡ 0 (mod 4) if k ≡ 2, 3 (mod 4) and v ≡ 2 (mod 4) if k ≡ 0, 1 (mod 4), so the
stated results follow immediately.
Remark 3.10. We observe that Theorem 3.2 and Corollary 3.4 provide stronger necessary
conditions for the existence of (k2 − k + 2, k)-MGR than those stated in Corollary 3.9.
For certain values of k, we are able to find “intervals” in which MGR cannot exist.
Define
Sk,ℓ = {k − 2ℓ+ 2 + 4i : 0 ≤ i ≤ ℓ− 1}
and define
Tk,ℓ = {k − 2ℓ+ 4i : 0 ≤ i ≤ ℓ− 1}.
Lemma 3.11. Suppose v = k2 − k + 2ℓ.
1. If v ≡ 2 (mod 4), then all elements of Sk,ℓ are ≡ 0, 1 (mod 4).
2. If v ≡ 0 (mod 4), then all elements of Tk,ℓ are ≡ 0, 1 (mod 4).
M. Buratti and D. R. Stinson: New results on modular Golomb rulers, optical orthogonal . . . 13
Proof. We prove 1. Suppose v = 2ℓ+k2−k ≡ 2 (mod 4). Then 2ℓ+k2−k−2−4i ≡ 0
(mod 4). It follows that k2 ≡ k − 2ℓ+ 2 + 4i (mod 4). Since k2 ≡ 0, 1 (mod 4) for all
integers k, the result follows.
The proof of 2. is similar.
Theorem 3.12. Let t be a positive integer.
1. If k = 4t2 + 4t+ 4, then there does not exist a (k2 − k + 4s, k)-MGR for all s such
that 1 ≤ s ≤ t.
2. If k = 4t2 + 4t+ 2, then there does not exist a (k2 − k + 4s, k)-MGR for all s such
that 1 ≤ s ≤ t.
3. If k = 4t2 + 3, then there does not exist a (k2 − k + 4s− 2, k)-MGR for all s such
that 1 ≤ s ≤ t.
4. If k = 4t2 + 1, then there does not exist a (k2 − k + 4s− 2, k)-MGR for all s such
that 1 ≤ s ≤ t.
Proof. We prove 1. Denote ℓ = 2s, where 1 ≤ s ≤ t and let v = k2 − k + 4s. Since
k ≡ 0 (mod 4), we have v ≡ 0 (mod 4). So we examine the elements in Tk,ℓ, which are
all congruent to 0 modulo 4 by Lemma 3.11. For the smallest element of Tk,ℓ, which is
k − 2ℓ, we have
k − 2ℓ ≥ k − 4t
= 4(t2 + t+ 1)− 4t
= 4t2 + 4
> (2t)2.
Similarly, for the largest element of Tk,ℓ, which is k − 2ℓ+ 4(ℓ− 1), we have
k − 2ℓ+ 4(ℓ− 1) ≤ k + 4t− 4
= 4(t2 + t+ 1) + 4t− 4
= 4t2 + 8t
< (2t+ 2)2.
Since all the elements of Tk,ℓ are congruent to 0 modulo 4 and they are between two
consecutive even squares, there cannot be any perfect squares in the set Tk,ℓ.
The proofs of 2., 3. and 4. are similar.
Example 3.13. If we take t = 3 in Theorem 3.12, we see that there does not exist a
(k2 − k + 4, k)-MGR, a (k2 − k + 8, k)-MGR or a (k2 − k + 12, k)-MGR when k =
50, 52. Further, there does not exist a (k2 − k + 2, k)-MGR, a (k2 − k + 6, k)-MGR or a
(k2 − k + 10, k)-MGR when k = 37, 39.
We will show that we can improve Theorem 3.7 when v ≡ 0 (mod 4). First we state
and prove a simple numerical lemma.
14 Ars Math. Contemp. 20 (2021) 1–27
Lemma 3.14. Let a be a positive integer. Then{
h(a− h) : 0 ≤ h ≤
⌊a
2
⌋}
=
{(a
2
)2
−
(a
2
− h
)2
: 0 ≤ h ≤
⌊a
2
⌋}
. (3.1)
Further, if a is even, then{
h(a− h) : 0 ≤ h ≤ a
2
}
=
{(a
2
)2
− h2 : 0 ≤ h ≤ a
2
}
. (3.2)
Proof. Clearly we have
h(a− h) =
(a
2
)2
−
(a
2
− h
)2
.
Therefore (3.1) holds. If a is even, then{(a
2
)2
−
(a
2
− h
)2
: 0 ≤ h ≤ a
2
}
=
{(a
2
)2
− h2 : 0 ≤ h ≤ a
2
}
,
and (3.2) holds.
Theorem 3.15. Suppose that X is a (v, k)-MGR with v = k2−k+2ℓ. Then the following
hold.
1. If v ≡ 0 (mod 8), then there exist integers i ∈ {0, 1, . . . , ℓ− 1} and j ∈ {0, 1, . . . ,
ℓ− 1− i} such that k− 2ℓ+4i is a perfect square and k− 2ℓ+2i+4j is a sum of
two squares.
2. If v ≡ 4 (mod 8), then there exist integers i ∈ {0, 1, . . . , ℓ− 1} and j ∈ {0, 1, . . . ,
ℓ− 1− i} such that that k− 2ℓ+ 4i is a perfect square and k− 2ℓ+ 2i+ 4j + 2 is
a sum of two squares.
Proof. Suppose v ≡ 0 (mod 8); then v/2 ≡ 0 (mod 4). From Lemma 3.6 and the
proof of Theorem 3.7, there are an even number, say 2i, of odd elements in L(X), where
0 ≤ i ≤ ℓ − 1. The number of elements ≡ 2 (mod 4) that are in L(X) is also even, say
2j, and we must have 0 ≤ j ≤ ℓ− 1− i.
Let a and b be the number of even and odd elements in X , respectively. We showed in
the proof of Theorem 3.7 that a and b are the solutions to the quadratic equation
x2 − kx+ v
4
− i = 0,
and hence a+ b = k, ab = v4 − i, and k
2 − v + 4i = k − 2ℓ+ 4i is a perfect square.
Let nα be the number of elements of X that are congruent to α modulo 4, for α =
0, 1, 2, 3. It is evident that n0 + n2 = a and that n1 + n3 = b. Thus, from (3.1) in
Lemma 3.14, we have
n0n2 ∈
{
h(a− h) : 0 ≤ h ≤
⌊a
2
⌋}
=
{(a
2
)2
−
(a
2
− h
)2
: 0 ≤ h ≤
⌊a
2
⌋}
and
n1n3 ∈
{
h(b− h) : 0 ≤ h ≤
⌊
b
2
⌋}
=
{(
b
2
)2
−
(
b
2
− h
)2
: 0 ≤ h ≤
⌊
b
2
⌋}
.
M. Buratti and D. R. Stinson: New results on modular Golomb rulers, optical orthogonal . . . 15
Multiplying by four, we get:
4n0n2 ∈
{
a2 − (a− 2h)2 : 0 ≤ h ≤
⌊a
2
⌋}
(3.3)
and
4n1n3 ∈
{
b2 − (b− 2h)2 : 0 ≤ h ≤
⌊
b
2
⌋}
. (3.4)
Now note that 2n0n2 + 2n1n3 is the number of differences in ∆X that are congruent
to 2 modulo 4, which of course is also equal to v4 − 2j. Thus, from (3.3) and (3.4), there
are integers h1, h2 such that 0 ≤ h1 ≤
⌊
a
2
⌋
, 0 ≤ h2 ≤
⌊
b
2
⌋
and
a2 − (a− 2h1)2 + b2 − (b− 2h2)2 =
v
2
− 4j. (3.5)
Using the facts that
a+ b = k and
ab =
v
4
− i,
we have
a2 + b2 = (a+ b)2 − 2ab
= k2 − v
2
+ 2i.
Substituting this into (3.5), we have
k2 − v
2
+ 2i− (a− 2h1)2 − (b− 2h2)2 =
v
2
− 4j,
or
k2 − v + 2i+ 4j = (a− 2h1)2 + (b− 2h2)2.
Since v = k2 − k + 2ℓ, we obtain
k − 2ℓ+ 2i+ 4j = (a− 2h1)2 + (b− 2h2)2.
We conclude that k − 2ℓ+ 2i+ 4j is a sum of two squares.
Suppose v ≡ 4 (mod 8). As before, there are an even number, say 2i, of odd elements
in L(X), where 0 ≤ i ≤ ℓ − 1. However, v/2 ≡ 2 (mod 4), so the number of elements
≡ 2 (mod 4) that are not in ∆X is an odd number, say 2j + 1, where 0 ≤ j ≤ ℓ− 1− i.
Reasoning exactly as in the case where v ≡ 0 (mod 8), we find that k − 2ℓ + 4i is a
perfect square and that k − 2ℓ+ 2i+ 4j + 2 is a sum of two squares.
We now give an application of Theorem 3.15.
Corollary 3.16. Suppose that k = n2 − 2ℓ + 4 where ℓ ≥ 1 and n ≥ ℓ + 1. Let v =
k2 − k + 2ℓ.
1. If v ≡ 0 (mod 8) and k − 2 is not the sum of two squares, then a (v, k)-MGR does
not exist.
16 Ars Math. Contemp. 20 (2021) 1–27
2. If v ≡ 4 (mod 8) and k is not the sum of two squares, then a (v, k)-MGR does not
exist.
Proof. We note that k − 2ℓ + 4(ℓ − 1) = n2 is a perfect square. We claim there are no
squares of the form k − 2ℓ + 4i where 0 ≤ i ≤ ℓ − 2. This is because the smallest such
integer is
k − 2ℓ = n2 − 4ℓ+ 4
≥ n2 − 4(n− 1) + 4
= n2 − 4n+ 8
= (n− 2)2 + 4.
Since all these integers have the same parity as n2 and they are not larger than k − 2ℓ +
4(ℓ − 1) = n2, the result follows. Therefore i = ℓ − 1 is the only value in [0, ℓ − 1] such
that k − 2ℓ+ 4i is a perfect square.
Now, in applying Theorem 3.15, we need to check that a certain condition holds for
0 ≤ j ≤ ℓ − 1 − i. Since i = ℓ − 1, we only need to consider j = 0. Theorem 3.15 then
states that a (v, k)-MGR does not exist if v ≡ 0 (mod 8) and k − 2ℓ+ 2(ℓ− 1) = k − 2
is not a sum of two squares; or if v ≡ 4 (mod 8) and k − 2ℓ + 2(ℓ − 1) + 2 = k is not a
sum of two squares. (It is not hard to verify that v ≡ 0 (mod 4), so either v ≡ 0 (mod 8)
or v ≡ 4 (mod 8).)
We give some examples to illustrate results that can be obtained using Corollary 3.16.
Example 3.17. Suppose we take n = 4t + 2 and ℓ = 5 in Corollary 3.16. Then v =
k2 − k + 10 ≡ 0 (mod 8). Here we have
k − 2 = (4t+ 2)2 − 10 + 4− 2 = 4(4t2 + 4t− 1).
This integer is not the sum of two squares because 4t2 + 4t− 1 ≡ 3 (mod 4). Hence, no
(k2 − k + 10, k)-MGR exists if k = 4(2t+ 1)2 − 6. The first values of k covered by this
result are k = 30, 94, 190, 318, 478, 670, 894, 1150, 1438, 1758.
Example 3.18. Suppose we take n = 4t + 2 and ℓ = 3 in Corollary 3.16. Then v =
k2 − k + 6 ≡ 0 (mod 8). Here we have
k − 2 = (4t+ 2)2 − 6 + 4− 2 = 16t2 + 16t.
This integer is the sum of two squares if and only if t2+t is the sum of two squares. Hence,
no (k2 − k + 10, k)-MGR exists if k = 4(2t + 1)2 − 2 and t2 + t is not the sum of two
squares. The first values of k covered by this result are k = 98, 194, 482, 674, 898, 1762,
2114, 2498, 2914 and 3362.
Example 3.19. Suppose we take n = 4t and ℓ = 5 in Corollary 3.16. Then v = k2 − k +
10 ≡ 4 (mod 8). Here we have
k = 16t2 − 6 = 2(8t2 − 3).
This integer is the sum of two squares if and only if 8t2 − 3 is the sum of two squares.
Hence, no (k2 − k+10, k)-MGR exists if k = (4t)2 − 6 and 8t2 − 3 is not the sum of two
squares. The first values of k covered by this result are k = 138, 5704, 1290, 2298, 2698,
3594, 5178, 6394, 7050 and 9210.
M. Buratti and D. R. Stinson: New results on modular Golomb rulers, optical orthogonal . . . 17
4 Nonexistence results for (v, k, 1)-OOC
In this section, we prove nonexistence results for some optimal (v, k, 1)-optical orthogonal
codes of size n > 1. Note that we are investigating the cases where v is even in this section.
Lemma 4.1. Suppose 1 ≤ ℓ ≤
(
k
2
)
and v = k(k − 1)n+ 2ℓ. Then a (v, k, 1)-OOC of size
n is optimal.
Proof. For v as given, we have⌊
v − 1
k(k − 1)
⌋
= n+
⌊
2ℓ− 1
k(k − 1)
⌋
.
However, 2ℓ− 1 < k(k − 1) because ℓ ≤
(
k
2
)
, so⌊
v − 1
k(k − 1)
⌋
= n.
Suppose X = {X1, . . . , Xn} is a (v, k, 1)-optical orthogonal code. We define ∆X and
the leave, L(X), in the obvious way:
∆X =
n⋃
i=1
{x− y mod v : x, y ∈ Xi, x ̸= y}
and
L(X) = Zv \∆X.
The following lemma is a straightforward generalization of Lemma 3.6.
Lemma 4.2. Suppose X is a (v, k, 1)-optical orthogonal code where v is even. Then
{0, v/2} ⊆ L(X). If v ≡ 0 (mod 4), then |L0(X)| and |L1(X)| are both even. If v ≡ 2
(mod 4), then |L0(X)| and |L1(X)| are both odd.
Theorem 4.3. Given v = k(k − 1)n+ 2ℓ with 1 ≤ ℓ ≤
(
k
2
)
, define the two sets
S =
{⌊v
4
⌋
− h : 0 ≤ h ≤ ℓ− 1
}
.
and
T =
{
h(k − h) : 0 ≤ h ≤
⌊
k
2
⌋}
.
Then a necessary condition for the existence of an optimal (v, k, 1)-OOC is that at least
one element of S is representable as a sum of n integers of T .
Proof. Note than an optimal (v, k, 1)-OOC will have size n, from Lemma 4.1. Assume
that X = {X1, . . . , Xn} is an (optimal) (v, k, 1)-OOC.
From Lemma 4.2, we see that v/2 ∈ L(X) and |L1(X)| has the same parity as v2 . Also,
as in the proof of Lemma 3.7, 0 ≤ |L1(X)| ≤ 2ℓ − 2. Thus, considering that the number
of odd elements in Zv is v/2, we see that the number of odd differences in
⋃n
i=1 ∆Xi is
twice an element of S.
Suppose that Xi contains exactly ai even elements, so k − ai is the number of odd
elements in Xi. Then the number of odd elements in ∆Xi is 2ai(k − ai), that is, twice
an element of T . It follows that at least one element of S is representable as a sum of n
integers belonging to T .
18 Ars Math. Contemp. 20 (2021) 1–27
Let us see some consequences of Theorem 4.3. As a first example, we consider the
cases where k = 3.
Corollary 4.4. An optimal (v, 3, 1)-OOC does not exist if v ≡ 14, 20 (mod 24).
Proof. When we take k = 3 in Theorem 4.3, we have T = {0, 2}. Suppose v is even and
we write v = 24t+2w, where 1 ≤ w ≤ 12. We express v in the form v = 6n+2ℓ, where
1 ≤ ℓ ≤ 3, obtaining the values of n and ℓ and the sets S that are shown in Table 2.
Table 2: Applications of Theorem 4.3 when k = 3.
v n ℓ S
24t+ 2 4t 1 {6t}
24t+ 4 4t 2 {6t, 6t+ 1}
24t+ 6 4t 3 {6t− 1, 6t, 6t+ 1}
24t+ 8 4t+ 1 1 {6t+ 2}
24t+ 10 4t+ 1 2 {6t+ 1, 6t+ 2}
24t+ 12 4t+ 1 3 {6t+ 1, 6t+ 2, 6t+ 3}
24t+ 14 4t+ 2 1 {6t+ 3}
24t+ 16 4t+ 2 2 {6t+ 3, 6t+ 4}
24t+ 18 4t+ 2 3 {6t+ 2, 6t+ 3, 6t+ 4}
24t+ 20 4t+ 3 1 {6t+ 5}
24t+ 22 4t+ 3 2 {6t+ 4, 6t+ 5}
24t+ 24 4t+ 3 3 {6t+ 4, 6t+ 5, 6t+ 6}
When v ≡ 14, 20 (mod 24), the set S consists of a single element, which is an odd
integer. Clearly it is not a sum of even integers, so we conclude from Theorem 4.3 that an
optimal (v, 3, 1)-OOC does not exist if v ≡ 14, 20 (mod 24).
Remark 4.5. It is well-known that an optimal (v, 3, 1)-OOC exists if and only if v ̸≡ 14, 20
(mod 24) (e.g., see [1, 2] for discussion about this result).
We adapt the argument used in Corollary 4.4 to prove a generalization that works for
odd integers k ̸≡ 1 (mod 8). First, we observe that, if k is odd, then all the elements of
T are even. So we obviously get a contradiction in Theorem 4.3 if the set S consists of a
single odd integer. This happens if ℓ = 1 (so v = nk(k − 1) + 2) and one of the following
two conditions hold:
1. nk(k − 1) ≡ 2 (mod 8) (v ≡ 0 (mod 4) in this case) or
2. nk(k − 1) ≡ 4 (mod 8) (v ≡ 2 (mod 4) in this case).
Since k is odd, we have k ≡ 1, 3, 5, 7 (mod 8). We consider each case separately.
k ≡ 1 (mod 8):
Here k(k − 1) ≡ 0 (mod 8), neither of 1. or 2. can hold.
k ≡ 3 (mod 8):
Here k(k−1) ≡ 6 (mod 8). For 1., we obtain 6n ≡ 2 (mod 8), so n ≡ 3 (mod 4)
and v = (4t+ 3)k(k − 1) + 2 for some integer t. It follows that
v ≡ 3k(k − 1) + 2 (mod 4k(k − 1)).
M. Buratti and D. R. Stinson: New results on modular Golomb rulers, optical orthogonal . . . 19
For 2., we obtain 6n ≡ 4 (mod 8), so n ≡ 2 (mod 4) and v = (4t+2)k(k−1)+2
for some integer t. It follows that
v ≡ 2k(k − 1) + 2 (mod 4k(k − 1)).
k ≡ 5 (mod 8):
Here k(k − 1) ≡ 4 (mod 8). For 1., we obtain 4n ≡ 2 (mod 8), which is
impossible. For 2., we obtain 4n ≡ 4 (mod 8), so n ≡ 1 (mod 2) and v =
(2t+ 1)k(k − 1) + 2 for some integer t. It follows that
v ≡ k(k − 1) + 2 (mod 2k(k − 1)).
k ≡ 7 (mod 8):
Here k(k−1) ≡ 2 (mod 8). For 1., we obtain 2n ≡ 2 (mod 8), so n ≡ 1 (mod 4)
and v = (4t+ 1)k(k − 1) + 2 for some integer t. It follows that
v ≡ k(k − 1) + 2 (mod 4k(k − 1)).
For 2., we obtain 2n ≡ 4 (mod 8), so n ≡ 2 (mod 4) and v = (4t+2)k(k−1)+2
for some integer t. It follows that
v ≡ 2k(k − 1) + 2 (mod 4k(k − 1)).
Summarizing the above discussion, we have the following theorem.
Theorem 4.6. There does not exist an optimal (v, k, 1)-OOC whenever one of the following
conditions hold:
• k ≡ 3 (mod 8) and v ≡ 3k(k − 1) + 2 (mod 4k(k − 1)).
• k ≡ 3 (mod 8) and v ≡ 2k(k − 1) + 2 (mod 4k(k − 1)).
• k ≡ 5 (mod 8) and v ≡ k(k − 1) + 2 (mod 2k(k − 1)).
• k ≡ 7 (mod 8) and v ≡ k(k − 1) + 2 (mod 4k(k − 1)).
• k ≡ 7 (mod 8) and v ≡ 2k(k − 1) + 2 (mod 4k(k − 1)).
The following results are immediate corollaries of Theorem 4.6.
Corollary 4.7. An optimal (v, 3, 1)-OOC does not exist if v ≡ 14, 20 (mod 24); an opti-
mal (v, 5, 1)-OOC does not exist if v ≡ 22 (mod 40); and an optimal (v, 7, 1)-OOC does
not exist if v ≡ 44, 86 (mod 168).
Example 4.8. As an example where Theorem 4.3 can be applied to an even value of k,
consider the case of an optimal (62, 6, 1)-OOC. Here we have 62 = 2× 6× 5 + 2× 1, so
n = 2 and ℓ = 1. The set S = {15} and T = {0, 5, 8, 9}. It is impossible to express 15 as
the sum of two numbers from T , so we conclude that an optimal (62, 6, 1)-OOC does not
exist.
We now prove some general nonexistence results.
20 Ars Math. Contemp. 20 (2021) 1–27
Theorem 4.9. Suppose 1 ≤ ℓ ≤
(
k
2
)
, and suppose an optimal (2k(k− 1) + 2ℓ, k, 1)-OOC
exists. Define the set R as follows:
R =

{⌊
k−ℓ+1
2
⌋
+ h : 0 ≤ h ≤ ℓ− 1
}
if k is even;
{k − ℓ+ 2h : 0 ≤ h ≤ ℓ− 1} if k is odd and ℓ is even;
{k − ℓ+ 2h+ 1 : 0 ≤ h ≤ ℓ− 1} if k and ℓ are both odd.
Then at least one integer in the set R can be expressed as the sum of two squares.
Proof. First, suppose k is even. Apply Theorem 4.3. We have v = 2k(k−1)+2ℓ and thus
we have
S =
{
k(k − 1)
2
+
⌊
ℓ
2
⌋
− h : 0 ≤ h ≤ ℓ− 1
}
.
From Lemma 3.14, we have
T =
{(
k
2
)2
− h2 : 0 ≤ h ≤ k
2
}
.
From Theorem 4.3, we have
k(k − 1)
2
+
⌊
ℓ
2
⌋
− h =
(
k
2
)2
− i2 +
(
k
2
)2
− j2
for integers h, i, j where 0 ≤ h ≤ ℓ− 1 and 0 ≤ i, j ≤ k/2. Simplifying, we obtain
k
2
−
⌊
ℓ
2
⌋
+ h = i2 + j2.
The result follows by noting that
k
2
−
⌊
ℓ
2
⌋
=
⌊
k − ℓ+ 1
2
⌋
since k is even.
Next, suppose k is odd and ℓ is even. Here v ≡ 0 (mod 4). We again apply Theo-
rem 4.3. Here we have
S =
{
k(k − 1) + ℓ
2
− h : 0 ≤ h ≤ ℓ− 1
}
and, from Lemma 3.14, we have
T =
{(
k
2
)2
−
(
k
2
− h
)2
: 0 ≤ h ≤ k − 1
2
}
.
From Theorem 4.3, we get
k(k − 1) + ℓ
2
− h =
(
k
2
)2
−
(
k
2
− i
)2
+
(
k
2
)2
−
(
k
2
− j
)2
M. Buratti and D. R. Stinson: New results on modular Golomb rulers, optical orthogonal . . . 21
for integers h, i, j where 0 ≤ h ≤ ℓ− 1 and 0 ≤ i, j ≤ (k − 1)/2. Simplifying, we have
2k(k − 1) + 2ℓ− 4h = 2k2 − (k − 2i)2 − (k − 2j)2.
Therefore,
(k − 2i)2 + (k − 2j)2 = 2(k − ℓ+ 2h),
and the result follows.
The final case is when k and ℓ are both odd. The proof for this case is very similar to
previous case.
Corollary 4.10. Suppose that k has prime decomposition that contains a prime p ≡ 3
(mod 4) raised to an odd power. Then an optimal (2k(k − 1) + 2, k, 1)-OOC does not
exist.
Proof. Suppose an optimal (2k(k − 1) + 2, k, 1)-OOC exists. Take ℓ = 1 in Theorem 4.9;
then h = 0 in the definition of the set R. It follows that, if k is even, then k/2 is the sum
of two squares; and if k is odd, then k is the sum of two squares. The desired result then
follows from Theorem 1.9.
Remark 4.11. The smallest applications of Corollary 4.10 are when k = 3 and k = 6.
We conclude that optimal (14, 3, 1)-OOC and optimal (62, 6, 1)-OOC do not exist. We
note that Corollary 4.4 also shows that an optimal (14, 3, 1)-OOC does not exist. Also, Ex-
ample 4.8 proved the nonexistence of an optimal (62, 6, 1)-OOC using a slightly different
argument. The next values of k covered by Corollary 4.10 are k = 7, 11, 12, 14, 15, 19,
21, 22, 23 and 24.
Now we prove a nonexistence result that holds for arbitrarily large values of ℓ.
Theorem 4.12. For any positive integer ℓ, there are infinitely many even integers k such
that an optimal (2k(k − 1) + 2ℓ, k, 1)-OOC does not exist.
Proof. Using Lemma 1.10, choose an even integer k such that
⌊
k−ℓ+1
2
⌋
+ h is not the sum
of two squares, for 0 ≤ h ≤ ℓ− 1. Then apply Theorem 4.9.
We next prove the nonexistence of certain optimal (3k(k − 1) + 2, k, 1)-OOC with k
even.
Theorem 4.13. There does not exist an optimal (3k(k − 1) + 2, k, 1)-OOC if
k = (4a+1(24c+ 7) + 2)/3 with a, c ≥ 0 or if k = 4a+1(8c+ 5) with a, c ≥ 0.
Proof. Assume that X is an optimal (3k(k − 1) + 2, k, 1)-OOC with k even. We apply
Theorem 4.3 with n = 3. Here, with the usual notation, we have
S =
{
3k2 − 3k + 2
4
}
if k ≡ 2 (mod 4), and
S =
{
3k2 − 3k
4
}
22 Ars Math. Contemp. 20 (2021) 1–27
if k ≡ 0 (mod 4). Also, as in the proof of Theorem 4.9, we have
T =
{(
k
2
)2
− h2 : 0 ≤ h ≤ k
2
}
.
It follows that the unique element in the set S must be a sum of three elements of T .
For k ≡ 2 (mod 4), we have
3k2 − 3k + 2
4
= 3
(
k
2
)2
− (h21 + h22 + h23)
for integers h1, h2, h3. It follows that (3k − 2)/4 is a sum of three squares, and hence
(3k − 2)/4 is not of the form 4a(8b+ 7) where a, b ≥ 0. Thus, if
3k − 2
4
= 4a(8b+ 7), (4.1)
an optimal (3k(k − 1) + 2, k, 1)-OOC does not exist. (4.1) holds if and only if
k =
4a+1(8b+ 7) + 2
3
.
In order for k to be an integer, b must be divisible by 3, say b = 3c. Therefore, if
k =
4a+1(24c+ 7) + 2
3
,
where a, c ≥ 0, an optimal (3k(k − 1) + 2, k, 1)-OOC does not exist.
The case k ≡ 0 (mod 4) is similar. Here, 3k/4 must be a sum of three squares, and
hence 3k/4 is not of the form 4a(8b+7). Therefore an optimal (3k(k− 1)+2, k, 1)-OOC
does not exist if
k =
4a+1(8b+ 7)
3
.
In order for k to be an integer, we must have b ≡ 1 (mod 3), say b = 3c + 1. Then
(8b + 7)/3 = 8c + 5. We conclude that an optimal (3k(k − 1) + 2, k, 1)-OOC does not
exist if
k = 4a+1(8c+ 5),
where a, c ≥ 0.
Finally, we prove the nonexistence of certain optimal (3k(k − 1) + 4, k, 1)-OOC with
k even.
Theorem 4.14. There does not exist an optimal (3k(k − 1) + 4, k, 1)-OOC if
k = (4a+3(24c+ 23)− 2)/3 with a, c ≥ 0 or if k = 4a+3(8c+ 5) with a, c ≥ 0.
Proof. We proceed as in the proof of Theorem 4.13, by applying Theorem 4.3 with n = 3.
Assume that X is an optimal (3k(k − 1) + 4, k, 1)-OOC with k even. We have
S =
{
3k2 − 3k − 2
4
,
3k2 − 3k − 2
4
+ 1
}
M. Buratti and D. R. Stinson: New results on modular Golomb rulers, optical orthogonal . . . 23
if k ≡ 2 (mod 4), and
S =
{
3k2 − 3k
4
,
3k2 − 3k
4
+ 1
}
if k ≡ 0 (mod 4). Also,
T =
{(
k
2
)2
− h2 : 0 ≤ h ≤ k
2
}
.
At least one element in the set S must be a sum of three elements of T .
Suppose k ≡ 2 (mod 4) and let n = (3k + 2)/4 − 1. Proceeding as in the proof
of Theorem 4.13, we see that one of n or n + 1 is the sum of three squares. However, if
n+ 1 = 4a(8b+ 7) where a ≥ 2, then Lemma 1.11 implies that neither n nor n+ 1 is the
sum of three squares. In this case, optimal (3k(k − 1) + 4, k, 1)-OOC does not exist. This
occurs when
3k + 2
4
= 4a(8b+ 7),
with a ≥ 2, or
k =
4a+1(8b+ 7)− 2
3
.
Since k is an integer, b ≡ 2 (mod 3), say b = 3c+ 2, and then
k =
4a+1(24c+ 23)− 2
3
,
where a ≥ 2. For k of this form, an optimal (3k(k − 1) + 4, k, 1)-OOC does not exist.
Suppose k ≡ 0 (mod 4) and let n = 3k/4 − 1. Here, by the same logic as above, an
optimal (3k(k − 1) + 4, k, 1)-OOC does not exist when
3k
4
= 4a(8b+ 7),
or
k =
4a+1(8b+ 7)
3
,
where a ≥ 2. Here, b ≡ 1 (mod 3), say b = 3c+ 1, and then
k = 4a+1(8c+ 5),
where a ≥ 2. For k of this form, an optimal (3k(k− 1)+ 4, k, 1)-OOC does not exist.
5 Other types of designs
In this section, we obtain necessary conditions for the existence of certain cyclic Steiner
2-designs and relative difference families using the techniques we have developed.
24 Ars Math. Contemp. 20 (2021) 1–27
5.1 Cyclic Steiner 2-designs
A Steiner 2-design of order v and block-size k, denoted as S(2, k, v), consists of a set of
k-subsets (called blocks) of a v-set (whose elements are called points) such that every pair
of points occurs in a unique block. An S(2, k, v) is cyclic if there is a cyclic permutation of
the v points that maps every block to a block.
It is well-known that a cyclic S(2, k, v) exists only if v ≡ 1 or k (mod k(k − 1)). A
cyclic S(2, k, v) with v ≡ 1 (mod k(k − 1)) is equivalent to a (v, k, 1)-OOC of size n;
in this case the leave is {0}. Further, a cyclic S(2, k, v) with v ≡ k (mod k(k − 1)) is
equivalent to a (v, k, 1)-OOC of size n whose leave is the subgroup of Zv of order k.
Assume that X = {X1, . . . , Xn} is an (k(k − 1)n + k, k, 1)-OOC of size n that is
obtained from a cyclic S(2, k, k(k − 1)n + k) with both k and n even. The leave L(X)
has exactly k/2 odd elements and therefore the number of odd differences in
⋃n
i=1 ∆Xi is
k(k − 1)n/2.
Reasoning as in the proof of Theorem 4.9, we see that k(k − 1)n/4 is the sum of n
integers in the set
T =
{(
k
2
)2
− h2 : 0 ≤ h ≤ k
2
}
.
Thus we have
kn
4
= h21 + h
2
2 + · · ·+ h2n
for a suitable n-tuple (h1, . . . , hn) of nonnegative integers, each of which does not exceed
k/2. Using Lagrange’s Four-square Theorem (Theorem 1.9), it is an easy exercise to see
that such an n-tuple certainly exists for n ≥ 4.
However, if n = 2, this is not always the case. Here we require
k
2
= h21 + h
2
2
for nonnegative integers h1, h2 ≤ k/2. As stated in Theorem 1.9, a positive integer can be
written as the as a sum of two squares if and only if its prime decomposition contains no
prime p ≡ 3 (mod 4) raised to an odd power. So we obtain the following result.
Theorem 5.1. If k is an even integer whose prime decomposition contains a prime p ≡ 3
(mod 4) raised to an odd power, then there does not exists a cyclic S(2, k, 2k(k− 1)+ k).
We can apply Theorem 5.1 with k = 6, 12, 14, 22, 24, 28, etc.
Now assume that X = {X1, . . . , Xn} is a (k(k − 1)n+ k, k, 1)-OOC that is obtained
from a cyclic S(2, k, k(k − 1)n + k) with k even and n odd. Here all the elements of the
leave of X are odd, and hence all (k(k− 1)n+ k)/2 odd elements of Zv have to appear in⋃n
i=1 ∆Xi. Reasoning as above, we see that
k(k−1)n+k
4 is the sum of n integers in the set
T =
{(
k
2
)2
− h2 : 0 ≤ h ≤ k
2
}
,
i.e.,
k(n− 1)
4
= h21 + h
2
2 + · · ·+ h2n
M. Buratti and D. R. Stinson: New results on modular Golomb rulers, optical orthogonal . . . 25
for a suitable n-tuple (h1, . . . , hn) of nonnegative integers not exceeding k/2. Again, such
a n-tuple exists by Lagrange’s Four-square Theorem if n ≥ 5.
But this is not always the case if n = 3. Here we require
k
2
= h21 + h
2
2 + h
2
3
for nonnegative integers h1, h2, h3 ≤ k/2.
Applying Legendre’s Three-square Theorem (Theorem 1.9), we have the following re-
sult.
Theorem 5.2. If k = 2a(8b+7) where a and b are nonnegative integers and a is odd, then
there does not exist a cyclic S(2, k, 3k(k − 1) + k).
We can apply Theorem 5.1 with k = 14, 46, 56, 62, etc.
5.2 Relative difference families
When G = Zv and the order of the subgroup H is equal to w, a (G,H, k, 1)-RDF is clearly
a (v, k, 1)-OOC whose leave is the subgroup of Zv of order w. In this case, some authors
(e.g., [22]) speak of a w-regular (v, k, 1)-OOC. Note that a w-regular (v, k, 1)-OOC is
optimal provided that w ≤ k(k − 1). Also, note that a k-regular (v, k, 1)-OOC gives rise
to a cyclic S(2, k, v).
Theorem 5.3. Let G be a group with a subgroup S of index 2 and let X be a (G,H, k, λ)-
relative difference family of size n, where |H| = w. If H is contained in S, then kn− λw
is a sum of n squares. If H is not contained in S, then kn is a sum of n squares.
Proof. Let us say that an element of G is even or odd according to whether it belongs to
or does not belong to S, respectively. Set X = {X1, . . . , Xn} and, for i = 1, . . . , n, let
ai and bi be the number of even and odd elements in Xi, respectively. The number of odd
elements in ∆Xi is 2aibi (note that here we are treating ∆Xi and ∆X as multisets since
differences may be repeated). Also, by definition, the number of odd elements in ∆X is λ
times the number of all odd elements of G \H .
If H , S are subgroups of a group G with |G : S| = 2, then either H ⊆ S or |H ∩ S| =
|H|/2. Hence, we have
n∑
i=1
2aibi =
λv
2
or
λ(v − w)
2
,
according to whether H is contained or not contained in S. Thus we have:
n∑
i=1
4aibi =
{
λv if H ⊆ S
λ(v − w) if H ̸⊆ S.
Now, given that ai + bi = k, we have
4aibi = 4ai(k − ai) = k2 − (k − 2ai)2.
Replacing this in the above formula and taking account of (1.1), we get
n∑
i=1
(k − 2ai)2 =
{
kn− λw if H ⊆ S
kn if H ̸⊆ S.
and the assertion follows.
26 Ars Math. Contemp. 20 (2021) 1–27
Theorem 5.3 is trivial for n ≥ 4 in view of Theorem 1.9. On the other hand, it gives
some important information for n = 1, 2, 3. We now discuss several consequences of
Theorem 1.9.
First, we point out a connection with the Bose-Connor Theorem (Theorem 3.3). Sup-
pose we take n = 1 in Theorem 5.3 and suppose H ⊆ S. Recall that S is a subgroup of
index 2. Denote |G| = v = uw, where |H| = w. Then Theorem 5.3 asserts that k − λw
must be a perfect square. This result can also be obtained from Theorem 3.3, as follows.
The development of the (G,H, k, λ)-relative difference family through the group G yields
a divisible design with λ1 = 0 and λ2 = λ. Since H and S are subgroups of G and H ⊆ S,
it must be the case that w | v2 , say v/2 = tw. Then u = v/w = 2t is even. Therefore
statement 1. of Theorem 3.3 applies, and k2−λv is a square. However, k(k−1)−λ(v−w)
from (1.1), so k2 − λv = k − λw, so we obtain the same result.
In the special case of the preceding result where w = 1, we see that k − λ is a square.
This also follows from the Bruck-Ryser-Chowla Theorem (as we already discussed in Ex-
ample 3.5 in the case where G is cyclic).
If we take n = 2 and w = 1, we see that, if X is a (v, k, λ)-DF with two base blocks
in a group with a subgroup of index 2, then 2k − λ is a sum of two squares (this result was
first shown in [17, Corollary 2.1]). Similarly, taking n = 3 and w = 1, we see that, if X is
a (v, k, λ)-DF with three base blocks in a group with a subgroup of index 2, then 3k− λ is
a sum of three squares (this result was first shown in [17, Corollary 2.2]).
Finally, n ∈ {2, 3} and w = k ≡ 0 (mod 2), then a cyclic S(2, k, k(k − 1)n + k)
exists only if k is a sum of n squares. This is equivalent to results obtained in Section 5.1.
6 Summary
We have proven a number of nonexistence results for infinite classes of modular Golomb
rulers, optical orthogonal codes, cyclic Steiner systems and relative difference families. We
note that very few results of this nature were previously known. Many of our new results
are based on counting even and odd differences and then applying some classical results
from number theory which establish which integers can be expressed as a sum of a two or
three squares.
ORCID iDs
Marco Buratti https://orcid.org/0000-0003-1140-2251
Douglas Robert Stinson https://orcid.org/0000-0001-5635-8122
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 29–36
https://doi.org/10.26493/1855-3974.2338.5df
(Also available at http://amc-journal.eu)
A family of fractal non-contracting weakly
branch groups
Marialaura Noce *
Georg-August-Universität Göttingen, Mathematisches Institut
Received 20 May 2020, accepted 21 July 2020, published online 13 July 2021
Abstract
We construct a new example of an infinite family of groups acting on a d-adic tree, with
d ≥ 2 that is non-contracting and weakly regular branch over the derived subgroup.
Keywords: Groups of automorphisms of rooted trees, branch groups.
Math. Subj. Class. (2020): 20E08
1 Introduction
Weakly branch groups were first defined by Grigorchuk in 1997 as a generalization of the
famous p-groups constructed by Grigorchuk himself [4, 5], and Gupta and Sidki [6]. These
groups possess remarkable and exotic properties. For instance, the Grigorchuk group is the
first example of a group of intermediate word growth, and amenable but not elementary
amenable. Also, together with the Grigorchuk group, other subgroups of the group of
automorphisms of rooted trees like the Gupta-Sidki p-groups and many groups in the family
of the so-called Grigorchuk-Gupta-Sidki groups have been shown to be a counterexample
to the General Burnside Problem.
For these reasons, (weakly) branch groups spread great interest among group theo-
rists, who have actively investigated further properties of these in the recent years: just-
infiniteness, fractalness, maximal subgroups, or contraction.
Roughly speaking, a group is said to be contracting if the sections of every element are
“shorter” than the element itself, provided the element does not belong to a fixed finite set,
called the nucleus (see the exact definition in Section 2).
*The author wants to thank Laurent Bartholdi for pointing out the existence of [9], and Gustavo A. Fernández-
Alcober and Albert Garreta for useful discussions. The author thanks the anonymous referee for helpful com-
ments. The author is supported by EPSRC (grant number 1652316), and partially by the Spanish Government,
grant MTM2017-86802-P, partly with FEDER funds.
E-mail address: mnoce@unisa.it (Marialaura Noce)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
30 Ars Math. Contemp. 20 (2021) 29–36
Even though in the literature there are many examples of weakly branch contracting
groups, not much is known about weakly branch groups that are non-contracting. In 2005
Dahmani [2] provided the first example of a non-contracting weakly regular branch au-
tomaton group. Another example with similar properties was constructed by Mamaghani
in 2011 [7]. Both are examples of groups acting on the binary tree. We also point out that
in [9] Sidki and Wilson proved in particular that every group acting on the binary tree with
finite abelianization (including non-contracting groups) embeds in a branch group. This
provides more examples of non-contracting branch groups acting on the binary tree.
For d ≥ 3, the Hanoi Towers group H(d) ≤ Aut Td (which represents the famous game
of Hanoi Towers on d pegs) is non-contracting and only weakly branch. To the best of our
knowledge if d > 3 it is not known if these groups can be branch. For more information on
the topic, see [3] and [10].
In this paper we explicitly construct an example of an infinite family of non-contracting
weakly branch groups acting on d-adic trees for any d ≥ 2. This result gives a wealth of
examples of groups with these properties. In the following we denote with Aut Td the
group of automorphisms of a d-adic tree.
Theorem 1.1. For any d ≥ 2, there exists a group M(d) ≤ Aut Td that is weakly regular
branch over its derived subgroup, non-contracting and fractal.
1.1 Organization
In Section 2 we give some definitions of groups acting on regular rooted trees and of prop-
erties like fractalness, branchness and contraction. In Section 3 we introduce these groups
and we prove the main theorem together with some additional results regarding the order
of elements of M(d).
2 Preliminaries
In this section we fix some terminology regarding groups of automorphisms of d-adic
(rooted) trees. For further information on the topic, see [1] or [8].
Let d be a positive integer, and Td the d-adic tree. We denote with Aut Td the group of
automorphisms of Td. We let Ln be the nth level of Td, and L≥n the levels of the tree from
level n and below.
The stabilizer of a vertex u of the tree is denoted by st(u), and, more generally, the
nth level stabilizer st(n) is the subgroup of Aut Td that fixes every vertex of Ln. If
G ≤ Aut Td, we define the nth level stabilizer of G as stG(n) = st(n) ∩ G. Notice that
stabilizers are normal subgroups of the corresponding group. We let ψ be the isomorphism
ψ : st(1) −→ Aut Td ×
d· · · ×Aut Td
g 7−→ (gu)u∈L1 ,
where gu is the section of g at the vertex u, i.e. the action of g on the subtree Tu that
hangs from the vertex u. Let Sd be the symmetric group on d letters. An automorphism
a ∈ Aut Td is called rooted if there exists a permutation σ ∈ Sd such that a permutes
rigidly the vertices of the subtrees hanging from the first level of the tree according to the
permutation σ, i.e. if v = xu ∈ V (Td), with x ∈ L1, then a(xu) = σ(x)u. We usually
identify a and σ.
M. Noce: A family of fractal non-contracting weakly branch groups 31
Notice that if g ∈ st(1) with ψ(g) = (g1, . . . , gd), and σ is a rooted automorphism,
then,
ψ(gσ) =
(
gσ−1(1), . . . , gσ−1(d)
)
. (2.1)
Any element g ∈ G can be written uniquely in the form g = hσ, where h ∈ st(1) and σ is
a rooted automorphism.
Notice also that the decomposition g = hσ, together with the action (2.1), yields iso-
morphisms
Aut Td ∼= st(1)⋊ Sd ∼=
(
Aut Td × d. . .×Aut Td
)
⋊ Sd
∼= Aut Td ≀ Sd ∼= ((· · · ≀ Sd) ≀ Sd) ≀ Sd.
(2.2)
Throughout the paper, we will use the following shorthand notation: let f ∈ Aut T of
the form f = gh, where g ∈ stG(1) and h is the rooted automorphism corresponding to
the permutation σ ∈ Sd. If ψ(g) = (g1, . . . , gd), we write f = (g1, . . . , gd)σ.
Definition 2.1. Let G ≤ Aut Td, and let V (Td) be the set of vertices of Td. Then:
(a) The group G is said to be self-similar if for any g ∈ G we have
{gu | g ∈ G, u ∈ V (Td)} ⊆ G.
In other words, the sections of g at any vertex are still elements of G. For example,
Aut Td is self-similar.
(b) A self-similar group G is said to be fractal if ψu(stG(u)) = G for all u ∈ V (Td),
where ψu is the homomorphism sending g ∈ st(u) to its section gu.
To prove that a group is self-similar it suffices to show that the condition above is
satisfied by the vertices of the first level of the tree (see [3, Proposition 3.1]). The situation
is similar in the case of fractal groups. More precisely, using Lemma 2.2, we deduce that
to show that a group G is fractal, it is enough to check the vertices in the first level of Td.
We recall that G is said to be level transitive if it acts transitively on every level of the tree.
Lemma 2.2 ([11, Lemma 2.7]). If G ≤ Aut Td is transitive on the first level and
ψx(stG(x)) = G for some x ∈ L1, then G is fractal and level transitive.
Here we present a family of non-contracting weakly branch groups. To this end, in the
following, we recall the corresponding two definitions.
Definition 2.3. A self-similar group G ≤ Aut Td is contracting if there exists a finite
subset F ⊆ G such that for every g ∈ G there is n such that gv belongs to F for all
vertices v of L≥n. Note that if you take two finite sets F1 and F2 satisfying the condition
on the sections above, then also F1∩F2 will satisfy the condition. For this reason, one can
consider the set that is intersection of such sets. This is called the nucleus of G and it is
denoted by N .
Definition 2.4. Let G be a self-similar subgroup of Aut Td. We say that G is weakly
regular branch over a subgroup K ≤ G if G is level transitive and we have
ψ(K ∩ stG(1)) ≥ K × · · · ×K.
If, additionally, K is of finite index in G, then G is said to be regular branch over K.
32 Ars Math. Contemp. 20 (2021) 29–36
3 The groups M(d)
Let d ≥ 2, and let Td be the d-adic tree. The group M(d) ≤ Aut Td is generated by d
elements m1, . . . ,md, where m1, . . . ,md are defined recursively as follows:
m1 = (1, . . . , 1,m1)(1 . . . d)
m2 = (1, . . . , 1,m2, 1)(1 . . . d− 1)
m3 = (1, . . . ,m3, 1, 1)(1 . . . d− 2)
...
md−1 = (1,md−1, 1, . . . , 1)(1 2)
md = (m1, . . . ,md).
For example, for d = 3, we have M(3) = ⟨m1,m2,m3⟩, where
m1 = (1, 1,m1)(1 2 3), m2 = (1,m2, 1)(1 2), m3 = (m1,m2,m3).
m1 :
(1 2 3)
(1 2 3)
(1 2 3)
...
m2 :
(1 2)
(1 2)
(1 2)
...
m3 :
1
(1 2 3) (1 2) 1
(1 2 3) (1 2)
1
...
Figure 1: The generators of M(3).
3.1 Proof of the main theorem
In this section we prove the main result of the paper. In order to ease notation, and unless
it is strictly necessary, we will simply write M to denote an arbitrary group M(d).
Proposition 3.1. The group M is fractal and level transitive.
M. Noce: A family of fractal non-contracting weakly branch groups 33
Proof. Notice that the group is transitive on the first level because the rooted part of the
generator m1 is (1 2 . . . d). Also, it is straightforward to see that the group is self-similar,
since the sections of every generator at the first level are generators of M. To see that M is
fractal, note that
md1 = (m1, . . . ,m1)
m
md−21
d = (m
m1
3 , . . . ,m2)
...
m
m21
d = (m
m1
d−1, . . . ,md−2)
mm1d = (m
m1
d , . . . ,md−1)
md = (m1, . . . ,md).
Then in the last component of the elements above we obtain all the generators of M. Using
Lemma 2.2, we conclude that M is level transitive and fractal.
Proposition 3.2. Let d ≥ 2. Then the group M(d) is weakly regular branch over its derived
subgroup M′(d).
Proof. We will distinguish the case d = 2, and d ≥ 3 separately. Let d = 2. The element
[m1,m2] is non-trivial since
[m1,m2] = (m
−1
1 m
−1
2 m
2
1,m
−1
1 m2),
and m−11 m2 /∈ stM(1). Then M(2)′ is non-trivial, and we have
[m21,m2] = (1, [m1,m2]). (3.1)
From (3.1) and since M(2)′ = ⟨[m1,m2]⟩M(2), we obtain that {1} ×M(2)′ ≤ ψ(M(2)′).
As M(2) is level transitive, we conclude that M(2)′ ×M(2)′ ≤ ψ(M(2)′), as desired.
Let d ≥ 3, and write M for M(d). First we show that M′ is non-trivial. Let us denote
σ = (1 2 . . . d) and τ = (1 2 . . . d− 1). We have
[m1,m2] = σ
−1(1, . . . , 1,m−11 )τ
−1(1, . . . , 1,m−12 ,m1)σ(1, . . . , 1,m2, 1)τ
= (1, . . . , 1,m−11 )
σ(1, . . . , 1,m−12 ,m1)
τσ(1, . . . , 1,m2, 1)
τσ [σ, τ ]
= (m−11 , 1, . . . , 1)(m1,m
−1
2 , 1, . . . , 1)(1, . . . , 1,m2)(1 2 d).
Hence, we obtain that
[m1,m2] = (1,m
−1
2 , 1, . . . , 1,m2)(1 2 d). (3.2)
By (3.2), we have [m1,m2] /∈ stM(1), thus M′ is non-trivial.
Now, for i = 1, . . . , d− 2, and j = i+ 1, . . . , d− 1, we have
[md+1−ii ,mj ]
md−11 = (1, . . . , 1, [mi,mj ]). (3.3)
Then in order to prove that {1} × · · · × {1} ×M′ ≤ ψ(M′ ∩ stM(1)), it only remains to
show that for any i = 1, . . . , d− 1, there exists x(i) ∈ M′ ∩ stM(1) such that
x(i) = (1, . . . , 1, [mi,md]).
34 Ars Math. Contemp. 20 (2021) 29–36
To find such x(i), we first observe that
[(md+1−ii )
mi−11 ,md] = (1, i. . ., 1, [mi,mi+1], . . . , [mi,md−1], [mi,md]).
In order to cancel all these commutators above except for the last component, we use (3.3),
and we observe that since M is level transitive, if we conjugate with a suitable power of
m1, we get [mi,mi+1]−1, . . . , [mi,md−1]−1 in each component. For example, if i = 2,
we have
[(md−12 )
m1 ,md] = (1, 1, [m2,m3], [m2,m4], . . . , [m2,md]).
By using the considerations above, we obtain that x(2) must be of the form
x(2) = [m3,m
d−1
2 ]
m21 [m4,m
d−1
2 ]
m31 . . . [md−1,m
d−1
2 ]
md−21 [(md−12 )
m1 ,md]
= (1, . . . , 1, [m2,md]).
To prove last part of the main theorem (that M(d) is non-contracting), we need some
preliminary tools. Namely, we show some results regarding the order of elements of M(d).
We will handle the case d = 2, and d > 2 separately. More precisely, we first prove
that M(2) is torsion-free, and then, for d > 2, we show that the groups M(d) are neither
torsion-free nor torsion, contrary to the case d = 2.
The following Remark 3.3 and Lemma 3.4 are key steps to prove that M(2) is torsion-
free. We write M for M(2).
Remark 3.3. Let h ∈ M′ with h = (h1, h2). Then h1h2 ∈ M′.
Proof. Consider the following map ρ:
ρ : stM(1) → M → M/M′
(h1, h2) 7→ h1h2 7→ h1h2.
Note that ρ is a homomorphism of groups since M/M′ is abelian. As stM(1)/Ker ρ is
abelian, M′ ≤ Ker ρ. This concludes the proof.
In the proof of next lemma, for a prime p we denote with νp(m) the p-adic valuation
of m, that is the highest power of p that divides m.
Lemma 3.4. We have M′ = (M′ ×M′)⟨[m1,m2]⟩. Furthermore
M/M′ ∼= ⟨m1M⟩ × ⟨m2M⟩ ∼= Z× Z.
Proof. Since M is weakly regular branch over M′ by Proposition 3.2, and
[m1,m2] = ([m1,m2]m
−1
2 m1,m
−1
1 m2),
we deduce that (m−12 m1,m
−1
1 m2) is an element of M
′. Furthermore, we claim that the
elements [m1,m2]y where y ∈ {m1,m2,m−11 ,m
−1
2 } are in ⟨[m1,m2]⟩ modulo M′ ×M′.
Indeed, we have
[m1,m2]
m1 = (m1
−2m2m1,m1
−1m−12 m1
2)
= ([m1
2,m2
−1]m2m1
−1, [m1,m2]m2
−1m1)
≡ (m1−1m2,m2−1m1) (mod M′ ×M′),
M. Noce: A family of fractal non-contracting weakly branch groups 35
and similarly for the other commutators. Thus M′ = (M′ ×M′)⟨[m1,m2]⟩, as required.
Now we claim that m1 is of infinite order. By way of contradiction suppose that, for
some k, m1 has order n = 2k, as m1 has order 2 modulo the first level stabilizer. We have
mn1 = (m
k
1 ,m
k
1) = (1, 1),
which yields a contradiction as k < n. This concludes the proof of the claim and implies
that also m2 is of infinite order, since m2 = (m1,m2).
Now we want to show that if mi1m
j
2 ∈ M′, then necessarily i = j = 0. As mi1m
j
2 ∈
M′ ≤ stM(1), then i must be even. By way of contradiction, we choose the element
mi1m
j
2 ∈ M′ subject to the condition that i is divisible by the least possible positive power
of 2, say 2a, for some a. In other words, ν2(i) = a. Then if mr1m
s
2 ∈ M′, necessarily
2a | r. Note that it cannot happen that r = 0 and s ̸= 0 as m2 is of infinite order. Now,
writing i = 2i1 for some i1, we have
mi1m
j
2 = (m1
i1+j ,m1
i1m2
j) ≡ [mk1 ,mk2 ] ≡ (m1km2−k,m1−km2k) (mod M′ ×M′).
This implies that m1i1+j−km2k ∈ M′ and m1i1+km2j−k ∈ M′. As 2a | i1 + j − k and
2a | i1 + k, then 2a divides also j. This is because 2a | 2i1 + j = i+ j and by hypothesis
2a | i. Finally, we also have m1i1+km2j−k ∈ M′, from which we get
m1
i1+km2
j−k =
(
m
i1+k
2 +j−k
1 ,m
i1+k
2
1 m
j−k
2
)
.
By Remark 3.3, we have mi1+j1 m
j−k
2 ∈ M′ which implies that 2a | i1 + j. As ν2(i1) =
a − 1 and 2a | j, then ν2(i1 + j) = a − 1, a contradiction as 2a | i1 + j. This completes
the proof.
As a consequence, we prove the following.
Proposition 3.5. The group M(2) is torsion-free.
Proof. Suppose by way of contradiction that there exists an element of finite order in M.
Since M/M′ ∼= Z × Z by Lemma 3.4, then this element must lie in M′ ≤ stM(1).
Suppose that among all elements of finite order, we take the element g that lies in
stM(n) \ stM(n + 1), with n minimum with this property. Write g = (g1, g2). As g is of
finite order, then also g1, g2 must be of finite order. By our minimality assumption of n, the
elements g1, g2 must lie at least in stM(n). This implies that g = (g1, g2) ∈ stM(n+ 1), a
contradiction to the fact that g ∈ stM(n) \ stM(n+ 1).
In the following we determine the order of some elements of M(d), for d > 2.
Proposition 3.6. Let d > 2. Then the group M(d) is neither torsion-free nor torsion.
Proof. For ease of notation we write M for M(d). We start by proving that the given
generators of M are of infinite order. Consider m1, and suppose by way of contradiction
that its order is n. Then if mn1 = 1, we obtain that m
n
1 must lie in stM(1). Also, its order
must be a multiple of d, say n = dk for some k, since m1 has order d modulo the first level
stabilizer. Since m1 = (1, . . . , 1,m1)(1 2 . . . d), we obtain
mn1 = (m
k
1 , . . . ,m
k
1) = (1, . . . , 1).
36 Ars Math. Contemp. 20 (2021) 29–36
This yields a contradiction since mk1 = 1 and k < n. Similar arguments can be used for
the generators m2, . . . ,md−1, and md has infinite order because md = (m1, . . . ,md).
Furthermore, by (3.2), we have
[m1,m2] = (1,m
−1
2 , 1, . . . , 1,m2)(1 2 d).
Thus it follows readily that [m1,m2]3 = 1. Hence M is not torsion-free.
We conclude the paper by proving the remaining part of the main theorem.
Proposition 3.7. The group M is non-contracting.
Proof. Suppose by way of contradiction that M is contracting with nucleus N . Notice that
the element mm1d stabilizes the vertex 1. As a consequence, by induction, m
m1
d fixes all the
vertices of the path v = 1 n. . .1 for all n ≥ 1. Also, (mm1d )v = m
m1
d . Clearly, this implies
that mm1d lies in N . Consider now a power k of m
m1
d . Arguing as before, we obtain again
that (mm1d )
k fixes v and its section at v is (mm1d )
k. Thus, (mm1d )
k ∈ N for any k ≥ 1.
This concludes the proof since mm1d has infinite order.
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 37–50
https://doi.org/10.26493/1855-3974.2046.cb6
(Also available at http://amc-journal.eu)
From Farey fractions to the Klein quartic
and beyond*
Ioannis Ivrissimtzis †
Department of Computer Science, Durham University, DH1 5LE, United Kingdom
David Singerman , James Strudwick
Mathematical Sciences, University of Southampton, SO17 1BJ, United Kingdom
Received 11 July 2019, accepted 21 September 2020, published online 14 July 2021
Abstract
In a paper published in 1878/79 Klein produced his famous 14-sided polygon repre-
senting the Klein quartic, his Riemann surface of genus 3 which has PSL(2, 7) as its au-
tomorphism group. The construction and method of side pairings are fairly complicated.
By considering the Farey map modulo 7 we show how to obtain a fundamental polygon
for Klein’s surface using arithmetic. Now the side pairings are immediate and essentially
the same as in Klein’s paper. We also extend his work from 7 to 11 as Klein also did in a
follow-up paper of 1879.
Keywords: Riemann surfaces, Klein quartic, regular maps, Farey tessellation, modular group, prin-
cipal congruence subgroups.
Math. Subj. Class. (2020): 30F10, 20H10, 51M20
1 Introduction
The Klein quartic was introduced in one of Felix Klein’s most famous papers, [5] of
1878/79. A slightly updated version appeared in Klein’s Collected Works [7], while for
a translation of this see the book The Eightfold Way, the Beauty of Klein’s Quartic Curve
edited by Silvio Levy [8]. This algebraic curve, whose equation is x3y + y3z + z3x = 0,
gives the compact Riemann surface of genus 3 with 168 automorphisms, the maximum
number by the Hurwitz bound.
*We thank the referees for their careful reading of this paper and their helpful suggestions.
†Coresponding author.
E-mail addresses: ioannis.ivrissimtzis@durham.ac.uk (Ioannis Ivrissimtzis), D.Singerman@soton.ac.uk
(David Singerman), J.Strudwick@soton.ac.uk (James Strudwick)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
38 Ars Math. Contemp. 20 (2021) 37–50
Let H denote the upper-half complex plane and let H∗ = H ∪ Q ∪ {∞}. Klein’s
surface is H∗/Γ(7), where Γ(7) is the principal congruence subgroup mod 7 of the classical
modular group Γ = PSL(2,Z). (For this concept see [4, p. 301].) Klein studies the
Riemann surface of the Klein quartic by constructing his famous 14-sided fundamental
region with its side identifications. See sections 11 and 12 of [5] for the construction and
between pages 448 and 449 of [5], page 126 of [7], or page 320 of [8] for the figure itself.
Our approach is to construct a fundamental region for Klein’s surface using the Farey
tessellation M3 of H∗, a triangular tessellation of H∗ which we define in §2, and which
was shown to be the universal triangular tessellation [10]. In §3 and §4, we study the level
n Farey map M3/Γ(n), through the correspondence of its directed edges with the elements
of Γ/Γ(n) and the correspondence of its vertices with the cosets of Γ1(n) in Γ. In §5 and
§6, we study the level 7 Farey map M3/Γ(7). As M3 ⊂ H∗, M3/Γ(7) ⊂ H∗/Γ(7), this
Farey map is embedded in the Klein surface. In a sense, we will show that this Farey map
is the Klein surface.
In §7 and §8, we review Klein’s original construction, computing Farey coordinates
on Klein’s 14-sided fundamental region and discussing the differences between the two
approaches. In volume 15 of Mathematische Annalen in 1879 [6], Klein extended his work
to study the surface H∗/Γ(11), which has PSL(2, 11) of order 660 as its automorphism
group and is somewhat more complicated. He did not draw a fundamental region for the
case n = 11 as he did for n = 7. However we are able to draw the corresponding Farey
map in §9.
2 The Farey map
The vertices of the Farey map M3 are the extended rationals, i.e. Q∪{∞} and two rationals
a
c and
b
d are joined by an edge if and only if ad − bc = ±1. These edges are drawn as
semicircles or vertical lines, perpendicular to the real axis, (i.e. hyperbolic lines). Here
∞ = 10 . This map has the following properties.
(a) There is a triangle with vertices 10 ,
1
1 ,
0
1 , called the principal triangle.
(b) The modular group Γ = PSL(2,Z) acts as a group of automorphisms of M3.
(c) The general triangle has vertices ac ,
a+b
c+d ,
b
d .
This forms a triangular tessellation of the upper half plane. Note that the triangle in (c) is
just the image of the principal triangle under the Möbius transformation corresponding to
the matrix
(
a b
c d
)
.
In [10] it was shown that M3 is the universal triangular map. This means that if M is
any triangular map on an orientable surface then M is the quotient of M3 by a subgroup
Λ of the modular group. A map is regular if its orientation preserving automorphism group
acts transitively on its darts, (i.e. directed edges) and M3/Λ is regular if and only if Λ is a
normal subgroup of Γ. The subgroup Λ here is called a map subgroup.
(In general if ∆(m,n) is the (2,m, n) triangle group, then every map of type (m,n)
has the form M̂/M where M̂ is the universal map of type (m,n) and M is a subgroup of
Γ. In our case we are thinking of the modular group Γ as being the (2, 3,∞) group. The
infinity here means that we are not concerned with the vertex valencies; we just require the
map to be triangular. For the general theory we refer to [3].)
We now consider the case when Λ = Γ(n), the principal congruence subgroup mod
n of the modular group Γ. The corresponding maps are denoted by M3(n). As Γ(n) is a
I. Ivrissimtzis et al.: From Farey fractions to the Klein quartic and beyond 39
Figure 1: The Farey map, (drawn by Jan Karabaš).
normal subgroup of Γ these maps are regular.
3 The map M3(n)
The map M3(n) is a regular map that lies on the Riemann surface H∗/Γ(n). The auto-
morphism group of M3(n) is Γ/Γ(n) ∼= PSL(2,Zn) whose order µ(n) for n > 2 is
µ(n) =
n3
2
∏
p|n
(
1− 1
p2
)
.
The product is taken over all prime factors of n, see [3, Chapter 6, Exercise 6L].
Also, µ(2) = 6.
Now µ(n) is the number of darts of M3(n) so the number of edges of this map is
µ(n)/2, and the number of faces is equal to µ(n)/3. Note that 10 is joined to
k
1 for k =
0, . . . , n− 1 so that 10 has valency n and by regularity every vertex has valency n. Thus the
number of vertices is equal to µ(n)/n. For example, µ(5) = 60, µ(7) = 168, µ(11) = 660,
so the numbers of vertices of M3(n), for n = 5, 7, 11, are 12, 24, 60, respectively. We can
now use the Euler-Poincaré formula to find the well-known formula for the genus g(n) of
M3(n);
g(n) = 1 +
n2
24
(n− 6)
∏
p|n
(
1− 1
p2
)
. (3.1)
3.1 Farey coordinates for M3(n)
If (a, c, n) = 1 then the projection of ac from M3 to M3(n) is denoted by [
a
c ], or simply
a
c when there is no room for ambiguity, To be precise, a Farey fraction
a
c is an equivalence
class of ordered pairs (a, c) ∈ Z2n with (a, c, n) = 1 under the equivalence relation (a, c) ≡
40 Ars Math. Contemp. 20 (2021) 37–50
(b, d) if b = ua, d = uc ∈ Zn and u = ±1 ∈ Zn. This is sometimes referred to as a Farey
coordinate of a vertex in M3(n).
See §4.1 for the case n = 5, where we give the Farey coordinates for the icosahedron.
4 The quasi-icosahedral structure of Farey maps
We now show that every Farey map has a quasi-icosahedral structure. Let us give some
definitions from [12].
1. The (graph-theoretic) distance δ(f1, f2) between two vertices f1 and f2 of a graph
is the least number of edges joining these two vertices.
2. A Farey circuit is a sequence of Farey fractions f1, f2, . . . , fk where fi is joined by
an edge to fi+1 with the indices taken mod k.
3. A pole of a Farey map is any vertex with coordinates a0 .
The following theorem was proved in [12].
Theorem 4.1. Let ac ,
b
d be distinct vertices of M3(p), where p is prime, and let ∆ =
ad− bc. Then:
δ
(
a
c
,
b
d
)
=

1 if and only if |∆| = 1,
2 if and only if |∆| ≠ 0, 1,
3 if and only if ∆ = 0.
Now let us call 10 the north pole N of M3(p). Then by the above theorem δ(N,
a
c ) = 1
if and only if c = ±1, δ(N, ac ) = 2 if and only if c ̸= 0,±1, and δ(N,
a
c ) = 3 if and only
if c = 0. That is, the vertices of M3(p) form four disjoint subsets: the north pole N at 10 , a
set of size n consisting of vertices whose graph-theoretic distance from N is 1, another set
of points at distance 2 from N , and other poles at distance 3 from N . In Theorem 4.2, we
will show that these two sets are in fact circuits. As the icosahedron has this property we
refer to these Farey maps as having a quasi-icosahedral structure.
(In [12] it was also shown that M3(n) has diameter 3 for all n ≥ 5.)
4.1 The icosahedron
M3(5) is an icosahedron [12] with vertex set{
1
0
,
2
0
,
0
1
,
1
1
,
2
1
,
3
1
,
4
1
,
0
2
,
1
2
,
2
2
,
3
2
,
4
2
}
;
see Figure 2. The north pole N at 10 , there is a Farey circuit of length 5 of points whose
denominator is equal to 1 and have distance 1 from N and a second circuit of length 5 of
points whose denominator is equal to 2 and have distance 2 from N . We also have the pole
2
0 at distance 3 from N .
For a quasi-icosahedral structure on M3(p) let N = 10 ∈ M3(p). The circuit of points
of distance 1 from N is
S1(p) =
0
1
,
1
1
, . . . ,
p− 1
1
.
The set of points at distance 2 from N is more complicated and we now construct it. To
make the calculation clearer we start with the example p = 7. From Theorem 4.1, we see
I. Ivrissimtzis et al.: From Farey fractions to the Klein quartic and beyond 41
Figure 2: Drawing of M3(5) with Farey coordinates.
that the points of distance 2 from 10 have the form
b
d where d = ±2 or ±3. Thus the points
1
3 ,
1
2 ,
2
3 ∈ S2(7) all have distance 2 from N . As the transformation t 7→ t+ 1 fixes N and
preserves distance, all points in S(7) + k have distance 2 from N , for k = 1, . . . , 6. Thus
we find the set
S2(7) =
1
3
,
1
2
,
2
3
,
4
3
,
3
2
,
5
3
,
0
3
,
5
2
,
1
3
,
3
3
,
0
2
,
4
3
,
6
3
,
2
2
,
0
3
,
2
3
,
4
2
,
3
3
,
5
3
,
6
2
,
6
3
consisting of points at distance 2 from N , see Figure 3. We now generalize this. Let p ≥ 5
be a prime and let
S(p) =
1
(p− 1)/2
,
1
(p− 3)/2
, . . . ,
1
3
,
1
2
,
2
3
, . . . ,
(p− 3)/2
(p− 1)/2
.
Then
Theorem 4.2. The concatenation of sequences
S2(p) = S(S + 1)(S + 2) . . . (S + p− 1),
where S = S(p), is the Farey circuit consisting of those points of distance 2 from N . The
length of S1(p) is p and the length of S2(p) is p(p− 4). There are (p− 1)/2 poles.
Proof. We first observe that the points in S2(p) do have distance 2 from N . Indeed, the
points 1k and
m−1
m for 2 ≤ k,m ≤
p−1
2 have distance 2 from N =
1
0 as
1
k ↔
0
1 and
m−1
m ↔
1
1 and none of these points have distance 1 from
1
0 . (The symbol ↔ means
adjacent to.)
The transformation t 7→ t+ 1 fixes 10 and preserves distance so that all points in S + k
have distance 2 from N = 10 . We now show that S2(p) is a Farey circuit. Clearly there are
edges between 1k and
1
k+1 for k ≥ 2 and between
k
k+1 and
k+1
k+2 for k ≥ 2. So, we only
42 Ars Math. Contemp. 20 (2021) 37–50
need to show that there is an edge between the last vertex in S + k and the first vertex in
S + k + 1. The last vertex of S + k is
k +
(p− 3)/2
(p− 1)/2
=
(p− 3 + kp− k)/2
(p− 1)/2
.
The first vertex of S + k + 1 is
k + 1 +
1
(p− 1)/2
=
(kp− k + p+ 1)/2
(p− 1)/2
.
As
[(p− 3 + kp− k)/2][(p− 1)/2]− [(kp− k + p+ 1)/2][(p− 1)/2] = −p+ 1,
we see that the last vertex of S + k is adjacent to the first vertex of S + k+1. Thus, S2(p)
is a Farey circuit consisting of points of distance 2 from 10 .
Now S1(p) clearly has p points, and the set S(p) has p−4 points, thus S2(p) has p(p−4)
points. The poles are 10 ,
2
0 , . . . with
k
0 =
−k
0 , and so the number of poles is
p−1
2 .
5 Drawing M3(7)
The map M3(7) has 24 vertices with Farey coordinates{
1
0
,
2
0
,
3
0
,
0
1
,
1
1
,
2
1
,
3
1
,
4
1
,
5
1
,
6
1
,
0
2
,
1
2
,
2
2
,
3
2
,
4
2
,
5
2
,
6
2
,
0
3
,
1
3
,
2
3
,
3
3
,
4
3
,
5
3
,
6
3
}
.
The first circuit is S1(7) = 01 ,
1
1 , . . . ,
6
1 , and we draw a polygonal curve C1(7), surrounding
1
0 , containing the points of S1(7). We draw a bigger simple closed curve C2(7), also
surrounding 10 , containing the points of S2(7). In Figure 3, C2(7) passes through the
points 13 ,
6
3 ,
6
2 ,
5
3 , . . . .
Finally, we can draw a simple closed curve C3(7) exterior to both C1(7) and C2(7)
which contains the poles 20 and
3
0 , see the dotted line in Figure 3. The pole
2
0 is a vertex
of seven triangles whose base is on the second circuit. One of these triangles is 63 ,
2
0 ,
1
3 and
the others are found by adding 1, 2, 3, 4, 5, 6 to these three points. For example, adding 1
to 63 ,
2
0 ,
1
3 gives
2
3 ( =
9
3 ),
2
0 ,
4
3 . (Adding the integer k has the geometric effect of rotating
M3(7) by 2πk .) The pole
3
0 is a vertex of seven quadrilaterals which are unions of two Farey
triangles, and also have one edge on C2(7). One of these is 13 ,
5
2 ,
3
0 ,
1
2 and we get the other
six by adding 1, 2, 3, 4, 5, 6. We end up with a 42-sided polygon pictured in Figure 3 (for
now ignore the dashed curves). It is interesting that exactly the same polygon was obtained
by E. Schulte and J. M. Wills in [9] by purely geometric methods.
6 The 14-sided polygon
We now show how to obtain a 14-sided polygon out of the Farey map M3(7) with the same
side-pairings as the Klein surface. As M3(7) has 42 edges and we need a 14-sided polygon
we define a new-edge to be a union of three consecutive edges which include vertices with
Farey coordinates 20 and
3
0 .
For example, our first new-edge goes from 20 to
5
3 to
3
2 to
3
0 and our second new edge
goes from 30 to
6
2 to
6
3 to
3
0 , see Figure 3. We now replace the new-edges by dashed lines.
I. Ivrissimtzis et al.: From Farey fractions to the Klein quartic and beyond 43
Figure 3: Drawing of M3(7) with Farey coordinates.
In Figure 3 the dashed line labelled 1 goes from a vertex labelled 20 to a vertex labelled
3
0 surrounding the vertices
5
3 and
3
2 of the first new edge, and similarly the dashed line
labelled 2 goes from 30 to
2
0 surrounding the vertices
6
2 and
6
3 . Notice that the dashed lines
are not part of the map M3(7), they are just a convenient way of representing our 14-
sided polygon. We can associate four Farey fractions to each dashed edge. For example,
associated to the dashed edge 1 we have the Farey fractions 20 ,
5
3 ,
3
2 ,
3
0 . We pair two new-
edges if their associated Farey fractions are the same. For example, consider the new-edge
labelled 6 in Figure 3. The associated Farey fractions are 30 ,
3
2 ,
5
3 ,
2
0 . These are the same
Farey fractions, but in reverse order as for the new-edge 1. This means we identify the
new edges 1 and 6 orientably. Similarly we get the other six identifications. Thus the
identifications are
1 ↔ 6, 3 ↔ 8, 5 ↔ 10, 7 ↔ 12, 9 ↔ 14, 11 ↔ 2, 13 ↔ 4.
This is exactly the same side-pairing as found by Klein from his 14-sided polygon which
shows that our 14-sided polygon does give the Klein quartic. Our way of finding the side
identifications is much more straightforward than the method used in Klein’s paper, which
we will summarize in §8.
7 Farey Coordinates for the Klein map
A regular map has type {m,n} if every face has size m and every vertex has valency n.
(We are following [1] here and not [3] where these numbers are reversed.) Now M3(n)
is a regular map of type {3, n} because 10 is adjacent to
0
1 , . . . ,
n−1
1 . Now M3(7) is the
44 Ars Math. Contemp. 20 (2021) 37–50
Klein map, or, in the standard notation in [1], the map {3, 7}8. The ‘8’ here is the length
of a Petrie polygon. (For Petrie polygons and how we find the lengths of Petrie polygons
using Farey fractions see [11].)
As noted in the introduction, the Klein map M3(7) is embedded in the Klein surface
H∗/Γ(7). The term “Klein map” comes from the drawing on page 320 of [7], or page 120
of [8], of Klein’s 14-sided polygon. After the given side identifications this does give a map
on a surface of genus 3. See Figure 4 (and just ignore the Farey coordinates in this diagram
for now). This is not the Klein map, for it is not regular, having vertices of different valency.
It consists of 336 triangles while the Klein map M3(7) has 56 triangles. Nevertheless, we
can easily obtain the Klein map from Figure 4. The vertices of the map are the vertices of
valency 14. Before we describe the Klein map structure on this surface we show how to
associate the 24 Farey fractions modulo 7 to the 24 vertices.
First, we assign the Farey coordinate 10 to the centre point. We note that there are
two circuits of seven vertices centred at 10 . We give the first circuit the Farey coordinates
0
1 ,
1
1 , . . . ,
6
1 . If we extend the perpendicular bisector from
1
0 to the hyperbolic line between
0
1 and
1
1 we get to another vertex of valency seven to which we assign the coordinate
0+1
1+1 =
1
2 . Similarly, we extend the perpendicular bisector from
1
0 to the hyperbolic line
between 11 and
1
2 to a vertex of valency seven which we give the Farey coordinate
3
2 . By
continuing, we find all vertices with Farey coordinates 12 ,
3
2 ,
5
2 ,
0
2 ,
2
2 ,
6
2 . Thus we have now
found all vertices with Farey coordinates xi for i = 1, 2 and we just have to find the vertices
with Farey ccordinates x0 or
x
3 which lie on the boundary of K. After Klein’s identifications
shown in Figure 3, we see that the 14 corners of K belong to two classes, which we can
label 20 ,
3
0 . Between any two of these vertices there is precisely one more vertex of M3(7).
(After side identifications these vertices also have valency 14.) We can assign to them
Farey coordinates of the form x3 just by reading them off from Figure 3. In fact, each
x
3
occurs exactly twice and we can now pair sides of K that have exactly the same value of x.
Again, this gives exactly the same side pairing as Klein found. We thus have two methods,
in sections 7 and 8, of using Farey coordinates to get Klein’s pairings just by observation.
Figure 4 gives a description of Klein’s work using Farey coordinates. We see that each
of the 14 sides of the boundary of K consists of a Farey edge and a non-Farey edge. The
segment from 20 to
x
3 is a Farey edge whilst the segment from
x
3 to
3
0 is not a Farey edge.
There is no automorphism of K mapping one segment to the other since all elements of Γ
map Farey edges to Farey edges. Note that by section 3, the Klein map has 24 vertices, 56
faces and 84 edges.
We now give the map structure. The vertices of the map are the points of valency 14 in
Figure 4, that is, those points that have been given Farey coordinates. An edge joins points
with Farey coordinates ac iand
b
d if and only if ad− bc ≡ 1 (mod 7). Three vertices with
Farey coordinates ac ,
b
d and
e
f form a triangular face if and only if e ≡ a + b (mod 7),
f ≡ c + d (mod 7). For example, there is a triangle with vertices 41 ,
4
3 and
6
3 for
6
3
represents the same point as 14 , for
1
4 =
−6
−3 .
8 What Klein did
Here we review Klein’s original construction of his fundamental domain of the congruence
subgroup Γ(7), and show how this construction can be interpreted in terms of the Farey
machinery we described above.
By the end of section 10 of [5] Klein had obtained the equation of his quartic curve and
I. Ivrissimtzis et al.: From Farey fractions to the Klein quartic and beyond 45
Figure 4: Farey coordinates on the Klein surface.
in section 11 he started to discuss the Riemann surface of this algebraic curve and also the
corresponding map. In fact, this was one of the first publications to use maps (or in today’s
language dessins d’enfants) in a profound way, pointing up the deep correspondence be-
tween maps and algebraic curves. While this correspondence was not properly understood
until Grothendieck’s Esquisse d’un programme some 105 years later [2], we note that in an
interesting anticipation of Grothendieck’s programme, Figure 2 of Klein’s follow-up paper
[6] shows the ten planar dessins of type (2, 3, 11) and degree 11.
Klein’s quite complicated construction of his fundamental domain comes from con-
sidering fundamental regions for subgroups of indices 7 and 8 in the modular group. In
section 12, he writes (in German) “In order not to make these considerations too abstract
I will resort to the ω-plane”; this is the upper-half plane on which the modular group acts.
46 Ars Math. Contemp. 20 (2021) 37–50
In Figure 6, he constructs a hyperbolic polygon corresponding to his 14-sided polygon
describing his surface. Then, in Figure 7, he draws semicircles (hyperbolic lines) in the
upper-half plane with rational vertices, which correspond to the edges of his 14-sided poly-
gon. Now consider this polygon as being inscribed in the unit disc so the vertices all lie on
the boundary circle. As the unit disc is conformally equivalent to the upper-half plane the
boundary circle corresponds to the real axis and so, every point of the circle has some real
coordinate. He starts with one edge (labelled 1) of his 14-sided polygon corresponding to
two consecutive edges of the polygon in the upper-half plane with vertices 27 ,
1
3 and
1
3 ,
3
7 .
(As we already noted above, 27 ,
1
3 is a Farey edge while
1
3 ,
3
7 is not, therefore we cannot
map one to the other by an element of Γ). A second edge (labelled 6) is given as the pair of
consecutive edges 187 ,
8
3 and
8
3 ,
19
7 . The Möbius transformation corresponding to the matrix(
113 −35
42 −13
)
in Γ(7) maps edge 1 (i.e. 27 ,
1
3 ,
3
7 ) to edge 6 (i.e.
18
7 ,
8
3 ,
19
7 ), and one more explicit example
of edge pairing is given. He states that in total seven such matrices can be found that give
all the side pairings. We feel that our technique of just using Farey coordinates is much
easier.
9 M3(11)
About a year after Klein wrote his paper [5] on the quartic curve, he wrote a further paper
[6] with the same title but with ‘siebenter’ replaced with ‘elfter’, i.e. ‘seventh’ replaced
with ‘eleventh’; basically, he was considering H∗/Γ(11). In that paper he did not draw a
diagram of the fundamental region equivalent to his drawing of K in [5]. Here we show
how to draw the Farey map M3(11) in a similar way to how we drew M3(7). This Farey
map will be embedded in the surface H∗/Γ(11).
The first circuit of vertices at distance 1 from 10 is
S1(11) =
0
1
,
1
1
, . . . ,
10
1
.
Now consider the sequence of vertices
S(11) =
1
5
,
1
4
,
1
3
,
1
2
,
2
3
,
3
4
,
4
5
;
and then the second circuit is
S2(11) = S(11) (S(11) + 1) . . . (S(11) + 10).
The orientation-preserving automorphism group of M3(11) is PSL(2, 11) of order 660
so the Farey map M3(11) has 660/2 = 330 edges, 660/3 = 220 triangles and 660/11 =
60 vertices. The Farey coordinates of the vertices are 10 ,
2
0 ,
3
0 ,
4
0 ,
5
0 and all Farey fractions
of the form rs for r = 0 to 10 and s = 1 to 5.
To draw the map we just need to find the 220 triangular faces. Because z 7→ z + 1 is
an automorphism of M3(11), which acts as a rotation about the centre 10 of the map, we
see that this map is divided into eleven congruent sectors each containing 220/11 = 20
triangles each. We construct one such sector W , shown in Figure 5, by starting from the
I. Ivrissimtzis et al.: From Farey fractions to the Klein quartic and beyond 47
Figure 5: The sector W .
central triangle 10 ,
0
1 ,
1
1 and adding 19 distinct triangles whose vertices lie in S(11). Exactly
8 of these 19 triangles have a vertex on the first circuit S1(11) ( 01 or
1
1 in particular) and
are uniquely determined. For the remaining 11 triangles, which either have three vertices
on S2(11), or two vertices on S2(11) and a pole vertex, there are several choices satisfying
the condition that they are distinct under rotation about 10 .
Figure 5 shows one such solution as the union of 20 triangles. The actual Farey coordi-
nates are
P0 =
1
0 P1 =
0
1 P2 =
1
5 P3 =
1
4 P4 =
1
3 P5 =
2
5
P6 =
1
2 P7 =
5
0 P8 =
6
2 P9 =
7
4 P10 =
3
5 P11 =
6
3
P12 =
4
0 P13 =
2
3 P14 =
6
4 P15 =
3
0 P16 =
3
4 P17 =
4
2
P18 =
4
5 P19 =
2
0 P20 =
6
5 P21 =
1
1
Each point Pi is labelled i in Figure 5 to reduce clutter.
Now let
W ∗ = W ∪ (W + 1) ∪ · · · ∪ (W + 10)
where W + k is defined as in Section 5, that is, geometrically, is the rotation of W by
2π
k . Then W
∗ is the union of 220 triangles as required and its boundary is a polygon with
11× 18 = 198 sides. A diagram of the map W ∗ is given in Figure 6.
Table 1 in the Appendix shows a list of the 198 boundary vertices of W ∗ arranged in
11 rows. The first row corresponds to W and the kth row is just the first row plus (k − 1).
We now notice that we have an orientable side pairing. For example, the first edge in row
1 going from 15 to
1
4 is paired with the edge in row 5 going from
1
4 to
1
5 , the next edge in
row 1 going from 14 to
1
3 is paired with the edge in row 8 going from
1
3 to
1
4 . Proceeding
in this way we find that all the 198 edges of the polygon are paired orientably which shows
that this polygon represents an orientable surface which must be M3(11). As the map W ∗
has 60 vertices, 220 edges, and 330 triangles, by the Euler-Poincaré formula the genus of
the surface is 26.
48 Ars Math. Contemp. 20 (2021) 37–50
Figure 6: The map W ∗.
ORCID iDs
Ioannis Ivrissimtzis https://orcid.org/0000-0002-3380-1889
David Singerman https://orcid.org/0000-0002-0528-5477
References
[1] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, volume 14
of Ergbnisse der Mathematik und ihre Grenzgebiete, Springer-Verlag, Berlin Heidelberg, 4th
edition, 1980, doi:10.1007/978-3-662-21943-0.
[2] A. Grothendieck, Esquisse d’un programme, in: L. Schneps and P. Lochak (eds.), Geomet-
ric Galois Actions, Volume 1: Around Grothendieck’s Esquisse d’un Programme, Cambridge
University Press, Cambridge, volume 242 of London Mathematical Society Lecture Note Se-
ries, pp. 5–48, 1997, doi:10.1017/cbo9780511758874.003, with an English translation on pp.
243–283.
[3] G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math.
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I. Ivrissimtzis et al.: From Farey fractions to the Klein quartic and beyond 49
[4] G. A. Jones and D. Singerman, Complex Functions: An Algebraic and Geometric Viewpoint,
Cambridge University Press, Cambridge, 1987, doi:10.1017/cbo9781139171915.
[5] F. Klein, Ueber die Transformation siebenter Ordnung der elliptischen Functionen, Math. Ann.
14 (1878), 428–471, doi:10.1007/bf01677143.
[6] F. Klein, Ueber die Transformation elfter Ordnung der elliptischen Functionen, Math. Ann. 15
(1879), 533–555, doi:10.1007/bf02086276.
[7] F. Klein, Gesammelte Mathematische Abhandlungen, Volume 3: Elliptische Funktionen, Ins-
besondere Modulfunktionen Hyperelliptische und Abelsche Funktionen Riemannsche Funktio-
nentheorie und Automorphe Funktionen, Springer, Berlin Heidelberg, 1923.
[8] S. Levy, The Eightfold Way: The Beauty of Klein’s Quartic Curve, volume 35 of Mathemat-
ical Sciences Research Institute Publications, Cambridge University Press, Cambridge, 1999,
http://library.msri.org/books/Book35/contents.html.
[9] E. Schulte and J. M. Wills, A polyhedral realization of Felix Klein’s map {3, 7}8 on a Riemann
surface of genus 3, J. London Math. Soc. 32 (1985), 539–547, doi:10.1112/jlms/s2-32.3.539.
[10] D. Singerman, Universal tessellations, Rev. Mat. Univ. Complut. Madrid 1 (1988), 111–123,
http://www.mat.ucm.es/serv/revmat/vol1-123/vol1-123h.html.
[11] D. Singerman and J. Strudwick, Petrie polygons, Fibonacci sequences and Farey maps, Ars
Math. Contemp. 10 (2016), 349–357, doi:10.26493/1855-3974.864.e9b.
[12] D. Singerman and J. Strudwick, The Farey maps modulo n, Acta Math. Uni. Com.
89 (2020), 39–52, http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/
amuc/article/view/913.
50 Ars Math. Contemp. 20 (2021) 37–50
A Appendix
Table 1: The boundary vertices of W ∗. The last vertex of a row is repeated as the first
vertex of the row below. Each row represents a sector; the first row represents sector W in
Figure 5. Vertices in bold belong to edges which are paired with edges in the first row W .
1
5
1
4
1
3
2
5
1
2
5
0
6
2
7
4
3
5
6
3
4
0
2
3
6
4
3
0
3
4
4
2
4
5
2
0
6
5
6
5
5
4
4
3
7
5
3
2
5
0
8
2
0
4
8
5
9
3
4
0
5
3
10
4
3
0
7
4
6
2
9
5
2
0
0
5
0
5
9
4
7
3
1
5
5
2
5
0
10
2
4
4
2
5
1
3
4
0
8
3
3
4
3
0
0
4
8
2
3
5
2
0
5
5
5
5
2
4
10
3
6
5
7
2
5
0
1
2
8
4
7
5
4
3
4
0
0
3
7
4
3
0
4
4
10
2
8
5
2
0
10
5
10
5
6
4
2
3
0
5
9
2
5
0
3
2
1
4
1
5
7
3
4
0
3
3
0
4
3
0
8
4
1
2
2
5
2
0
4
5
4
5
10
4
5
3
5
5
0
2
5
0
5
2
5
4
6
5
10
3
4
0
6
3
4
4
3
0
1
4
3
2
7
5
2
0
9
5
9
5
3
4
8
3
10
5
2
2
5
0
7
2
9
4
0
5
2
3
4
0
9
3
8
4
3
0
5
4
5
2
1
5
2
0
3
5
3
5
7
4
0
3
4
5
4
2
5
0
9
2
2
4
5
5
5
3
4
0
1
3
1
4
3
0
9
4
7
2
6
5
2
0
8
5
8
5
0
4
3
3
9
5
6
2
5
0
0
2
6
4
10
5
8
3
4
0
4
3
5
4
3
0
2
4
9
2
0
5
2
0
2
5
2
5
4
4
6
3
3
5
8
2
5
0
2
2
10
4
4
5
0
3
4
0
7
3
9
4
3
0
6
4
0
2
5
5
2
0
7
5
7
5
8
4
9
3
8
5
10
2
5
0
4
2
3
4
9
5
3
3
4
0
10
3
2
4
3
0
10
4
2
2
10
5
2
0
1
5
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 51–68
https://doi.org/10.26493/1855-3974.1996.db7
(Also available at http://amc-journal.eu)
On the incidence map of incidence structures*
Tim Penttila
School of Mathematical Sciences, The University of Adelaide,
Adelaide, South Australia, 5005 Australia
Alessandro Siciliano
Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della
Basilicata, Viale dell’Ateneo Lucano 10, 85100 Potenza, Italy
Received 30 April 2019, accepted 24 September 2020, published online 20 July 2021
Abstract
By using elementary linear algebra methods we exploit properties of the incidence map
of certain incidence structures with finite block sizes. We give new and simple proofs of
theorems of Kantor and Lehrer, and their infinitary version. Similar results are obtained
also for diagrams geometries.
By mean of an extension of Block’s Lemma on the number of orbits of an automor-
phism group of an incidence structure, we give informations on the number of orbits of:
a permutation group (of possible infinite degree) on subsets of finite size; a collineation
group of a projective and affine space (of possible infinite dimension) over a finite field on
subspaces of finite dimension; a group of isometries of a classical polar space (of possible
infinite rank) over a finite field on totally isotropic subspaces (or totally singular in case of
a orthogonal space) of finite dimension.
Furthermore, when the structure is finite and the associated incidence matrix has full
rank, we give an alternative proof of a result of Camina and Siemons. We then deduce that
certain families of incidence structures have no sharply transitive sets of automorphisms
acting on blocks.
Keywords: Incidence structure, incidence map, diagram geometry.
Math. Subj. Class. (2020): 05B20, 05B05
*The authors would like to thank the anonymous referee for her/his comments as they greatly improved the
first version of the paper.
E-mail addresses: tim.penttila@adelaide.edu.au (Tim Penttila), alessandro.siciliano@unibas.it (Alessandro
Siciliano)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
52 Ars Math. Contemp. 20 (2021) 51–68
1 Introduction
An incidence structure is a triple I = (P,B, I) where P and B are disjoint sets and I is
a subset of P × B. The elements of P are called points, those of B blocks and I defines
the following incidence relation: the point P and the block B are incident if and only if
(P,B) ∈ I, and we will write P IB. The incidence structure I has finite block sizes if
{P ∈ P : P IB} has finite size for all B ∈ B; I is finite if P and B, and hence also I,
are finite sets. An automorphism of an incidence structure is a pair of permutations (π, β),
with π acting on P and β on B, such that P IB if and only if Pπ IBβ , for all P ∈ P and
B ∈ B. The group of all automorphisms is denoted by Aut I.
A finite incidence structure can be represented by a (0, 1)-matrix A with rows indexed
by points and columns indexed by blocks, and with the (P,B)-entry equal to 1 if and only
if P is incident with B. The incidence matrix A have been studied by many authors at least
since the 1960s, and most of their investigations were on the rank of A. Dembowski in [12,
p. 20] showed that the rank of the incidence matrix defined by the natural incidence relation
of points versus i-dimensional subspaces of a finite d-dimensional projective or affine space
is the number of points of the geometry. This result was generalized by Kantor in [14].
He showed that the incidence matrix defined by the incidence between the i-dimensional
subspaces and the j-dimensional subspaces of a finite d-dimensional projective or affine
space, with 0 ≤ i < j ≤ d − i − 1, has full rank. Analogous results for the incidence
matrices of all k-subsets versus all l-subsets of a m-set and for the incidence matrices
arising from finite polar spaces were proved by Lehrer [16].
A decomposition of an incidence structure I = (P,B, I) is a partition of P into point
classes together with a partition of B into block classes. A decomposition is said to be
block-tactical if the number of points in a point class which lie in a block depends only on
the class in which the block lies. When the incidence structure is finite then the fundamental
Block’s Lemma [2, Theorem 2.1] states that in a block-tactical decomposition the number
of point classes differs from the number of block classes by at most the nullity of the
incidence matrix of the structure. A principle example of block tactical decomposition is
obtained by taking as the point and the block classes the orbits of any automorphism group
of the structure. So, Block’s Lemma naturally leads to consideration of the rank of the
incidence matrix in order to study the number of orbits of an automorphism group of an
incidence structure.
When I = (P,B, I) is finite, and both permutation representations of any automor-
phism of I are regarded as linear representations of the automorphism group, then the in-
cidence matrix A of I is an intertwining operator between the linear representations of the
automorphism group on P and B. Using this fact, Camina and Siemons [11] showed that
when A has maximum rank then the permutation representation on points is a subrepresen-
tation of the permutation representation on blocks. This containment relation implies the
non-existence of sharply 1-transitive sets of automorphisms on blocks unless the number
of points divides the number of blocks [19].
The aim of this paper is to bring together all the previous questions by providing a
unified treatment. Our approach is different from those adopted by the authors referred to
above: the main idea is to exploit properties of the incidence map of incidence structures
by using elementary linear algebra methods. We find a new and simpler proof of Kantor’s
and Lehrer’s theorems, beside giving the infinitary version of these results. We also provide
some geometric version of the main result in [9] on the number of orbits of a permutation
group on unordered sets by mean of an extension of Block’s Lemma [2] on the number of
T. Penttila and A. Siciliano: On the incidence map of incidence structures 53
orbits of an automorphism group of an incidence structure. Furthermore, when the structure
is finite and the associated incidence matrix has full rank, we give an alternative proof of
the result of Camina and Siemons [11].
We now give a summary of the present paper. In Section 2 we prove that the incidence
map of certain (possibly infinite) incidence structures is one-to-one. The keystone is a re-
sult (Lemma 2.6) about the kernel of the incidence map from i-dimensional subspaces to
(i+1)-dimensional subspaces of a finite d-dimensional projective space, where incidence is
the inclusion relation. By replacing the dimension with size of a set and the Gaussian coef-
ficients with binomial coefficients, we get the analogous result for the incidence map from
k-sets to (k + 1)-sets of an m-set, where incidence is the inclusion relation. This leads
to an alternative proof of both of Kantor’s theorems, on the incidence structures arising
from projective and affine spaces, and of Lehrer’s theorem [16] on the incidence structures
arising from subsets. These results are summarized in Theorem 2.7. In Section 3 we illus-
trate some applications of Theorem 2.7. Under the hypothesis that every block is incident
with a finite number of points we prove the infinitary version of the above results. From
Kantor’s theorem for projective spaces, and because of its infinitary version, we prove that
the Lehrer result about incidence structures in finite classical polar spaces [16] holds also
in case of polar spaces of infinite rank. Similar results are obtained for diagram geometries
associated to certain finite Chevalley groups. If ∆ denotes the diagram of the geometry,
then by using [7, Theorem 2] we show that the k-varieties give rise to full substructures
of the incidence structure of i-varieties versus j-varieties of the geometry, provided i and
k lie in distinct connected components of ∆ − {j}. This gives plenty of scope to apply
the main result (Lemma 3.1) of this section. It is conceivable that the weak conclusion that
there are as many j-varieties as i-varieties could be useful to diagram geometers. Section 4
is related with Block’s Lemma. In the function space and incidence map setting we prove a
slight extension of this fundamental result. We then apply it to obtain informations on the
number of orbits of: a permutation group (of possible infinite degree) on subsets of finite
size; a collineation group of a projective and affine space (of possible infinite dimension)
over a finite field on subspaces of finite dimension; a group of isometries of a classical polar
space (of possible infinite rank) over a finite field on totally isotropic subspaces (or totally
singular in case of a orthogonal space) of finite dimension. We point out that the result
on permutation groups was obtained by Cameron in [9], where the theorem of Livingstone
and Wagner [17] is proved to hold also for permutation groups of infinite degree. Section 5
is all in the finite setting. We provide an alternative proof of the result of Camina and
Siemons [11] which states that if the incidence map of a finite incidence structure is one-
to-one, then the permutation representation on points of any given automorphism group is
a subrepresentation of the representation on blocks with equal or greater multiplicity. We
then deduce that certain families of incidence structures have no sharply transitive sets of
automorphisms acting on blocks.
Although some of the results presented here have been obtained by other authors and
appear scattered over a large number of papers, in our opinion it is difficult to find a con-
venient reference for this knowledge with a presentation that doesn’t assume a lot of the
reader. This work can be considered as an attempt to providing such a reference.
54 Ars Math. Contemp. 20 (2021) 51–68
2 The rank of incidence maps
In order to treat our arguments by linear algebra methods, we introduce the incidence map
of a finite incidence structure. Let I = (P,B, I) be an incidence structure. The point space
of I is the vector space QP of all functions P → Q; the block space of I is the vector
space QB of all functions B → Q. When I has finite block sizes, we define the (linear)
incidence map α : QP → QB of I by the rule
(fα)(B) =
∑
P IB
f(P ),
for all B ∈ B and f ∈ QP .
For any subset Y of a given set X the characteristic function χY ∈ QX of Y is defined
as follows:
χY (x) =
{
1 for x ∈ Y ;
0 for x ∈ X \ Y.
With this notation, the set {χ{P} : P ∈ P} is a basis for QP and {χ{B} : B ∈ B} is a
basis for QB; we refer to each of these bases as the natural basis of the corresponding space.
If I is finite the matrix of the map α with respect to these bases is precisely the incidence
matrix of I, with multiplication being on the right (i.e., vectors regarded as rows).
We now exhibit some properties of the incidence maps of the incidence structures aris-
ing from subspaces of a finite dimensional projective space over a finite field.
Let PG(d, q) be the projective space of dimension d over the finite field with q elements.
For 0 ≤ i ≤ d − 1, let Fi denote the set of all i-dimensional subspaces (or i-subspaces,
for short) of PG(d, q). For i ̸= j we consider the incidence structure I = (P,B, I) where
P = Fi, B = Fj and the incidence relation I is given by set-theoretic inclusion.
The following notation will be adopted in the rest of the paper:
• Vi denotes the vector space QFi of functions from Fi to Q;
• αi,j denotes the incidence map from Vi to Vj , with i ̸= j;
• W−1 = V−1 = {∅};
• Wi denotes the kernel of αi,i−1, for i ≥ 0.
With the above notation, αi,i is the identity map on Vi. For any Si ∈ Fi, the coordinate
array of χ{Si}αi,j , whose entries are indexed by elements of Vj , is precisely the i-th row
of the incidence matrix A of αi,j . In other words, if i > j then the image under αi,j of
χ{Si} is the characteristic function of the set of j-subspaces contained in Si. Similarly,
if i < j then the image under αi,j of χ{Si} is the characteristic function of the pencil of
j-subspaces passing through Si.
In the following we need the q-analogs of binomial coefficients, which are defined by[
n
k
]
q
=
k−1∏
i=0
(qn−i − 1)/(qk−i − 1)
for non-negative integers n, k with n ≥ k. Note that
[
n
k
]
q
is the number of (k−1)-subspaces
of PG(n− 1, q).
T. Penttila and A. Siciliano: On the incidence map of incidence structures 55
Lemma 2.1. Let −1 ≤ i ≤ j ≤ k ≤ d− 1. Then
αi,jαj,k =
[
k − i
j − i
]
q
αi,k.
Proof. By applying directly the definition of αi,j we see that
(fαi,jαj,k)(Sk) =
∑
Si⊆Sj⊆Sk
f(Si)
holds for all f ∈ Vi. The result now follows by recalling that the number of j-subspaces
in PG(d, q) through any given i-subspace which is in turn contained in a k-subspace is[
k−i
j−i
]
q
.
Lemma 2.2. For i = −1, . . . , d,
Vi =
i⊕
j=−1
Wjαj,i. (2.1)
(Note that some of the summands may be 0).
Proof. For i = −1 the result is trivial. For i = 0, . . . , d − 1, we note that Vi is a vector
space over a field of characteristic zero. Then the inner product defined by
⟨g, h⟩i =
∑
Si∈Fi
g(Si)h(Si), (2.2)
for all g, h ∈ Vi, is a non-degenerate bilinear form. Since, in the natural bases of Vi
and Vj , the matrix of αi−1,i is the transpose of the matrix of αi,i−1, then ⟨fαi−1,i, g⟩i =
⟨f, gαi,i−1⟩i−1, for all f ∈ Vi−1 and g ∈ Vi, i.e. the incidence map αi−1,i and the dual
map αi,i−1 are adjoint.
We now show that Vi = Wi⊕Imαi−1,i. Let ⊥i denote the polarity defined by the inner
product ⟨−,−⟩i. Since Vi is finite dimensional, then Vi = Imαi−1,i⊕ (Imαi−1,i)⊥i . Fur-
thermore, for all g ∈ Wi and f ∈ Vi−1, ⟨fαi−1,i, g⟩i = ⟨f, gαi,i−1⟩i−1 = 0 holds,
giving Imαi−1,i ⊆ W⊥ii , or equivalently, Wi ⊆ (Imαi−1,i)⊥i . Conversely, if g ∈
(Imαi−1,i)
⊥i , then 0 = ⟨fαi−1,i, g⟩i = ⟨f, gαi,i−1⟩i−1, for all f ∈ Vi−1. By the non-
degeneracy of ⟨−,−⟩i−1, we get gαi,i−1 = 0, and hence g ∈ Wi.
We now use induction on i. For i = −1 we have V−1 = W−1. Assume the statement
holds for Vi−1, that is Vi−1 =
⊕i−1
j=−1 Wjαj,i−1. As Vi = Imαi−1,i⊕Wiαi,i, to conclude
the proof we only need to prove that Imαi−1,i =
⊕i−1
j=−1 Wjαj,i. But this easily follows
from Lemma 2.1 since
Imαi−1,i = Vi−1αi−1,i =
i−1⊕
j=−1
Wjαj,i−1αi−1,i =
i−1⊕
j=−1
Wjαj,i.
Remark 2.3. We point out that the bilinear form defined by (2.2) is an appropriate one for
the permutation module Vi, in that permutations of the characteristic functions of single-
tons are isometries of the form. In the basis consisting of the characteristic functions of
singletons, this is just a way of saying that permutation matrices are orthogonal in the usual
sense of the term, that is PPT = I .
56 Ars Math. Contemp. 20 (2021) 51–68
Lemma 2.4. For i = 0, . . . , d− 1,
αi,i+1αi+1,i = αi,i−1αi−1,i +
([
d− i
1
]
q
−
[
i+ 1
1
]
q
)
αi,i.
Proof. Let Si, S′i ∈ Fi. For any given Si+1 ∈ Fi+1 we have
(χ{Si}αi,i+1)(Si+1) =
{
1 if Si ⊂ Si+1;
0 otherwise.
It easily follows that
(χ{Si}αi,i+1αi+1,i)(S
′
i) =
∑
Si+1⊃S′i
(χ{Si}αi,i+1)(Si+1)
is the number of (i+ 1)-subspaces containing both Si and S′i. This number equals
0 if dim (Si ∩ S′i) < i− 1;
1 if dim (Si ∩ S′i) = i− 1;[
d−i
1
]
q
if S′i = Si.
Applying similar arguments we see that (χ{Si}αi,i−1αi−1,i)(S
′
i) is the number of (i− 1)-
subspaces contained in both Si and S′i. This number is
0 if dim (Si ∩ S′i) < i− 1;
1 if dim (Si ∩ S′i) = i− 1;[
i+1
1
]
q
if S′i = Si.
The result then follows.
Lemma 2.5. For j = −1, . . . , i,
(αi,i+1αi+1,i)|Wjαj,i =
i∑
k=j
([
d− k
1
]
q
−
[
k + 1
1
]
q
)
αi,i.
Proof. We use induction on i. For i = −1 we have W−1 = V−1 = {∅} by definition. We
also note that
[
d+1
1
]
q
= (qd+1 − 1)/(q− 1) is the number of points in PG(d, q), that is the
size of F0. Then,
(α−1,0α0,−1)|V−1 = (qd+1 − 1)/(q − 1)α−1,−1 =
[
d+ 1
1
]
q
α−1,−1.
Now let i ≥ 0. For j = i, the result follows immediately from Lemma 2.4.
Let j < i. By Lemma 2.4 we have
(αi,i+1αi+1,i)|Wjαj,i = (αi,i−1αi−1,i)|Wjαj,i +
([
d− i
1
]
q
−
[
i+ 1
1
]
q
)
αi,i|Wjαj,i .
T. Penttila and A. Siciliano: On the incidence map of incidence structures 57
To conclude the proof it is enough to show that
(αi,i−1αi−1,i)|Wjαj,i =
i−1∑
k=j
([
d− k
1
]
q
−
[
k + 1
1
]
q
)
αi,i.
By the inductive hypothesis
(αi−1,iαi,i−1)|Wjαj,i−1 =
i−1∑
k=j
([
d− k
1
]
q
−
[
k + 1
1
]
q
)
αi−1,i−1,
and Lemma 2.1 gives αj,i−1αi−1,i =
[
i−j
i−j−1
]
q
αj,i =
[
i−j
1
]
q
αj,i. Hence, we may write
wjαj,iαi,i−1αi−1,i =
[
i− j
1
]−1
q
wjαj,i−1(αi−1,iαi,i−1)αi−1,i
=
i−1∑
k=j
([
d− k
1
]
q
−
[
k + 1
1
]
q
)[
i− j
1
]−1
q
wjαj,i−1αi−1,i
=
i−1∑
k=j
([
d− k
1
]
q
−
[
k + 1
1
]
q
)
wjαj,i,
for wj ∈ Wj . This implies
(αi,i−1αi−1,i)|Wjαj,i =
i−1∑
k=j
([
d− k
1
]
q
−
[
k + 1
1
]
q
)
αi,i,
which is the desired result.
Lemma 2.6. Let i = 0, . . . , d− 1. Then
kerαi,i+1 =
{
0 for i < d−12 ;
Wd−i−1αd−i−1,i for i ≥ d−12 .
Proof. It is clear that kerαi,i+1 ≤ ker (αi,i+1αi+1,i). In addition,
dimker (αi,i+1αi+1,i) = dimkerαi,i+1 + dim (kerαi+1,i ∩ Imαi,i+1).
From the proof of Lemma 2.2, we get kerαi+1,i ∩ Imαi,i+1 = 0. Therefore kerαi,i+1 =
ker (αi,i+1αi+1,i).
From Lemmas 2.2 and 2.5, the eigenvalues of αi,i+1αi+1,i are the integers
i∑
k=j
([
d− k
1
]
q
−
[
k + 1
1
]
q
)
, (2.3)
for j = −1, . . . , i, with the j-th eigenvalue corresponding to the summand Wjαj,i in the
decomposition (2.1) of Vi. For i < (d− 1)/2 all these integers are non-zero, and therefore
kerαi,i+1 = 0.
58 Ars Math. Contemp. 20 (2021) 51–68
Let i ≥ (d − 1)/2. Two cases are treated separately according as d is odd or even.
Let d be odd and assume i = (d − 1)/2. It is easily seen that the only zero eigenvalue of
αi,i+1αi+1,i is for j = i = d− i− 1, as d− (d− 1)/2 = (d− 1)/2 + 1. Therefore,
kerα d−1
2 ,
d+1
2
= W d−1
2
α d−1
2 ,
d−1
2
.
Now let i = (d − 1)/2 + δ, for some integer δ > 0. We note that the summand with
k = (d− 1)/2 in the expression (2.3) is zero. A straightforward calculation shows that for
sufficiently small j, the summand with k = (d−1)/2− l in (2.3) erases with the summand
with k = (d − 1)/2 + l, for 1 ≤ l ≤ δ. This implies that the only zero eigenvalue of
αi,i+1αi+1,i is for j = (d − 1)/2 − δ = d − i − 1. Hence, the kernel of αi,i+1αi+1,i is
Wd−i−1αd−i−1,i.
For d even, the above approach still works up to some differences. For completeness,
we give all details.
If d is even, we write i = ⌈d−12 ⌉+ δ, for some integer δ ≥ 0. For sufficiently small j,
the summand with k = ⌈d−12 ⌉− l−1 in the expression (2.3) erases with the summand with
k = ⌈d−12 ⌉+ l, for 0 ≤ l ≤ δ. This implies that the only zero eigenvalue of αi,i+1αi+1,i is
for j = ⌈d−12 ⌉−δ−1 = d−i−1. Hence the kernel of αi,i+1αi+1,i is Wd−i−1αd−i−1,i.
The above Lemmata lead to the following fundamental theorem whose proof is new
and, in our opinion, more elementary than those provided in [14] and [16].
Theorem 2.7. The incidence map of the following incidence structures is one-to-one:
(i) i-sets versus j-sets of a d-set, with i < j and i+ j ≤ d < ∞.
(ii) i-spaces versus j-spaces of PG(d, q), with 0 ≤ i < j ≤ d− 1 and i+ j < d < ∞.
(iii) i-flats versus j-flats of the affine space AG(d, q) of dimension d over the finite field
with q elements, with 0 ≤ i < j ≤ d− 1 and i+ j < d < ∞.
Proof. We first give the proof of (ii). We need to prove that kerαi,j = 0, for 0 ≤ i < j ≤
d− 1 and i+ j < d. We use induction on j − i.
If j − i = 1 then kerαi,i+1 = 0, by Lemma 2.6 as i < (d − 1)/2. Now let j − i > 1
and assume kerαi′,j′ = 0 for any pair (i′, j′) with 0 ≤ i′ < j′ ≤ d − 1, i′ + j′ < d
and j′ − i′ < j − i. By Lemma 2.1, we have kerαi,j = kerαi,i+1αi+1,j . In addition
dimkerαi,i+1αi+1,j = dimkerαi,i+1 + dim (Imαi,i+1 ∩ kerαi+1,j).
Assume i + j < d − 1 so i < (d − 1)/2 and i + 1 + j < d. Then kerαi,i+1 = 0 by
Lemma 2.6, and kerαi+1,j = 0 by inductive hypothesis. Hence kerαi,j = 0 in this case.
Now assume i + j = d − 1. We will prove the result by calculating the dimension of
Imαi,d−i−1. By Lemma 2.1 and 2.2 we have
Imαi,d−i−1 = Viαi,d−i−1 =
i⊕
k=−1
Wkαk,d−i−1.
By the previous part, the map αk,d−i−1 is one-to-one for k = −1, . . . , i−1 as k+d−i−1 <
d − 1. Then dim Wkαk,d−i−1 = dim Wk, with Wk = kerαk,k−1. By the arguments in
the proof of Lemma 2.2 we get dim Wk = dim Vk − dim Imαk−1,k. By Lemma 2.6
T. Penttila and A. Siciliano: On the incidence map of incidence structures 59
the map αk−1,k is one-to-one for k = −1, . . . , i − 1 as k − 1 < (d − 1)/2. Therefore
dim Imαk−1,k = dimVk−1. This implies
dim Wkαk,d−i−1 = dim Wk
= dim Vk − dimVk−1
=
[
d+ 1
k + 1
]
q
−
[
d+ 1
k
]
q
,
for k = −1, . . . , i− 1. Therefore
dim Imαi,d−i−1 = dimViαi,d−i−1
= 1 +
i−1∑
k=0
([
d+ 1
k + 1
]
q
−
[
d+ 1
k
]
q
)
+ dimWiαi,d−i−1
=
[
d+ 1
i
]
q
+ dimWiαi,d−i−1.
Still by the proof of Lemma 2.2, we may write Vi = Imαi−1,i ⊕Wi, where αi−1,i is
one-to-one as i < (d− 1)/2. Hence,
dimWi = dimVi − dimVi−1 =
[
d+ 1
i+ 1
]
q
−
[
d+ 1
i
]
q
.
This implies
dimWiαi,d−i−1 =
[
d+ 1
i+ 1
]
q
−
[
d+ 1
i
]
q
− ε,
for some ε ≥ 0. Thus
dim Imαi,d−i−1 = dimViαi,d−i−1
=
[
d+ 1
i
]
q
+ dimWiαi,d−i−1
=
[
d+ 1
i
]
q
+
([
d+ 1
i+ 1
]
q
−
[
d+ 1
i
]
q
− ε
)
=
[
d+ 1
i+ 1
]
q
− ε.
As dimVi =
[
d+1
i+1
]
q
, then dimkerαi,d−i−1 = ε. At this point to finish the proof we
need to evaluate dimWiαi,d−i−1. We have Imαi,d−i−1 ≤ Vd−1−1, and dimVd−i−1 =
dimVi by duality. Note that Wiαi,d−i−1 is a component of Vd−i−1 by Lemma 2.1. Then
dimVd−i−1 − dimWiαi,d−i−1 = dimVi − dimWiαi,i =
[
d+ 1
i
]
q
.
Hence ε = 0 and this concludes the proof of (ii).
Similar arguments can be used to prove (i). We just need to replace the projective
dimension with size of set minus one and the q-binomial coefficients with binomial coeffi-
cients.
60 Ars Math. Contemp. 20 (2021) 51–68
We now prove (iii). Let αAi,j denote the incidence map of the i-flats versus the j-flats
of AG(d, q). Embed AG(d, q) in PG(d, q) identifying every k-flat of AG(d, q) with the
k-dimensional spaces of PG(d, q) it spans. Let H denote the hyperplane at infinity of
AG(d, q). Let f ∈ kerαAi,j and g be the extension of f on Vi defined as follows:
g(Si) =
{
f(Si) if Si ̸⊆ H;
0 if Si ⊆ H.
Then
(gαi,j)(Sj) =
∑
Si⊆Sj
g(Si) =
{
(fαAi,j)(Sj) if Sj ̸⊆ H;
0 if Sj ⊆ H.
Since f ∈ kerαAi,j , we get g ∈ kerαi,j . By (ii) g = 0 and hence f = 0.
Remark 2.8. For 2i + 1 ≤ d, the summands Wjαj,i in the decomposition of Vi given
in Lemma 2.2, are all the irreducible constituents of the permutation representation of
PGL(d, q) on Fi. To see this, set G = PGL(d, q). From the proof of Lemma 2.1 we
have Vi = Imαi−1,i ⊕ Wi. The map αi−1,i is one-to-one, so the number of irreducible
components in its image is precisely the number of the irreducible components of the per-
mutation QG-module Vi−1. This number is i + 1, being the dimension of the intersection
of two (i− 1)-subspaces a complete invariant. This shows that the modules in question are
pairwise non-isomorphic, and irreducible. This was proved by Steinberg [22] using deeper
representation theory.
An analogous result holds for the permutation QG-module defined by the symmetric
group Sym(n) acting on the m-sets, with 2m ≤ n. Here the size of set minus one replaces
the projective dimension, and binomial coefficients replace q-binomial coefficients.
Remark 2.9. For 2i+1 ≤ d, the summand Wjαj,i, for j = 0, . . . , i, in the decomposition
of Vi given in Lemma 2.2, is the restriction over the rationals of the (j + 1)-th eigenspace
of the Bose-Mesner algebra of the association scheme on Fi [13, Theorem 2.7]. For a
thorough treatment on association schemes we refer the reader to [1, 4].
3 Some applications of Theorem 2.7
The incidence structure J = (Q, C, J) is said to be a substructure of I = (P,B, I) if
Q ⊆ P , C ⊆ B and J = I ∩ (Q × C). The substructure J of I is said to be full if
{P ∈ P : P IC} ⊆ Q, for all C ∈ C.
Lemma 3.1. Let I = (P,B, I) be an incidence structure with finite block sizes. Suppose
that there is a set F of full substructures of I, all of whose incidence maps are one-to-one,
and such that, for any P ∈ P there exists J ∈ F such that P is a point of J . Then the
incidence map of I is one-to-one.
Proof. Let αI be the incidence map of I and f ∈ kerαI . For any given P ∈ P , let
J = (Q, C, J) ∈ F such that P ∈ Q. Let αJ be the incidence map of J . Set g = f |Q.
Since J is full we have
(gαJ )(C) =
∑
Q∈Q
Q JC
g(Q) =
∑
R∈P
R IC
f(R) = (fαI)(C),
T. Penttila and A. Siciliano: On the incidence map of incidence structures 61
for all C ∈ C. Since f ∈ ker (αI) we have (gαJ )(C) = 0, for all C ∈ C, that is
g ∈ kerαJ . Thus g = 0, and therefore f(P ) = g(P ) = 0. Since P is arbitrary, it follows
that f = 0.
The above Lemma allows to get the infinitary version of Theorem 2.7; this means that
the incidence structures involved are over a set with infinite size (in case (i)), or a space
with infinite dimension (in case (ii) and (iii)).
Theorem 3.2. The incidence map of the following structures is one-to-one:
(i) i-sets versus j-sets of an infinite set, with i < j < ∞.
(ii) i-spaces versus j-spaces of a projective space of infinite dimension over a finite field,
with i < j < ∞.
(iii) i-flats versus j-flats of an affine space of infinite dimension over a finite field, with
i < j < ∞.
Proof. We apply Lemma 3.1 and Theorem 2.7 to the above structures by taking the set F
of full substructures as follows: all subsets of size i + j for statement (i), all subspaces of
dimension i+j+1 for statement (ii), all flats of dimension i+j+1 for statement (iii).
Theorem 3.3. Let A be a classical polar space of (possible infinite) rank m over a finite
field. Then the incidence map of totally isotropic subspaces (or totally singular in case of
a orthogonal space) of A of algebraic dimension k versus singular subspaces of algebraic
dimension l is one-to-one, if k < l < ∞ and k + l ≤ m.
Proof. Let I = (P,B, I) be the incidence structure defined by the subspaces of algebraic
dimension k versus subspaces of algebraic dimension l of A. Let F be the family of
all the subspaces of A of algebraic dimension k + l. Since every element J of F is a
full substructure of I, we may apply Theorem 2.7 (ii), or Theorem 3.2 (ii) for the infinitary
version, with i = k−1, j = l−1 and d = k+l−1. Thus we get that the incidence map αJ
of J is one-to-one. The result then follows by applying Lemma 3.1 to the family F .
Remark 3.4. For the case of finite rank the above theorem is due to Lehrer [16, The-
orem 5.3]. Note that Lehrer mistakenly asserts that the incidence map of the incidence
structure of singular 1-spaces versus singular (n− 1)-spaces of the O+(2n, q) polar space
is not one-to-one. This error is caused by confusing the O+(2n, q) polar space with the
Dn(q) building.
In the following we apply Lemma 3.1 to the incidence structures known as diagram
geometries. For a thorough treatment on diagram geometries we refer the reader to [7, 8];
our notation is taken from [7].
Let Γ = (S, I,∆, τ) be a diagram geometry of finite rank with diagram ∆, and I =
(P,B, I) be the incidence structure where P is the set of all i-varieties and B the set of all
j-varieties of S; I is the restriction of I on P × B. Assume that blocks in I have finite size
and let k ∈ ∆ \ {j} such that i and k lie in distinct components of the diagram ∆ − {j}.
We now show that the set of k-varieties of S gives rise to a family F of full substructures
of I with the property that for any point (i-variety) P of I there exists J ∈ F such that P
is a point of J .
62 Ars Math. Contemp. 20 (2021) 51–68
For any given k-variety Λ of S, set JΛ = (PΛ,BΛ, IΛ) where PΛ and BΛ are the set of
all i-varieties and j-varieties of S incident to Λ in Γ, respectively; IΛ is the restriction of I
on PΛ × BΛ.
Let B be a j-variety in BΛ and let ΓB be the residue of B in Γ, that is the diagram
geometry (S′, I′,∆′, τ ′) where S′ is the set of all varieties of S of type m ∈ ∆\{j} which
are incident with B, the incidence relation I′ is the restriction of I to S′, ∆′ = τ(S′) and τ ′
is the restriction of τ to S′. It is known that the diagram of ΓB is ∆− {j} [7, Theorem 1].
Therefore the i-varieties of S′ are precisely all elements (i-varieties) of PΛ that are incident
with B in JΛ. In addition, as Λ is incident with B, it is a k-variety of S′. Since i and k
lie in distinct components of ∆ − {j}, by [7, Theorem 2] every i-variety of S′ is incident
with every k-variety, in particular every i-variety of S′ is incident with Λ. This implies that
{P ∈ P : P I B} is a subset of PΛ. From the arbitrariness of B in BΛ it follows that JΛ is
a full substructures of I.
Let F be the family of the substructures JΛ, for all k-varieties Λ of S. Since the type
map τ take all values of ∆ on every maximal flag of Γ then for every i-variety P of S
there exists a k-variety Λ such that P is a point of JΛ. These considerations together with
Lemma 3.1 led to the following result.
Theorem 3.5. Let Γ = (S, I,∆, τ) be the diagram geometry underlying the buildings of
types F4, E6, E7 and E8. Then the incidence map of i-varieties versus j-varieties of Γ is
one-to-one in the following cases:
(i) F4:
1 2 3 4
(i, j) = (1, 2), (4, 3).
(ii) E6:
1 2 3
4
5 6
(i, j) = (1, 2), (1, 3), (2, 3), (6, 5), (6, 3), (5, 3).
(iii) E7:
1 2 3
4
5 6 7
(i, j) = (1, 2), (1, 3), (2, 3), (7, 6), (7, 5), (7, 3), (6, 5), (6, 3), (5, 3).
(iv) E8:
1 2 3
4
5 6 7 8
(i, j) = (1, 2), (1, 3), (2, 3), (8, 7), (8, 6), (8, 5), (8, 3), (7, 6), (7, 5), (7, 3), (6, 5), (6, 3).
Proof. Consider the diagram Γ = (S, I,∆, τ) for F4, and take (i, j) = (1, 2), k = 3. Let
F be the family of full substructures arising from the 3-varieties of S constructed as above.
The points and blocks of any JΛ ∈ F are precisely the 1- and 2-varieties of S incident
T. Penttila and A. Siciliano: On the incidence map of incidence structures 63
with Λ. By [7, Theorem 1], these are precisely the 1- and 2-varieties of the residue R(Λ)
of Λ in Γ, whose diagram is
1 2 4
Note that every 1- and 2-variety is incident with every 4-variety. This implies that the
set of the 1- and 2-varieties of S incident with Λ form a finite projective plane, whose
incidence map is injective by a result of Bruck and Ryser [5] and Bose [3]. Lemma 3.1
yields that the incidence map of 1-varieties versus 2-varieties of S is one-to-one in this
case. Very similar argument can used with (i, j) = (4, 3) and k = 2.
Now consider the diagram Γ = (S, I,∆, τ) for E6, and take (i, j) = (1, 2), k = 4.
As above the points and blocks of any JΛ ∈ F are precisely the 1- and 2-varieties of S
incident with Λ, and these are precisely the 1- and 2-varieties of the residue R(Λ) of Λ in
Γ, whose diagram is
1 2 3 5 6
This implies that R(Λ) has the geometry of a PG(5, q). We now apply Theorem 2.7
to conclude that the incidence map of JΛ is incidence, and Lemma 3.1 yields that the
incidence map of 1-varieties versus 2-varieties of S is one-to-one in this case. Very similar
arguments apply for the remaining cases, and for the buildings E7, E8.
4 An extension of Block’s Lemma
An automorphism of the incidence structure I = (P,B, I) is a mapping g of P ∪ B such
that g defines permutations on P and B such that P IB if and only if P gIBg . The group of
all automorphisms of I is denoted by Aut I.
A decomposition of an incidence structure I = (P,B, I) is a pair (X ,Y), with X a
partition of P and Y a partition of B. A decomposition (X ,Y) of an incidence structure
with finite block sizes is block-tactical if
|{P ∈ X : P IB1}| = |{P ∈ X : P IB2}|,
for all X ∈ X , Y ∈ Y , B1, B2 ∈ Y . An example of block tactical decomposition of an
incidence structure I is obtained by taking the orbits on points and blocks of a subgroup of
Aut I.
With a decomposition (X ,Y) of I = (P,B, I) we associate the following subspaces of
the point space QP and the block space QB of I: the point class space VX of all functions
on P constant on each X ∈ X , and the block class space VY of all functions on B constant
on each Y ∈ Y .
Lemma 4.1. A decomposition (X ,Y) of an incidence structure I = (P,B, I) with finite
block sizes and incidence map α is block-tactical if and only if VXα ⊆ VY .
Proof. Suppose (X ,Y) is block-tactical and f ∈ VX . For each X ∈ X , let PX be a fixed
chosen point in X . As f is constant on X , then f(P ) = f(PX) for all P ∈ X . Let Y ∈ Y
64 Ars Math. Contemp. 20 (2021) 51–68
and B1, B2 ∈ Y . Then |{Q ∈ X : QIB1}| = |{Q ∈ X : QIB2}| and therefore
(fα)(B1) =
∑
P IB1
f(P ) =
∑
X∈X
∑
P∈X
P IB1
f(P )
=
∑
X∈X
|{Q ∈ X : QIB1}|f(PX)
=
∑
X∈X
|{Q ∈ X : QIB2}|f(PX)
=
∑
X∈X
∑
P∈X
P IB2
f(P ) =
∑
P IB2
f(P ) = (fα)(B2).
Hence fα is constant on Y . So fα ∈ VY , giving VXα ⊆ VY .
Conversely, suppose that VXα ⊆ VY . Let X ∈ X and χX ∈ QP denote the charac-
teristic function of X . Then, χX can be naturally considered as an element of VX , thus
χXα ∈ VY by hypothesis. Therefore, we have
|{P ∈ X : P IB1}| = (χXα)(B1) = (χXα)(B2) = |{P ∈ X : P IB2}|,
for each Y ∈ Y and B1, B2 ∈ Y . Hence (X ,Y) is block-tactical.
The following result is a slight extension of a fundamental result of R. E. Block [2,
Theorem 2.1] often known as “Block’s Lemma”.
Lemma 4.2. Let I = (P,B, I) be an incidence structure with finite block sizes and (X ,Y)
a block-tactical decomposition of I. Let α denote the incidence map of I. Then
dimVX ≤ dimVY + dim (kerα).
Proof. By Lemma 4.1, we have VXα ⊆ VY , so dim (VXα) ≤ dimVY . Now dimVX =
dim (VXα) + dim (VX ∩ kerα) ≤ dimVY + dim (kerα).
Theorem 4.3. Let G be one of the following groups:
(i) a permutation group of finite degree d;
(ii) a group of collineations of PG(d, q), d < ∞;
(iii) a group of affine collineations of AG(d, q), d < ∞;
(iv) a group of semi-linear isometries of a classical polar space of finite rank d over a
finite field.
For any given non-negative integer i < d, let ni denote the number of orbits on i-sets for
(i), on subspaces of dimension i for (ii), on flats of dimension i for (iii), on totally isotropic
subspaces (or totally singular in case of a orthogonal space) of dimension i for (iv). Then
ni ≤ nj , for i < j and i+ j < d.
T. Penttila and A. Siciliano: On the incidence map of incidence structures 65
Proof. Let Xi be the set of the orbits of G on the corresponding family of objects indexed
by i. For any i < j < d, put (X ,Y) = (Xi,Xj). The set of all characteristic functions χX ,
X ∈ X , is a basis for VX . Hence, dimVX = |X | = ni. Similarly, dimVY = |Y| = nj ,
and Lemma 4.2 gives |X | ≤ |Y| + dim (kerα) since the point- and block-orbits of any
subgroup of the full automorphism group of an incidence structure form a block-tactical
decomposition. The result is obtained by applying Theorems 2.7 and 3.3.
The following is the infinite version of the previous result.
Theorem 4.4. Let G be one of the following groups:
(i) a permutation group of infinite degree;
(ii) a group of collineations of a projective space of infinite dimension over a finite field;
(iii) a group of affine collineations of an affine space of infinite dimension over a finite
field;
(iv) a group of semi-linear isometries of a classical polar space of infinite rank over a
finite field.
For any given non-negative integer i, let ni denote the number of orbits on i-sets for (i),
on subspaces of dimension i for (ii), on flats of dimension i for (iii), on totally isotropic
subspaces (or totally singular in case of a orthogonal space) of dimension i for (iv). Let l
be the least index such that nl is infinite. Then n0 ≤ n1 ≤ · · · ≤ nl−1 and nk is infinite for
all k ≥ l.
Proof. Let Xi be the set of the orbits of G on the corresponding family of objects indexed
by i.
Let i < j ≤ l − 1. We apply very similar arguments as in the proof of Theorem 4.3
to the block-tactical decomposition (X ,Y) = (Xi,Xj). Then Theorems 3.2 and 3.3 give
ni ≤ nj .
Let l ≤ i < j < ∞. Since the incidence map of the incidence structure associated with
(Xi,Xj) has trivial kernel by Theorems 3.2 and 3.3, we may apply Proposition 2.1 in [9]
(where ρ is the incidence relation).
Remark 4.5. Theorem 4.4 (i) is due to Cameron [9, Theorem 2.2].
Remark 4.6. By using the Generalized Continuum Hypothesis it is possible to give a slight
improvement of the previous result when ni and nj , i < j, are infinite.
From Lemma 4.2 we get dimVXi ≤ dimVXj + dim (kerα), and it is known that
dimV = |V | when V is an infinite dimensional vector space over an infinite field F such
that |V | > |F |.
Set ni = ℵβi , βi ≥ 0. Thus, |VXi | = |QXi | = ℵ
ℵβi
0 = ℵβi+1 = 2ℵβi > ℵ0 = |Q|
by the Generalized Continuum Hypothesis. Therefore, dimVXi = 2
ℵβi , and similarly,
dimVXj = 2
ℵβj . Hence Lemma 4.2 yields
2ℵβi ≤ 2ℵβj + dim (kerα).
Theorems 3.2 and 3.3 yield 2ℵβi ≤ 2ℵβj , and the Generalized Continuum Hypothesis
implies ℵβi ≤ ℵβj , that is ni ≤ nj .
66 Ars Math. Contemp. 20 (2021) 51–68
Remark 4.7. In the paper [18], examples of infinite Desarguesian projective planes with
collineation groups having three orbits on points and two on lines are provided, solving a
problem posed by Cameron [10] and attributed to Kantor.
5 Incidence structures and permutation representations
Block’s Lemma leads to consideration of kerα. It is particularly nice when kerα is trivial,
and the following lemma also emphasizes this case.
Lemma 5.1. Let I = (P,B, I) be a finite incidence structure whose incidence map is one-
to-one. For any given automorphism group G of I the permutation representation of G
on P is a subrepresentation of the permutation representation of G on B (considered as
linear representation over a field of characteristic zero).
Proof. The point space QP is the permutation Q-module for G on P , and the block space
QB is the permutation Q-module for G on B. Since G preserves the incidence, we have
(fgα)(B) =
∑
P IB
fg(P ) =
∑
P IB
f(P g
−1
) =
∑
P IBg−1
f(P ) = (fα)(Bg
−1
) = (fα)g(B),
for all f ∈ QP and g ∈ G. Therefore, α is a QG-homomorphism from QP to QB. As
α is one-to-one, the permutation representation of G on P is a subrepresentation of the
permutation representation of G on B (over Q). For other fields of characteristic zero, we
need only tensor up.
Lemma 5.2. Let G be a group acting as a transitive permutation group on a finite set X
of size n. Let S be a subset of G such that
∑
s∈S s is mapped to the 0-matrix under every
irreducible non-principal representation. Then |X| divides |S|.
Proof. Let {x1, . . . , xn} be the natural basis of the permutation QG-module on X . The
matrix representation with respect this basis of any element s ∈ G on the trivial module is
1/|X|J , where J is the all-one n × n matrix. This implies that the matrix representation
of the endomorphism
∑
s∈S s on the trivial module is |S|/|X|J .
On the other hand, the matrix representation of
∑
s∈S s in the basis {x1, . . . , xn} is
PS =
∑
s∈S P (s), where P (s) is the permutation matrix representing s ∈ G. Note that
the entries in PS are positive integers. Since
∑
s∈S s is mapped to the 0-matrix under
every irreducible non-principal representation, we have PS = |S|/|X|J . The result then
follows.
Theorem 5.3 ([19]). Let I = (P,B, I) be a finite incidence structure with incidence map
one-to-one. If the automorphism group of I contains a subset which is sharply transitive
on blocks, then |P| divides |B|.
Proof. Set G = Aut I and S ⊂ G be sharply transitive on blocks. Hence, |S| = |B|.
By [19, Lemma 1], the endomorphism
∑
s∈S s of the permutation QG-module QB on
blocks is mapped to the 0-matrix under every irreducible non-principal representation. By
Lemma 5.1, every irreducible submodule of QP is a submodule of QB with less or equal
multiplicity. Hence,
∑
s∈S s acting on QP is mapped to the 0-matrix under every irre-
ducible non-principal representation in QP . By Lemma 5.2, we have |P| divides |B|.
T. Penttila and A. Siciliano: On the incidence map of incidence structures 67
Corollary 5.4. Let I = (P,B, I) be a finite incidence structure with incidence map one-
to-one and automorphism group G acting transitively on blocks. If |P| does not divide |B|,
then G does not contain a subset acting sharply transitive on blocks.
The above result can be restated as follows.
Corollary 5.5. Let I = (P,B, I) be a finite incidence structure with incidence incidence
map one-to-one and automorphism group Aut I acting transitively on blocks. Let H de-
note the one-block stabilizer in Aut I. If |P| does not divide |B|, then the permutation
representation of Aut I on the cosets of H contains no sharply transitive subset.
Remark 5.6. Corollary 5.4 applies to the following incidence structures as their incidence
map is one-to-one: combinatorial designs, linear spaces and circular spaces (see [6]); in-
cidence structures in projective and affine spaces (see [14] and Theorem 2.7); incidence
structures in classical polar spaces (see [16] and Theorem 3.3); incidence structures on
subsets ([9, 14, 15, 20] and Theorem 2.7); nonbipartite graphs (see [21]).
ORCID iDs
Alessandro Siciliano https://orcid.org/0000-0002-6042-3377
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 69–87
https://doi.org/10.26493/1855-3974.2226.e93
(Also available at http://amc-journal.eu)
On plane subgraphs of complete topological
drawings*
Alfredo Garcı́a Olaverri † , Javier Tejel Altarriba ‡
Departamento de Métodos Estadı́sticos and IUMA, University of Zaragoza, Spain
Alexander Pilz §
Institute of Software Technology, Graz University of Technology, Austria
Received 22 January 2020, accepted 15 October 2020, published online 17 August 2021
Abstract
Topological drawings are representations of graphs in the plane, where vertices are
represented by points, and edges by simple curves connecting the points. A drawing is
simple if two edges intersect at most in a single point, either at a common endpoint or at a
proper crossing. In this paper we study properties of maximal plane subgraphs of simple
drawings Dn of the complete graph Kn on n vertices. Our main structural result is that
maximal plane subgraphs are 2-connected and what we call essentially 3-edge-connected.
Besides, any maximal plane subgraph contains at least ⌈3n/2⌉ edges. We also address the
problem of obtaining a plane subgraph of Dn with the maximum number of edges, proving
that this problem is NP-complete. However, given a plane spanning connected subgraph of
Dn, a maximum plane augmentation of this subgraph can be found in O(n3) time. As a
side result, we also show that the problem of finding a largest compatible plane straight-line
graph of two labeled point sets is NP-complete.
Keywords: Graph, topological drawing, plane subgraph, NP-complete problem.
Math. Subj. Class. (2020): 05C10, 68R10
*This project has received funding from the European Union’s Horizon 2020 research and innovation pro-
gramme under the Marie Skłodowska-Curie grant agreement No. 734922.
†Supported by MINECO project MTM2015-63791-R and Gobierno de Aragón Grant E41-17 (FEDER).
‡Supported by MINECO project MTM2015-63791-R, Gobierno de Aragón Grant E41-17 and project
PID2019-104129GB-I00 / AEI / 10.13039/501100011033 of the Spanish Ministry of Science and Innovation.
§Supported by a Schrödinger fellowship of the Austrian Science Fund (FWF): J-3847-N35.
E-mail addresses: olaverri@unizar.es (Alfredo Garcı́a Olaverri), jtejel@unizar.es (Javier Tejel Altarriba),
apilz@ist.tugraz.at (Alexander Pilz)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
70 Ars Math. Contemp. 20 (2021) 69–87
1 Introduction
In a topological drawing (in the plane or on the sphere) of a graph, vertices are represented
by points and edges by simple curves connecting the corresponding pairs of points.Usually,
we only consider drawings satisfying some natural non-degeneracy conditions, in particular
a drawing is called simple (or a good drawing) if two edges intersect at most in a single
point, either at a common endpoint or at a crossing in their relative interior. When all
the edges of a topological drawing are straight-line segments, then the drawing is called a
rectilinear drawing or geometric graph.
In this paper we consider only simple topological drawings of the complete graph Kn
on n vertices. Simple topological drawings of complete graphs have been studied exten-
sively, mainly in the context of crossing number problems. It is well known that a drawing
minimizing the number of crossings has to be simple, and besides, if n ≥ 8, the drawings
of Kn minimizing that crossing number are not rectilinear. We refer the reader to [1, 3, 4]
for recent advances on the Harary-Hill conjecture on the minimum number of crossings of
drawings of Kn, and to the survey [22] for some variants on this crossing number problem.
The problem of enumerating all the non-isomorphic drawings of Kn has been studied
in [2, 12, 13, 18] (two drawings are isomorphic if there is a homeomorphism of the sphere
that transforms one drawing into the other).
Let Dn be a simple topological drawing of Kn. Herein, we consider graphs in con-
nection with their drawings, and in particular when addressing subgraphs of Kn we also
consider the associated sub-drawing of Dn. We are interested in crossing-free edge sets F
in Dn, and we will say that F is a plane subgraph of Dn. Crossing-free edge sets in Dn
have attracted considerable attention, in part because problems on embedding graphs on a
set of points usually generalize to finding plane subgraphs of Dn. For instance, the problem
of computing the maximum number of plane Hamiltonian cycles that a simple drawings Dn
can contain, is a generalization of the same problem considering only rectilinear drawings
of Kn. And this last is the (open) problem of computing the maximum number of simple
n-gons that can be formed on n points in the plane.
There are relatively few results on plane subgraphs of Dn. It is well known that in any
drawing Dn of Kn, there are plane subgraphs with 2n− 3 edges, and that there are at most
2n−2 edges uncrossed by any other edge [6, 8, 19]. Pach, Solymosi, and Tóth [14] showed
that any Dn has Ω
(
log1/6(n)
)
pairwise disjoint edges. This bound was subsequently
improved in [5, 16, 23]. The current best bound of Ω(n1/2−ϵ) is by Ruiz-Vargas [20].
However, the much stronger conjecture that any simple drawing Dn of Kn contains a plane
Hamiltonian cycle remains unproved, although it has been verified for n ≤ 9, see [2].
In the course of their work on disjoint edges and empty triangles in Dn, Fulek and
Ruiz-Vargas [6] showed the following lemma.1
Lemma 1.1 (Fulek and Ruiz-Vargas [6]). Between any plane connected subgraph F of Dn
and a vertex v not in F , there exist at least two edges from v to F that do not cross F .
This result can be used to build large plane subgraphs. For instance, we can begin with
F consisting of only one edge, then for each vertex v not in F , we add to F the edges
from v to F not crossing F . In this way, we will obtain a maximal plane subgraph: a plane
subgraph F such that any edge e /∈ F crosses some edge of F .
1Their lemma is actually more general. It does not require F and v to be elements of a drawing of Kn, but
rather of a drawing that contains all edges from v to vertices of F .
A. Garcı́a Olaverri et al.: On plane subgraphs of complete topological drawings 71
In Section 2 of this work, we extend that Lemma 1.1 to arbitrary (not necessarily con-
nected) plane subgraphs. Further, in Section 3, we prove that any plane subgraph of Dn
can be augmented to a 2-connected plane subgraph of Dn. A consequence of this result is
that maximal plane subgraphs contain at least min(⌈3n/2⌉, 2n− 3) edges, and this bound
is tight. Maximal plane subgraphs of Dn have other interesting properties. For example,
we show that, when removing two edges from a maximal plane subgraph, it either stays
connected or one of the two components is a single vertex. Another consequence of the
previous results is that for every vertex v of a drawing Dn, there is a plane subgraph of Dn
consisting of the n-vertex star of edges incident to v, plus the edges of a spanning tree on
the n− 1 vertices of V \ {v}.
The problem setting changes when we want our plane graphs not only to be maximal,
but also to contain the maximum number of edges. While for geometric graphs, every max-
imal plane subgraph is a triangulation and thus also has a maximum number of edges, the
situation is different for plane subgraphs of Dn. In Section 4, we will prove that comput-
ing a plane subgraph of Dn with maximum number of edges is an NP-complete problem.
However, if a connected plane spanning subgraph F is given, we can adapt a classic algo-
rithm from computational geometry to show that a maximum plane augmentation of F can
be found in O(n3) time.
As a side result, we also show that the problem of finding a largest compatible plane
graph on two labeled point sets is NP-complete.
Finally, going back to Lemma 1.1, we give an O(n) algorithm to compute all the edges
from a vertex v to a plane connected subgraph F that do not cross F .
2 Adding a single vertex
We now discuss a generalization of Lemma 1.1 to arbitrary plane subgraphs. This general-
ization will also follow independently from Theorem 3.1. Still, the following proposition
gives further insight on the position of the uncrossed edges around the vertex v, which
might help in the construction of algorithms.
We assume that a simple topological drawing Dn of Kn in the plane is given, with
vertex set V = {v1, . . . , vn}. If x1, x2 are two points on an edge e of Dn (not necessar-
ily the endpoints of e), by line x1x2 we mean the portion of the curve e of the drawing
placed between the points x1 and x2. For a vertex v, the star graph with center v is the
subgraph formed by the edges connecting v to all the other vertices. We denote this set of
edges by S(v), usually call rays to these edges emanating from v, and we suppose that the
rays of S(v) are (circularly) clockwise ordered. By the clockwise range [vp, vq] of S(v)
we mean the ordered set of rays placed clockwise between vp and vq, including rays vp
and vq. When vp or vq or both are not included in that ordered set of rays, we will use
(vp, vq], [vp, vq) or (vp, vq), respectively. In the same way, we can define counterclock-
wise ranges.
In the rest of this section, we suppose F is a given plane subgraph of Dn and v a vertex
not in F . In the figures, we use red color for the edges of F , so we usually call them red
edges. We say a ray vr of S(v) is uncrossed if it does not cross any edge of F .
Suppose that the ray vr crosses some edge of F , let e = pq be the first edge of F
crossed by vr, and let x be the first crossing point. Without loss of generality, we can
suppose that the rays vr, vp, and vq appear in this clockwise order in S(v). See Figure 1.
We define the clockwise range Rcw of rays centered at v corresponding to the crossing
72 Ars Math. Contemp. 20 (2021) 69–87
v
pq
r
qp
r
RcwRccw
x x
Rccw
Rcw
yy′
ll′
l′l
y′y
v
Tcw
Tcw
Figure 1: The clockwise and counterclockwise ranges of a first crossing.
x in the following way: if no ray in the clockwise range (vp, vq) crosses the edge pq
between x and p, then Rcw is the range (vr, vp]; otherwise, (some rays in the clockwise
range (vp, vq) cross the line xp), Rcw is the clockwise range (vr, vl], where vl is the last
ray in (vp, vq) crossing the line xp. That implies that if vl crosses xp at the point y, among
the intersection points of rays in (vp, vq) with the line xp, the closest to x is y. See Figure 1.
Analogously, the range Rccw is defined either as the counterclockwise range (vr, vq] if no
edge in the counterclockwise range (vq, vp) crosses the line xq, or as the counterclockwise
range (vr, vl′], where vl′ is the ray in the counterclockwise range (vq, vp) crossing the line
xq in a point y′ closest to x. By definition, the rays vr, vp, vl, vl′, vq appear clockwise in
that order around v. Observe that Rcw and Rccw are disjoint sets and they are also nonempty,
as vp is in Rcw and vq is in Rccw. The following result generalizes Lemma 1.1.2
Proposition 2.1. Suppose the ray vr first crosses the edge e of F at the point x. Let Rcw
and Rccw be the clockwise and counterclockwise ranges of rays of v corresponding to that
crossing. Then, each one of these two ranges contains an uncrossed ray. As a consequence,
S(v) contains at least two uncrossed rays.
Proof. We prove the statement for Rcw, the proof for Rccw is identical.
Observe that by definition, no red edge can cross the line vx, and a ray in the clockwise
range (vq, vr] cannot cross the line xp. Let y be the crossing point between the red edge
e = pq and the ray vl. When Rcw is (vr, vp] (i.e., no ray in the clockwise range (vp, vq)
crosses xp), then we identify the points p, l and y. The lines vx, xy and yv define a simple
closed curve, that divides the plane into two regions Tcw, Tcw, where Tcw is the region not
containing the point q.
From the definition of Tcw, it follows that a ray containing a point placed in the interior
of Tcw must be in the range Rcw. Besides, a red edge can cross the boundary of that region
only through the line yv, and hence, if a red edge e crosses yv, one endpoint of e must be
inside Tcw the other one in Tcw.
The proof is done by induction on |Rcw|, the number of rays in that range. So, first sup-
pose that the only ray in the range Rcw is the ray vp. In this case, Tcw is the region bounded
by the closed curve vx, xp, pv not containing the point q. This region cannot contain any
2Like Lemma 1.1, this result is more general. It does not require F and v to be elements of a drawing of Kn,
but rather of a drawing that contains all edges from v to vertices of F .
A. Garcı́a Olaverri et al.: On plane subgraphs of complete topological drawings 73
vertex r′ of F , because then vr′ would be in Rcw, therefore vp must be uncrossed. This
proves the base case of the induction.
Now suppose that the proposition has been proved for any clockwise range containing
less than |Rcw| rays. Let vr′ be the first ray of Rcw. Of course, if the ray vr′ is uncrossed,
the proof is done, so we can suppose that the ray vr′ is first crossed by a red edge e′ at a
point x′. We are going to prove that the clockwise range R′cw corresponding to the crossing
x′ is strictly contained in Rcw. Then, by induction, R′cw contains an uncrossed ray, and thus
also Rcw. To prove that R′cw ⊂ Rcw strictly, it is enough to prove that the clockwise last ray
of R′cw is contained in Rcw.
Let us first analyze Case A: when the edge e′ is precisely the edge e. See Figure 2.
In this case, if x′ is between x and y, then the clockwise range R′cw corresponding to the
v
pq
r
x
v
pq
r
x
r′
r′y
l
x′ y x′
Figure 2: Case A, the ray vr′ first crosses the edge e = pq.
new crossing point x′ is precisely Rcw minus its first ray vr′. And if x′ is between y and p,
then all the points of the line x′p (including point p) are in the interior of the region Tcw,
therefore the corresponding last ray of R′cw has to be in Rcw. Thus, in both subcases is
R′cw ⊂ Rcw strictly.
Suppose now Case B: when e′ ̸= e. See Figure 3. Clearly, at least one endpoint of e′ is
v
pq
r
x
Subcase B1
v
pq
r
x
Subcase B2
r′
x′
x′
p′
q′
Figure 3: Case B, the ray vr′ first crosses an edge e′ ̸= e.
in Tcw, as otherwise the ray vl would be crossed twice by e′. Hence, either both endpoints
are in Tcw, subcase B1, or one of them is in Tcw and the other one in Tcw, subcase B2. In
subcase B1, the entire edge e′ = p′q′ must be inside Tcw. Therefore, any ray containing a
74 Ars Math. Contemp. 20 (2021) 69–87
point of e′ must be in Rcw. In particular, the last ray of R′cw must be in Rcw, and hence,
R′cw is strictly contained in Rcw.
In subcase B2, an endpoint, p′, of e′ is inside Tcw and the other, q′, is in Tcw. Observe
that the ray vp′ must be in Rcw, however the ray vq′ cannot be in Rcw because the range
(vr, vr′) is empty, and a ray in [vr′, vl] finishing at q′ has to cross the edge e′. See Figure 3.
Therefore, the rays vr′, vp′, vq′ appear clockwise around v in this order. Hence, the last
ray of R′cw is either vp
′ or a ray crossing the line x′p′. In any case, as the line x′p′ is inside
Tcw, this last ray of R′cw has to be in Rcw. This completes the proof.
3 Structure of maximal plane subgraphs
Let Dn be an arbitrary simple drawing of Kn. In this section, we identify several structural
properties of maximal plane subgraphs of Dn, using Lemma 1.1 or Proposition 2.1 as
our main tool. Maximal plane subgraphs turn out to be 2-connected. While there are
examples of maximal plane subgraphs that are not 3-connected, we elaborate further on
the structure, showing that a maximal plane subgraph is either 3-edge-connected or has a
vertex of degree 2.
Theorem 3.1. A maximal plane subgraph of Dn is spanning and 2-connected.
Proof. The proof is by induction on n. The result is obviously true for n ≤ 3. For n > 3,
assume there exists a maximal plane subgraph F that is not 2-connected, and let us see that
a contradiction is reached.
We first claim that, under this assumption, F has no vertices of degree less than 3.
Suppose the contrary, that the vertex v has degree ≤ 2. Let F ′ be the subgraph of F
obtained after removing the vertex v, and let F ′ be a maximal plane subgraph (in the
drawing Dn−{v} of Kn−1) containing F ′. By the induction hypothesis, F ′ is 2-connected.
We observe that v cannot have (in F ) degree less than 2, since applying Lemma 1.1 to v
and F ′ would give two edges at v not crossing F , contradicting the maximality of F .
So suppose v has degree 2. As we assume that F is not 2-connected, F ′ cannot be 2-
connected. However, F ′ is 2-connected, and hence there exists an edge e′ in F ′ − F ′. By
the maximality of F , e′ must cross at least one edge vw of F incident to v. But applying
Lemma 1.1 to v and F ′ gives at least two edges incident to v not crossing F ′. These two
edges and also vw do not cross F , contradicting the maximality of F . Therefore, the claim
follows.
Assume now that F is not connected. Let C1, C2 be two connected components of F .
As all vertices have (in F ) degree at least 3, C1 cannot be an outerplanar graph, and it has
more than one face. Without loss of generality, we can suppose that C2 is in the unbounded
face of C1. Let v1 be an interior vertex of C1, F ′ the graph obtained from F by removing
v1, and f1 the face of F ′ containing v1. The face containing C2 remains unchanged by the
removal of v1. By induction, F ′ can be completed to a 2-connected plane graph F ′, and
due to the maximality of F , all the edges in F ′ − F ′ should be in the face f1. But then, as
C2 is outside f1, F ′ could not be connected, a contradiction. Thus, F has to be connected.
By a similar reasoning we arrive at our contradiction to F not being 2-connected. A
block is a 2-connected component of a graph, and a leaf block is a block with only one cut
vertex. Since F is not 2-connected, it has at least two leaf blocks B1 and B2. As all vertices
have degree at least 3, B1 cannot have all its vertices on the same face. Again, without loss
of generality, we can suppose B2 is in the outer face of B1, and there is an interior vertex
A. Garcı́a Olaverri et al.: On plane subgraphs of complete topological drawings 75
v1 of B1. Removing v1 from F , we obtain a plane graph F ′ that has a face f1 containing
v1, and F ′ is contained in a maximal plane graph F ′ that is 2-connected. Again, by the
maximality of F , all the edges in F ′ − F ′ must be in f1, implying that B2 is still a block
of F ′, contradicting the fact that F ′ is 2-connected. Hence, F must be 2-connected.
Theorem 3.1 can be used to obtain more properties of maximal plane subgraphs.
Lemma 3.2. If a maximal plane subgraph F of Dn contains a vertex v of degree 2, then
the subgraph of F obtained after removing v is also maximal in Dn − {v}.
Proof. Suppose the contrary. Remove v from F to obtain F ′ and let F ′ be a maximal plane
graph containing F ′. As F is maximal but F ′ is not, F ′ − F ′ must contain an edge e′ that
crosses some edge vw of F . But by Lemma 1.1 there are at least two edges from v to F ′.
These two edges and also vw do not cross F , contradicting the maximality of F .
Proposition 3.3. Any maximal plane subgraph F of Dn with n ≥ 3 must contain at least
min(⌈3n/2⌉, 2n− 3) edges. This bound is tight.
Proof. Suppose that n > 3 and F0 = F has a vertex v0 with degree 2. By removing this
vertex we obtain another maximal plane graph F1 (maximal on n−1 points), and if F1 is in
the same conditions (with at least three vertices and a vertex v1 of degree 2), by removing
v1 we obtain a new maximal plane graph F2, and so on. We finish this process in a step k
because either Fk only has three points, or all the points of Fk have degree at least 3. In the
first case, the original graph F contains n = k + 3 vertices and 2k + 3 edges, so 2n − 3
edges. In the second case, F must contain at least 2k + ⌈3(n − k)/2⌉ edges, this amount
reaching its minimum value when k = 0.
Finally, let us see that the bound is tight. If 2 ≤ n ≤ 6, then a straight-line drawing
on n points in convex position gives the bound 2n − 3 ≤ ⌈3n/2⌉. If n > 6 and n is an
even number, a drawing like the one shown in Figure 4 proves that the bound ⌈3n/2⌉ is
tight. The drawing is done on n = 2(k + 1) points in convex position, that clockwise are
u0
u1 uk
v1 vk
vk+1
Figure 4: A drawing of Kn. The missing edges should be drawn as straight-line segments
inside the convex hull of the set of points. The black edges form a maximal plane subgraph
with ⌈3n/2⌉ edges.
76 Ars Math. Contemp. 20 (2021) 69–87
denoted by u0, u1, u2, . . . , uk, vk+1, vk, . . . , v2, v1. Let C denote the convex hull of that
set of points. All the edges of Dn are drawn straight-line except for the 2(k − 1) edges
uivi+1, viui+1, i = 1, . . . , k − 1, and the edge u0vk+1, that are drawn outside C as shown
in Figure 4. Observe that the 2(k − 1) edges uivi+1, viui+1, i = 1, . . . , k − 1, are the
diagonals of the (k−1) quadrilaterals uiui+1vi+1vi, with uivi+1 only crossing viui+1 and
u0vk+1, for i = 1, . . . , k − 1. Clearly, straight-line edges can cross at most once, and the
edges placed outside C, by construction, cross at most once. The graph F formed by the
2(k + 1) edges on the boundary of C, the k edges uivi, i = 1, . . . , k, and the edge u0vk+1
is clearly plane and maximal, since the other straight-line edges cross at least one edge
uivi, and the non-straight-line edges cross the edge u0vk+1.
If n is odd, we can add to the previous set a point u′0 between u0 and u1, very close
to segment u0u1, but keeping all the 2k + 3 points in convex position. By connecting u′0
with straight lines to the rest of the points, we obtain a simple topological drawing of Kn
on this set of n = 2k+ 3 points, and a new maximal plane graph is obtained by adding the
edges u0u′0,u
′
0u1 to the above graph F . This new maximal plane subgraph also has ⌈3n/2⌉
edges.
We mention another interesting implication of Theorem 3.1. For a vertex v, we can
augment the star S(v) to a 2-connected plane graph F , and since F \ {v} is connected, it
contains a spanning tree. So we have
Corollary 3.4. For each vertex v there exists a spanning tree Tv of V \ {v}, such that the
edges of S(v) ∪ Tv form a plane subgraph of Dn.
Our next results are about diagonals on plane cycles. Let C = (v1, v2, . . . , vk) be a
plane cycle of Dn. A diagonal of C is an edge of Dn connecting two non-consecutive
vertices of C. It was previously known that, even for the case where there are diagonals
intersecting both faces of C, there are at least ⌈k/3⌉ of them not crossing C (cf. [17,
Corollary 6.6]). Proposition 3.3, applied to the subdrawing induced by the vertices of C,
directly implies the following result.
Corollary 3.5. Let C = (v1, v2, . . . , vk) be a plane cycle of Dn, with k ≥ 6. Then, there
exists a set D of diagonals, with |D| ≥ ⌈k/2⌉, such that the subgraph C ∪D is plane.
It turns out that the structure of the diagonals of a cycle, as shown in the next lemma, is
useful for our further results.
Lemma 3.6. Let C = (v1, v2, . . . , vk) be a plane cycle of Dn, k ≥ 3, dividing the plane
into two faces f1 and f2. If there is no diagonal of C entirely in f1, then all the diagonals
of C are entirely in f2.
Proof. The proof is by induction on k. For k < 5 the statement is obvious, so suppose
k ≥ 5 and consider only the subdrawing Dk induced by the vertices of C. Suppose C ∪D
is a maximal plane graph of Dk, so necessarily, D consists of diagonals placed on f2.
Let d be a diagonal of D connecting two vertices at minimum distance on the graph C.
Lemma 3.2 implies that in a maximal plane subgraph, vertices with degree 2 cannot be
adjacent. Therefore, diagonal d has to connect two vertices at distance 2 on C. Without
loss of generality, suppose d = vkv2 and let ∆ be the triangle vkv1v2. Then, the cycle
C1 = (v2, v3, . . . , vk) with k − 1 vertices has the faces f ′1 = f1 +∆ and f ′2 = f2 −∆.
A. Garcı́a Olaverri et al.: On plane subgraphs of complete topological drawings 77
We claim that there cannot be diagonals of C1 entirely in f ′1. Such a diagonal e en-
tirely in f ′1 would have to intersect ∆. Then, adding e to C ∪ {vkv2} and removing all
edges crossed by e, we would obtain a plane graph F in which v1 has degree 0 or 1. By
Lemma 1.1, there must be another edge between v1 and C1, and this edge would be a di-
agonal of C entirely in f1, a contradiction. Thus, by induction, any diagonal vivj of C1 is
entirely in f ′2 and hence also in f2.
It remains to see that the diagonals with endpoint v1 are also in f2. By our induction
hypothesis, the diagonal v2v4 is in f ′2 and thus also in f2. Hence, arguing as before on the
cycle C3 = (v1, v2, v4, . . . , vk), we deduce that all the diagonals of C3 incident to v1 must
be in f2. So it remains to see that the diagonal v1v3 is also in f2. But v3v5 is also in f ′2, so
it is in f2, and again applying the same reasoning on the cycle (v1, v2, v3, v5, . . . , vk), all
the diagonals of this cycle not incident to v4 have to be in f2.
To prove the next result, we recall some definitions and properties of any 2-connected
graph G = (V,E). Two vertices v1, v2 are called a separation pair of G if the induced
subgraph G \ {v1, v2} on the vertices V \ {v1, v2} is not connected. Let G1, . . . , Gl be the
connected components of G \ {v1, v2}, with l ≥ 2. For each i ∈ {1, . . . , l}, let G∗i be the
subgraph of G induced by V (Gi) ∪ {v1, v2}. Observe that G∗i contains at least one edge
incident to v1 and at least another edge incident to v2.
Theorem 3.7. Let F be a maximal plane subgraph of Dn, n ≥ 3. Then, for each separation
pair v1, v2 of F , at least one of the subgraphs F
∗
i must be 2-connected.
Proof. Suppose that v1, v2 is a separation pair of F , and let F 1, F 2, . . . , F l be the con-
nected components of F \ {v1, v2}, l ≥ 2. Since F is 2-connected, the graph F \ {v2} is
connected with v1 as a cut vertex. As F is plane, we can suppose that v1 is in the outer face
of F \ {v2} (v2 must be inside that face) and that clockwise around vertex v1 first there
appear the edges from v1 to some vertices of the component F 1, then edges connecting v1
to some vertices of F 2 and so on. See Figure 5.
v1
v2
F ∗3 F
∗
2 F ∗1 u1u2
u3
R1R2R3
Figure 5: A plane graph with separating pair v1, v2 and three subgraphs F ∗i , none of them
2-connected. This plane graph cannot be maximal.
Now suppose that none of the subgraphs F
∗
i is 2-connected. Then each subgraph F
∗
i
contains at least one cut vertex ui. Since F i is connected and there exist edges in F
∗
i
incident to v1 and v2, vertex ui is different from v1 and v2. On the other hand, a connected
component C of F
∗
i \{ui} must contain at least one of v1 or v2 because otherwise, C would
be a connected component of F \ {ui}, contradicting that F is 2-connected. Therefore,
F
∗
i \ {ui} has exactly two components, one containing v1, the other one containing v2.
78 Ars Math. Contemp. 20 (2021) 69–87
This also implies that the edge v1v2 of Dn cannot belong to F , and that the cut-vertex ui
is in the outer face of F
∗
i (and hence in the outer face of F \ {v2}) since v1 and v2 are in
the outer face of F \ {v2}. See Figure 5.
In the graph F \ {v2}, around the vertex v1, the edges to vertices of F1 first appear,
then the edges to vertices of F2 and so on. Therefore, when we add to that graph the vertex
v2 and all the edges connecting v2 to each component Fi to obtain F , v1 and v2 must be
in the faces Ri of F defined as the regions placed between the last edge from v1 to F i and
the first edge from v1 to F i+1, for i = 1, . . . , l, and the vertex ui must be in the faces Ri
and Ri−1. However, by the maximality of F , no edge of Dn is entirely in any of those
faces Ri. Then, Lemma 3.6 implies that no point of the edge v1v2 of Dn can be inside any
face Ri. See Figure 5. Thus, v1v2 must begin between two edges v1v, v1v′ with both v
and v′ belonging to a common connected component F i. However, since ui belongs to the
faces Ri−1 and Ri, any curve from v1 to v2 passes either through the point ui or through
the interior of Ri−1 or Ri, which contradicts either the simplicity of Dn or Lemma 3.6.
Therefore, if none of the subgraphs F
∗
i is 2-connected, F cannot be maximal.
We call a graph essentially 3-edge-connected if it stays connected after removing any
two edges not sharing a vertex of degree 2 (i.e., the graph either stays connected or one
component is a single vertex). Theorem 3.7 implies that a maximal plane subgraph is
essentially 3-edge-connected:
Theorem 3.8. Any maximal plane subgraph, F , of a simple topological drawing of Kn is
essentially 3-edge-connected.
Proof. If the removal of two edges v1v2 and v′1v
′
2 from the plane subgraph F results in two
non-trivial components C1, C2 (see Figure 6), then v1, v′2 is a separation pair of F , that has
as induced subgraphs C1∪{v′1v′2} and C2∪{v1v2}, neither of which is 2-connected. Then,
by Theorem 3.7, F cannot be maximal.
v1
R1
R2
C1
C2
v2
v′1 v′2
Figure 6: A graph that is not essentially 3-edge-connected. The induced subgraphs of the
separation pair v1, v′2 are subgraph C1 plus edge v
′
1v
′
2 and subgraph C2 plus edge v1v2. By
Lemma 3.6, the edge v1v′2 of Dn cannot enter either the R1 face or the R2 face, which is
impossible in any good drawing.
4 Adding the maximum number of edges
Now, assume that a plane subgraph of Dn is given, and we want to add the maximum
number of edges keeping plane the augmented graph. Clearly, the decision of adding one
A. Garcı́a Olaverri et al.: On plane subgraphs of complete topological drawings 79
edge will in general block other edges from being added. We will see that the complexity
of an algorithm solving this problem highly depends on whether the given subgraph is
connected or not.
Before talking about algorithms and their complexity we have to talk about what infor-
mation of the drawing Dn we will need to compute plane subgraphs. For each vertex v, the
clockwise cyclic order of edges of S(v) is usually given as a permutation of V \ {v} (that
is to be interpreted circularly) of the second vertices of all edges of S(v). That permutation
of V \ {v} is called the rotation of v, and the rotation system of a drawing Dn consists of
the collection of the rotations of each vertex v of Dn. It is well-known that from the in-
formation provided by the rotation system, one can determine whether two edges cross or
not, and therefore, that information is enough to compute plane subgraphs. See [7, 11, 15].
From the rotation system, we can also compute (in O(n2) time) the inverse rotation system
that, for each vertex vi and index j, j ̸= i, gives the position of vj in the rotation of vi.
When we say that a drawing Dn is given, we mean that we know the rotation system
and the inverse rotation system of Dn. Using these two structures, one can determine
whether two edges cross, in which direction an edge is crossed, and in which order two
non-crossing edges cross a third one in constant time [11].
Theorem 4.1. Let F be a connected spanning plane subgraph of Dn. Then there is an
O(n3) time algorithm to augment F to a plane subgraph F ′ of Dn with the maximum
number of edges.
Proof. As F is plane and thus contains a linear number of edges, we can identify all the
edges of Dn not crossed by F in O(n3) time. This also gives us, for each such edge, the
face of F in which it is contained, and we can also compute for each face f of F the set ∆f
of edges of Dn entirely inside f . Clearly, each face of F can be considered independently,
adding the maximum number of edges in it.
Let f be a face of F . For simplicity, we assume f to be bounded by a simple cycle
(v1, . . . , vk). Other cases can be solved similarly by an appropriate “splitting” of edges
having f on both sides. Disregarding Dn, consider the rectilinear drawing Dk obtained
from k points p1, . . . , pk placed on a circle C, and assign to each edge pipj of Dk weight
0 if vivj is in ∆f , weight 1 otherwise. Observe that two edges of ∆f cross properly, if and
only if, the corresponding 0-weight edges in circle C cross properly. It is well-known that
a minimum-weight triangulation in Dk can be obtained in O(k3) time [9] by a dynamic
programming algorithm, and this triangulation gives a plane set of 0-weight edges with
maximum cardinality. Hence, the corresponding edges of ∆f form a plane set of edges
entirely inside face f with maximum cardinality.
In contrast to this result, the problem becomes NP-complete when the subgraph F is
not connected.
Theorem 4.2. Given a simple topological drawing Dn of Kn and a cardinality k′, it is
NP-complete to decide whether there is a plane subgraph that has at least k′ edges.
Proof. We give a reduction from the independent set problem on segment intersection
graphs (SEG problem), which is known to be NP-complete [10]: Given a set S of s seg-
ments in the plane that pairwise either are disjoint or intersect in a proper crossing, and an
integer k > 0, is there a subset of k disjoint segments?
80 Ars Math. Contemp. 20 (2021) 69–87
For each instance of a SEG problem, we are going to build, in polynomial time, a
drawing Dn of Kn and an integer k′ such that the instance of the SEG problem has a Yes
answer, if and only if, the drawing Dn contains a plane subgraph with k′ edges.
Let vi, ti be the endpoints of each segment si, i = 1, . . . , s, of S. We can suppose that
these endpoints are in general position and that their convex hull is a triangle. Thus, for
each endpoint vi, we can find a disc Bi centered at vi, such that any straight line connecting
two endpoints of S different from vi does not cross Bi.
In each disc Bi, we place two points ui, wi very close to the segment viti, in such a
way that when connecting the point vi with straight-line segments to all the other points, the
segments viui, viti, viwi are clockwise consecutive. In other words, the clockwise wedge
defined by the half-lines viui, viwi only contains the endpoint ti. See Figure 7.
ui
vi wi
ti
ui
vi wi
ti
Figure 7: Drawings Dn (left) and Dn (right). The gray wedge only contains the endpoint
ti. In Dn, the dashed edges need to take a detour to avoid intersecting the edge uiwi twice.
Consider the rectilinear drawing Dn obtained by connecting the n = 4s points vi, ui,
wi, ti. In Dn, maximal plane graphs are triangulations, but we are going to consider only
the family Γ of plane triangulations of Dn containing the 2s edges uivi, wivi. The weight
of a triangulation of Γ is the number of edges viti that it contains. Clearly, in the set S
there are k disjoint segments, if and only if, there is a triangulation in Γ with weight k.
Now, consider the drawing Dn obtained from Dn doing the following changes:
For i = 1, . . . , s, only the edges of the star S(ui) crossing viwi, the edges of the star
S(wi) crossing uivi, and the edge uiwi are modified.
Suppose that in S(ui) after uivi are clockwise the edges uip1, . . . , uipk, uiwi, where
each uipj has to cross viwi. Let vipi1 , vipi2 . . . , vipik be the clockwise ordered edges
of S(vi) with endpoint one of the vertices pi. Then, we modify Dn by redrawing uipi1
following first the line uivi until point vi, then turning around vi and following the line
vipi1 , in such a way that in the rotation of ui the new edge uipi1 is placed just before uivi.
See Figure 7, right. The new drawing obtained is simple, because no edge crosses both uivi
and vipi1 , edges uipj cannot cross vipi1 and none edge of S(pi1) can cross uivi. Moreover,
the number of crossings in the edge viwi has decreased by one. We repeat the same process
for the edge uipi2 (the new edge uipi2 is placed just before uivi in the rotation of ui ), then
uipi3 , and so on. The same process can be done with the edges wiqj crossing uivi. See
Figure 7, right. Finally, we can redraw uiwi in the same way, following the edge uivi then
A. Garcı́a Olaverri et al.: On plane subgraphs of complete topological drawings 81
turning around vi following edge viwi. If we do this process for all the edges crossing viui
or uiwi, i = 1, . . . , s, at the end we obtain the simple drawing Dn. By construction, in
Dn, neither the edges viui nor the edges viwi are crossed by any other edge.
Now, let us see that Dn has a triangulation of the family Γ of weight k, if and only if,
Dn has a plane subgraph of size k′, with k′ = 3n− 6− (s− k) = 11s− 6 + k. Suppose
Dn has a triangulation F with weight k. This means that F contains (s − k) edges uiwi.
By removing from F these uiwi edges, we obtain a plane set F ′ of edges, where no edge
of F ′ has been modified to obtain the drawing Dn. Therefore, the edges of F ′ also form a
plane subgraph in Dn of size 3n− 6− (s− k) = 11s− 6 + k.
Conversely, suppose Dn contains a plane subgraph with 3n−6−(s−k) edges. Since the
edges uivi, wivi are not crossed by any edge of Dn, they must belong to any maximal plane
graph of Dn. Therefore, Dn has a plane subgraph F containing all the edges uivi, viwi
and of size k′ ≥ 3n − 6 − (s − k). As the wedge viwi, viui only contains point ti, if the
edge viti is not in F , then, the face of F containing the edges viui and viwi cannot be
a triangle. But, if a plane graph on n vertices contains more than (s − k) non-triangular
faces, its maximum number of edges is < 3n− 6− (s− k). As a consequence, viti is not
in F for at most (s − k) indices i, or equivalently, the plane subgraph F contains at least
k edges viti. This means that we can obtain a triangulation of the family Γ of weight k by
including k of these non-crossing edges.
Note that in the straight-line setting, we can always draw a triangulation of the underly-
ing point set, which contains the maximum number of edges. However, this is not the case
for simple topological drawings. We were not able to come up with a reduction solving the
following problem.
Problem 4.3. What is the complexity of deciding whether a given Dn contains a triangu-
lation, i.e., a plane subgraph whose faces are all 3-cycles?
Our reduction can also be adapted for a related problem on compatible graphs. We leave
the realm of general simple topological drawings and consider the following problem in the
more specialized setting of geometric graphs (rectilinear drawings). Let P = {p1, . . . , pn}
and P ′ = {p′1, . . . , p′n} be two sets of points in the plane. A planar graph is compatible
if it can be embedded on both P and P ′ in a way that there is an edge pipj if and only if
there is an edge p′ip
′
j . Saalfeld [21] asked for the complexity of deciding whether two such
point sets (with a given bijection between them) have a compatible triangulation. We will
say that triangulations F of P and F ′ of P ′ have k′ compatible edges when there exists a
subset of k′ edges pipj of F , such that their images, edges p′ip
′
j , are edges of F ′.
We can show the NP-completeness of the following optimization variant of the prob-
lem. (However, as the similar Open Problem 4.3, Saalfeld’s problem remains unsolved.)
Theorem 4.4. Given two point sets P = {p1, . . . , pn} and P ′ = {p′1, . . . , p′n} and the in-
dicated bijection between them, as well as a cardinality k′, the problem of deciding whether
P and P ′ admit two triangulations with k′ compatible edges is NP-complete.
Proof. We follow the idea of the proof of Theorem 4.2, and use a reduction from the SEG
problem. Suppose that an instance of the SEG problem is given: a set S of s segments in
the plane that pairwise either are disjoint or intersect in a proper crossing, and an integer
k > 0. We will build two sets of points P = {p1, . . . , pn} and P ′ = {p′1, . . . , p′n} and
obtain an integer k′ such that the SEG problem has answer Yes if and only if, P and P ′
admit two triangulations with k′ compatible edges.
82 Ars Math. Contemp. 20 (2021) 69–87
Let P be the set of n = 5s points formed by the vi, ti, ui, wi (i = 1, . . . , s) points
obtained from S as in the above Theorem 4.2, plus s points ṽi, where each point ṽi is
placed inside the triangle viuiwi very close to the point vi, to the right of the oriented line
viti, in such a way that in the wedge defined by the half-lines ṽiui, ṽiwi the only point of
P is ti, and the wedges uivi, uiṽi and wiṽi, wivi do not contain points of P . See Figure 8,
left. By construction, any triangulation F of the set of points P must contain the edge viṽi.
Observe that if the edge tivi is in F , then the edge tiṽi has to be also in F . Also note that
uiwi is only crossed by the edges tivi and tiṽi.
ui
vi wi
ti
ṽi
p
ui
vi wi
ti
ṽ′i
p
Bi Bi
Figure 8: The point sets P (left) and P ′ (right).
In the same way, let P ′ be the set of n = 5s points vi, ti, ui, wi, ṽ′i (i = 1, . . . , s), where
now each point ṽ′i is placed outside the triangle viuiwi, very close to the intersection point
of uiwi with viti, to the right of the line viti, and satisfying that any clockwise triangle
uiwip contains inside the point ṽ′i. See Figure 8, right. The bijection between the points
of P and P ′ is the obvious one, to each point ṽi of P corresponds point ṽ′i of P
′, for any
other point its image is itself.
To prove the statement of the theorem, it is enough to prove the following:
If in the set S there are k disjoint segments, then there are triangulations F and F ′
of the sets P and P ′, respectively, with k′ = 3n − 6 − (s − k) compatible edges. And
reciprocally, if F and F ′ contain k′ = 3n− 6− (s− k) compatible edges, then S contains
k disjoint segments.
Suppose first that S contains a set D of k disjoint segments viti. Let P0 be the set of 4s
common points of P and P ′ (all the points vi, ui, wi, ti). We build a triangulation F0 of P0
in the following way. If viti is in D, then we include the edges viti, viui, viwi, tiui, tiwi
in F0. If viti is not in D, then we include the edges uivi, viwi, wiui in F0. After that, we
add edges in an arbitrary way until obtaining a triangulation F0 of P0. Now, to obtain F
and F ′, we add the points ṽi and ṽ′i to F0 and retriangulate the triangular faces where they
are. If the edge viti is in D, then the points ṽi, ṽ′i are both in the triangle uiviti. So, by
adding the point ṽi and the three edges ṽiui, ṽivi, ṽiti to F0, or the point ṽ′i and the three
A. Garcı́a Olaverri et al.: On plane subgraphs of complete topological drawings 83
edges ṽ′iui, ṽ
′
ivi, ṽ
′
iti we continue with all the edges being compatible. However, if the edge
viti is not in D, then the point ṽi is in the triangle uiviwi, but the point ṽ′i is in a triangle
uiwipi. Then, we obtain a triangulation F of P by adding the edges ṽiui, ṽivi, ṽiwi, and
a triangulation F ′ of P ′ by adding the edges ṽ′iui, ṽ
′
ipi, ṽiwi. Now, the images of the
edges ṽivi of F , edges ṽ′ivi, are not in F ′ (there the edges ṽ
′
ipi appear instead). This
situation occurs (s − k) times, so the number of compatible edges between F and F ′ is
3n− 6− (s− k).
Conversely, suppose P and P ′ contain triangulations F and F ′ with k′ = 3n − 6 −
(s − k) compatible edges. If ṽiti is not in F , then the edges ṽivi and uiwi must be both
in F , because the edge uiwi can be crossed only by the edges tivi and tiṽi. However, in
set P ′, always the edge ṽ′ivi is crossed by the edge uiwi. Then, for each index i such that
ṽiti is not in F , one of the edges ṽivi or uiwi of F is not in F ′. Therefore, this situation
can happen at most (s− k) times, that is, the triangulation F must contain at least k edges
ṽiti. But if k segments ṽiti are disjoint, also their corresponding viti edges are disjoint.
Therefore, S has to contain at least k disjoint segments.
Finally, let us analyze the complexity of augmenting a plane subgraph F of Dn until
obtaining a maximal plane subgraph. Since F has O(n) edges, the set of edges of S(v) not
crossing F can be trivially found in O(n2) time. This directly implies an O(n3) algorithm
to obtain a maximal plane graph containing F : For i = 1, . . . , n, update F by adding
the edges of S(vi) non-crossing F , not in F . The following result implies that, if F is
connected, finding a maximal plane subgraph containing F can be done in O(n2) time.
Theorem 4.5. Given a simple topological drawing of Kn, a connected plane subgraph F ,
and a vertex v, we can find the edges from v to F not crossing F in O(n) time.
Proof. Notice that as F is a plane graph, we can compute in linear time, for each vertex w
the clockwise order of the edges of F incident to w, the faces of F , and for each face f ,
the clockwise cyclic list of edges and vertices found along its boundary.
Suppose first that the vertex v is not in F , and let vw1 be the first edge in the rotation of
v with one endpoint in F . The algorithm runs in three stages. In the first stage, it starts by
finding the edge of F , edge e1 = u0u1, that intersects vw1 closest to v along vw1. When
the first intersection point occurs precisely at the vertex w1, we take e1 as the first edge
of F that follows, counterclockwise, to w1v in S(w1). Using the rotation system and its
inverse, this edge e1 of F can be found in linear time, since |F | ∈ O(n). It also gives us
the face f of F containing the vertex v inside.
For simplicity, let us suppose that f is a bounded face and that the boundary of f is a
simple cycle, formed by the edges e1 = u0u1, e2 = u1u2, em = um−1u0. We will later
discuss the general case. Notice that if the edges vwi, vwj , vwk are in this clockwise order
in S(v), their corresponding first crossing points xi, xj , xk with F are found in a clockwise
walk of the boundary of f in that same clockwise order. See the right bottom drawing of
Figure 9.
In the second stage, the algorithm simulates a clockwise walk x1u1, u1u2, . . . ,uk−1uk ,
. . . of the boundary of f starting at point x1, the first crossing point of vw1 with e1 = u0u1,
and simultaneously a clockwise walk vw1, vw2, . . . , vwi, . . . on the edges of the star S(v),
beginning with the edge vw1. In each step, the algorithm makes progress in at least one
of the two walks, by adding the following edge on the boundary to the boundary walk or
passing to explore the following edge of S(v). In this process the algorithm will keep a list
σ with some of the explored edges of S(v).
84 Ars Math. Contemp. 20 (2021) 69–87
In a generic step, the edges Si = (vw1, vw2, . . . , vwi) of S(v), and the portion of the
boundary of f , Wk = (x1u1, u1u2, . . . , uk−1uk), have been visited, and the two following
invariants hold:
(A) The first crossing of the edge vwi is not on Wk−1 = (x1u1, u1u2, . . . , uk−2uk−1)
(the walk Wk minus its last edge).
(B) The list σ contains an ordered list (vui1 , vui2 , . . . , vuis) of the explored edges of
S(v) finishing at some of the vertices uj , 1 ≤ j < k, satisfying:
(B1) All the explored edges of S(v) not placed in σ cross the boundary of f .
(B2) The first crossing point of each edge vuj of σ with the boundary of f is either
uj or is placed clockwise after uj .
Initially, if x1 is an interior point of the edge e1 = u0u1, then Wk = (x1u1), Si =
(vw1, vw2) and the list σ is empty. If x1 coincides with the vertex u0, then Wk = (u0u1),
Si = (vw1, vw2) and the list contains the edge vu0. In both cases invariants (A) and (B)
are satisfied (the walk Wk−1 is empty or consists of only one vertex).
f
v
f
v1
2
3
4
5
6 7
8
9
10
f
v
1
2
3
4
5
6
7
8
9
10
v
f
Figure 9: Top left: A vertex v inside the face f . Only the edges vui, with ui incident to the
face f , can be uncrossed by F . Top right: A clockwise walk along the boundary of the face
f . Bottom left: In a walk along the boundary of f , the first crossing points of the edges
of S(v) are found in the same order as the edges of S(v). Bottom right: An equivalent
drawing to the top left figure with the boundary of f being a simple cycle. Some vertices,
like (2, 4, 9), can correspond to the same vertex of the first drawing.
In this second stage the algorithm proceeds as follows:
A. Garcı́a Olaverri et al.: On plane subgraphs of complete topological drawings 85
• If vwi crosses the last edge of Wk, edge ek, or if wi is not a vertex of f , iterate
considering the clockwise successor vwi+1 of vwi in the rotation of v.
As the first crossing of vwi must be on the edge ek or a posterior edge et, t > k, also
the first crossing of vwi+1 must be on ek or a posterior edge. Thus invariant (A) is
kept. On the other hand, observe that σ does not change, vwi must not be included
in σ (it crosses f ), and Wk is not modified. Therefore invariant (B) is also kept.
• If vwi does not cross ek and wi is a vertex of f , wi ̸= uk, then, add the following
edge ek+1 on f to Wk, keeping the same edge vwi of S(v).
Invariant (A) is kept, because the first crossing point of vwi cannot be on Wk. In-
variant (B) is also kept, because σ is not modified.
• If vwi does not cross ek and wi = uk, then, add vwi to the list σ, pass to explore the
following edge vwi+1 of S(v) and add the following edge ek+1 on f to Wk.
Again, invariant (A) is kept, because the first crossing of vwi+1 must be after uk.
On the other hand, the first crossing point of vwi is either uk or it is placed after uk,
hence property (B) is kept.
This second stage of the algorithm ends when all the edges of S(v) and f have been
explored. The last edge of the boundary of f being either u0x1 or um−1u0. Therefore, at
the end, invariant (B) implies that σ will contain the uncrossed edges of S(v) plus some
crossed edges vui of S(v) satisfying that the first crossing (on the boundary of f ) is placed
after the endpoint ui of that edge.
In each step of this stage, a new edge in the boundary of f , a new edge of S(v), or both
edges become explored. As the number of edges in f and in S(v) is linear, this second
stage of the algorithm runs in O(n) time.
In the third stage, the algorithm repeats counterclockwise the above stage considering
only the edges in σ. That means, it explores counterclockwise the boundary of f (in the
order x1u0, u0um−1, . . . , and counterclockwise the edges of S(v) placed in σ (so, in the
order vuis , vuis−1 , . . . ). In this third stage, in linear time, a new list σ is obtained. By
invariant (B1), all the uncrossed edges of S(v) have to be in σ. And by invariant (B2), if
vui is in σ, its first crossing point cannot be clockwise nor counterclockwise before ui, so
it has to be ui. Therefore, σ will contain the uncrossed edges of S(v).
In general, the boundary of face f is not a simple cycle, some edges of f can be incident
to f for both sides, so they appear twice in a walk along the boundary of f . However, this
general case can be transformed to the previous case by standard techniques, as done in [6]
in their proof of the general case of Lemma 1.1. In Figure 9, the bottom right figure shows
how to transform the drawing of the top left figure, to obtain an equivalent drawing where
the boundary of f is a simple cycle. When the face f is the unbounded face the algorithm
is totally analogous.
Finally, let us consider the case when the vertex v is in F . Then, vertex v can be
incident to several faces f1, . . . , fl, l ≥ 1. Again, for simplicity, suppose that the boundary
of each one of these faces is a simple cycle. For each face fi, if vwi1 , vwi2 are the two
edges incident to vertex v in fi, we can compute by the above method the uncrossed edges
of S(v) placed inside fi, using only the edges of S(v) placed clockwise between vwi1
and vwi2 .
86 Ars Math. Contemp. 20 (2021) 69–87
5 Conclusion
In this paper, we considered maximal and maximum plane subgraphs of simple topological
drawings of Kn. It turns out that maximal plane subgraphs have interesting structural prop-
erties. These insights could be useful in improving the bounds on the number of disjoint
edges in any such drawing, continuing this long line of research.
Also, algorithmic questions arise. For example, Proposition 2.1 ensures that there are
always two edges connecting a vertex v to a not necessarily connected plane graph F in Dn
without crossings. Moreover, the set of edges of S(v) not crossing F can be trivially found
in O(n2) time. This leads to the following question.
Problem 5.1. Given a not necessarily connected plane graph F in Dn, plus a vertex v not
in F , can the edges of S(v) incident to but not crossing F be found in o(n2) time?
ORCID iDs
Alfredo Garcı́a Olaverri https://orcid.org/0000-0002-6519-1472
Alexander Pilz https://orcid.org/0000-0002-6059-1821
Javier Tejel Altarriba https://orcid.org/0000-0002-9543-7170
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 89–102
https://doi.org/10.26493/1855-3974.2154.cda
(Also available at http://amc-journal.eu)
Graphical Frobenius representations
of non-abelian groups*
Gábor Korchmáros
Dipartimento di Matematica, Informatica ed Economia, Università della Basilicata,
Contrada Macchia Romana, 85100 Potenza, Italy
Gábor P. Nagy
Department of Algebra, Budapest University of Technology and Economics,
Egry József utca 1, H-1111 Budapest, Hungary, and
Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary
Received 14 October 2019, accepted 19 October 2020, published online 18 August 2021
Abstract
A group G has a Frobenius graphical representation (GFR) if there is a simple graph Γ
whose full automorphism group is isomorphic to G acting on the vertices as a Frobenius
group. In particular, any group G with a GFR is a Frobenius group and Γ is a Cayley
graph. By very recent results of Spiga, there exists a function f such that if G is a finite
Frobenius group with complement H and |G| > f(|H|) then G admits a GFR. This paper
provides an infinite family of graphs that admit GFRs despite not meeting Spiga’s bound.
In our construction, the group G is the Higman group A(f, q0) for an infinite sequence of
f and q0, having a nonabelian kernel and a complement of odd order.
Keywords: Cayley graph, Frobenius group, Suzuki 2-group, Frobenius graphical representation.
Math. Subj. Class. (2020): 20B25, 05C25
1 Introduction
Graphs and their automorphism groups have intensively been investigated especially for
vertex-transitive (and hence regular) graphs. Many contributions have concerned vertex-
transitive graphs with large automorphism groups compared to the degree of the graph, and
have in several cases relied upon deep results from group theory, such as the classification
of primitive permutation groups.
On the other end, the smallest vertex-transitive automorphism groups of graphs occur
*Support provided by NKFIH-OTKA Grants 114614, 115288 and 119687.
E-mail addresses: gabor.korchmaros@unibas.it (Gábor Korchmáros), nagyg@math.bme.hu (Gábor P. Nagy)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
90 Ars Math. Contemp. 20 (2021) 89–102
when the group is regular on the vertex-set. A group is said to have a graphical regular rep-
resentation (GRR) if there exists a graph whose (full) automorphism group is isomorphic to
G acting regularly on the vertex-set. Actually, almost all finite groups have GRRs. In fact,
all the few exceptions were found in the 1970-80s by a common effort of G. Sabidussi,
W. Imrich, M. E. Watkins, L. A. Nowitz, D. Hetzel, C. D. Godsil, and L. Babai, see [2,
Section 1]. Since regular automorphism groups of a graph are those which are vertex tran-
sitive but contain no non-trivial automorphism fixing a vertex, a natural next choice as a
small vertex-transitive automorphism group of a graph may be a Frobenius group on the
vertex-set: an automorphism group of a graph that is vertex-transitive but not regular and
only the identity fixes more than one vertex. It is well known that any group may be a
Frobenius group in at most one way. Furthermore, each graph Γ with a (sub)group G of
automorphisms acting regularly on the vertex-set is a Cayley graph Cay(G,S).
All these give a motivation for the study of Frobenius groups G which have a graphical
Frobenius representation (GFR), that is, there exists a graph whose (full) automorphism
group is isomorphic to G acting on the vertex-set as a Frobenius group. The systematic
study of the GFR problem was initiated by J. K. Doyle, T. W. Tucker and M. E. Watkins in
their recent paper [2]. As pointed out by those authors, the GFR problem is largely not anal-
ogous to the GRR problem since all groups have a regular representation whereas Frobenius
groups have highly restricted algebraic structures. Nevertheless, they conjectured that like
the GRR-case, “all but finitely many Frobenius groups with a given Frobenius complement
have a GFR”. Very recently Spiga [9, 10] was able to prove that conjecture for Cayley
graphs and Cayley digraphs (digraphical Frobenius representations, or DFRs). Spiga com-
bined combinatorial properties of Cayley graphs with some deeper results on the 1-point
stabilizers of primitive permutation groups to obtain a function f : N → N such that if G
is a finite Frobenius group with complement H satisfying |G| > f(|H|), then G admits a
GFR; see [10, Theorem 1.1], [9, Theorem 1], and Section 9.
It is apparent from Spiga’s work and from the results, examples and classification of
smaller groups with GFRs in [2], see in particular [2, Theorem 5.3 and Remark 5.4], that
an interesting open problem is the explicit constructions of GFR for Frobenius groups G
which do not meet Spiga’s bound and whose kernel H is a non-abelian 2-group.
In this paper we provide such a construction. Our choice of Frobenius groups is in-
fluenced by Higman’s classification of Suzuki 2-groups [5], as we take for G the group
A(f, q0) from Higman’s list where q0 and q = 2f are 2-powers. The group A(f, q0) is a
subgroup of G of GL(3,Fq) whose main properties are recalled in Section 2. We build a
Cayley graph Γu on the Frobenius kernel K of G, with a certain inverse closed subset S of
K as connecting set, constructed from an element u ∈ Fq . We show that G has GFR on Γu
provided that q = 2f , q0 and u are carefully chosen.
Our notation and terminology are standard. For the definitions and known results on
Frobenius groups which play a role in the present paper, the reader is referred to [2].
2 The group A(f, q0)
Let Fq be the finite field of order q = 2f with f ≥ 4, and let q0 = 2f0 be another power of
2 smaller than q. For a, c ∈ Fq and λ ∈ F∗q , we write
Φa,c =
1 0 0a 1 0
c aq0 1
 , Ψλ =
1 0 00 λ 0
0 0 λq0+1
 .
G. Korchmáros and G. P. Nagy: Graphical Frobenius representations of non-abelian groups 91
We define the groups
K = {Φa,c | a, c ∈ Fq},
H = {Ψλ | λ ∈ F∗q}.
Then, K is a 2-group of order q2 and H is a cyclic group of order q − 1. Moreover,
H normalizes K, and its action fixes no nontrivial element in K. Their closure group is
HK, and denoted by A(f, q0) in Higman’s paper [5]. For brevity, we write G in place of
A(f, q0). With this change G = HK. Since H ∩Hg = 1 holds for any g ∈ G \H , G is
a Frobenius group in its action on the set G/H of right cosets of H . The point stabilizer
is H and K is a regular normal subgroup. It may be noticed that when q = 2q20 then G is
similar to the 1-point stabilizer of the Suzuki group Sz(q) in its double transitive action on
q2 + 1 points. A straightforward computation shows that the H-orbits on K are
Ωu = {Φa,uaq0+1 | a ∈ F∗q}, u ∈ Fq, (2.1)
and
Ω∞ = {Φ0,c | c ∈ F∗q}.
3 A Cayley graph arising from G
For every u ∈ Fq , we may build a Cayley graph in the usual way:
Γu = Cay(K,Ωu ∪ Ωu+1).
Since Ωu ∪ Ωu+1 is H-invariant, the group G induces automorphisms of Γu. This allows
us to look at (the matrix group) G as a Frobenius group on K = V (Γu). Our aim is to
show that if q, q0 and u ∈ Fq are carefully chosen then Aut(Γu) coincides with G. Define
the set Uq,q0 of elements u ∈ Fq which satisfy both conditions:
(U1) u = (1 + ηq0)/(η + ηq0) for some primitive element η of Fq;
(U2) the polynomial Xq0+1 + uXq0 + (u+ 1)X + 1 has no roots in Fq .
Then such an appropriate choice of the triple (q, q0, u) is given in the following theo-
rem.
Theorem 3.1. Assume that q − 1 and q20 − 1 are relatively prime. Then
(i) Γu is connected Cayley graph.
(ii) If, in addition, u ∈ Uq,q0 , then Aut(Γu) = G, that is, G has a graphical Frobenius
representation on Γu.
The question whether Theorem 3.1 provides an infinite family is also answered posi-
tively.
Theorem 3.2. For infinitely many 2-powers q it is true that whenever the 2-power q0 sat-
isfies gcd(q − 1, q20 − 1) = 1, the set Uq,q0 is not empty.
Computer calculations show that for many u ̸∈ Uq,q0 , the graph Γu is still a GFR for G.
Hence, the conditions (U1) and (U2) are only needed for our proofs. However, Uq,q0 = ∅
implies φ(q)/q ≤ 3, which happens extremely rarely for q = 2f , f odd; see Section 5.
92 Ars Math. Contemp. 20 (2021) 89–102
4 Some more properties of the abstract structure of the group G
Lemma 4.1. The following hold in K:
(i) Φ2a,c = Φ0,aq0+1 and Φ
−1
a,c = Φa,c+aq0+1 .
(ii) Φ−1a,cΦ
−1
b,dΦa,cΦb,d = Φ0,aq0b+abq0 .
(iii) Ω∞ consists of central involutions of K.
(iv) For each u ∈ Fq , we have Ω−1u = Ωu+1.
Proof. Straightforward matrix computation.
Lemma 4.2. Assume that gcd(q20 − 1, q − 1) = 1. Then the following hold:
(i) K ′ = Z(K) = {1} ∪ Ω∞.
(ii) K ′ and K/K ′ are elementary abelian 2-groups of order q.
(iii) For u ∈ Fq , the set Ωu generates K.
(iv) H acts transitively (hence irreducibly) on the nontrivial elements of K ′ and K/K ′.
(v) The subgroup H is maximal in HK ′, which is maximal in G.
Proof. By the assumption, the map a 7→ a + aq0 has kernel F2, and, a 7→ aq0+1 is a
bijection of F∗q . Hence, any element of Fq can be written in the form aq0b + abq0 , which
implies (i). For a ∈ F∗q , we have Φ2a,uaq0+1 = Φ0,aq0+1 . Thus, Ω∞ ⊆ ⟨Ωu⟩ and (iii)
follows. The rest is straightforward computation.
Notice that Lemma 4.2(iii) yields Theorem 3.1(i).
5 On Conditions (U1) and (U2)
A natural key question regarding the applicability of Theorem 3.1 is the existence of some
q such that Uq,q0 is not empty, that is, Fq contains an element u satisfying both Conditions
(U1) and (U2). Theorem 3.2 states that infinitely many such q exist and we are going to
show how to prove it using Euler’s phi function and the Möbius function. For this purpose,
we need some algebraic preparatory results stated in the next lemmas.
Lemma 5.1. Let q = 2f be a power of 2 with odd exponent f . There exist at least
2(q + 1)/3 elements u ∈ Fq such that Xq0+1 + uXq0 + (u+ 1)X + 1 has no roots in Fq .
Proof. Define the rational function
U(x) =
xq0+1 + x+ 1
xq0 + x
.
Clearly, 0 and 1 are never roots of Xq0+1 + uXq0 + (u + 1)X + 1. Moreover, Xq0+1 +
uXq0 + (u+ 1)X + 1 has a root in Fq if and only if u = U(x) for some x ∈ Fq \ {0, 1}.
Since U(0) = U(1) = ∞ and
U(x) = U
(
x+ 1
x
)
= U
(
1
x+ 1
)
identically, we have |U(Fq \ {0, 1})| ≤ (q − 2)/3. Here we use the fact that F4 is not a
subfield of Fq and x, (x+ 1)/x, 1/(x+ 1) are distinct elements of Fq .
G. Korchmáros and G. P. Nagy: Graphical Frobenius representations of non-abelian groups 93
Lemma 5.2. For infinitely many odd integers n, inequality φ(2n − 1)/(2n − 1) > 1/3
holds.
Proof. The claim follows from the asymptotic formula of [7, Theorem 3]
1
M
∑
1≤m≤M
φ(2m − 1)
2m − 1
= µ+O(M−1 logM),
with µ is given by the absolute convergent series
µ =
∑
d odd
µ(d)
dtd
≈ 0.73192,
where td is the multiplicative order of 2 modulo d, and µ(d) is the Möbius function; see
[11, Theorem 4.1].
We give a second, elementary proof based on Fermat’s Little Theorem. We show that
for primes p, φ(2p − 1)/(2p − 1) → 1. Let r1, . . . , rk be the different prime factors of
2p − 1. For i = 1, . . . , k, let mi be the order of 2 modulo ri. Then mi | p and p = mi.
Moreover, 2ri−1 ≡ 1 (mod ri) implies p | (ri−1). In fact, p | (ri−1)/2 and ri = 2sip+1
holds for some integer si ≥ 1. This implies
k < log2p(2
p − 1) < p
log2 p
.
Hence,
1 >
φ(2p − 1)
2p − 1
=
k∏
i=1
(
1− 1
ri
)
>
(
1− 1
2p
) p
log2 p
,
where the latter term converges to 1. This proves our claim.
Remark 5.3. As pointed out in [8], much more is true: [7] implies that given any ε > 0,
there is a c > 0 such that φ(2n − 1)/(2n − 1) > c apart from a set of n with upper density
< ε.
We are in a position to prove Theorem 3.2. By Lemma 5.2, it suffices to show that for
an arbitrary odd integer f with φ(2f − 1)/(2f − 1) > 1/3, q = 2f fulfills the conditions
of Theorem 3.2. Fix such an f and choose an arbitrary integer f0, coprime to f . Then
q0 = 2
f0 satisfies gcd(q − 1, q20 − 1) = 1. By the choice of f , Fq has more than (q − 1)/3
primitive elements. In our case, x 7→ xq0−1 is bijective in Fq , hence the maps
η 7→ η′ = 1 + η
q0
η + ηq0
, u 7→ u′ = 1 +
(
u
u+ 1
) 1
q0−1
are well-defined inverses to each other. Now, the claim follows from Lemma 5.1.
6 Incidences
Recall that Γu denotes the Cayley graph Cay(K,Ωu ∪Ωu+1), where the vertices of Γu are
the elements of K and Ωu is defined in (2.1). The identity Φ0,0 of K will also be denoted
by ε. The group G = HK acts on K, the action is induced as follows: The elements of
94 Ars Math. Contemp. 20 (2021) 89–102
K act in the right regular action and the elements of H act by conjugation. In the sequel,
we identify G with its permutation action on K, whereby some caution is required since
for a subset X of K, the point-wise stabilizer of X in G and the centralizer of X in G are
in general different. As a permutation group, G is a subgroup of the automorphism group
Aut(Γu), and H is its cyclic subgroup of order q − 1, fixing ε and preserving both Ωu and
Ωu+1. Formally, ε is viewed as an element of Aut(Γu); nevertheless, we will also use the
notation id to denote the trivial automorphism of Aut(Γu).
For any two elements Φa,c, Φb,d ∈ K with Φa,cΦ−1b,d ∈ Ωu, we introduce the directed
edge notation Φa,c
u−→ Φb,d in Γu and we refer to it as a u-edge. It should be noticed
that our notation is not the usual one for Cayley digraphs, where the arrows point in the
opposite direction. An obvious observation is that the following are equivalent:
(i) Φa,c
u−→ Φb,d,
(ii) Φa,cΦ−1b,d ∈ Ωu,
(iii) c+ d = (a+ b)q0(ua+ (u+ 1)b),
(iv) c+ d = u(a+ b)q0+1 + aq0b+ bq0+1.
Now we collect some incidences in Γu which play a role in our proof.
Lemma 6.1. Assume gcd(q − 1, q20 − 1) = 1 and define
η = 1 +
(
u
u+ 1
) 1
q0−1
for u ∈ Fq \ {0, 1}. Then the following hold in Γu for a, b ̸= 0:
Φa,uaq0+1
u−→ Φb,ubq0+1 ⇐⇒ b =
a
η
, (6.1a)
Φa,uaq0+1
u+1−→ Φb,ubq0+1 ⇐⇒ b = aη, (6.1b)
Φa,(u+1)aq0+1
u−→ Φb,(u+1)bq0+1 ⇐⇒ b = a ·
η
1 + η
, (6.1c)
Φa,(u+1)aq0+1
u+1−→ Φb,(u+1)bq0+1 ⇐⇒ b = a ·
1 + η
η
, (6.1d)
Φa,uaq0+1
u−→ Φb,(u+1)bq0+1 ⇐⇒ b =
a
1 + η
, (6.1e)
Φa,uaq0+1
u+1−→ Φb,(u+1)bq0+1 ⇐⇒
(a
b
)q0+1
+ u
(a
b
)q0
+ (u+ 1)
(a
b
)
+ 1 = 0.
(6.1f)
Proof. (6.1a): Since Γu has no loops, we may assume a ̸= b.
Φa,uaq0+1
u−→ Φb,ubq0+1 ⇐⇒ uaq0+1 + ubq0+1 = (a+ b)q0(ua+ (u+ 1)b)
⇐⇒ 0 = (u+ 1)aq0b+ uabq0 + bq0+1
⇐⇒ 0 = (u+ 1)
(a
b
)q0
+ u
(a
b
)
+ 1
⇐⇒ 0 = (u+ 1)
(a
b
+ 1
)q0
+ u
(a
b
+ 1
)
G. Korchmáros and G. P. Nagy: Graphical Frobenius representations of non-abelian groups 95
⇐⇒
(a
b
+ 1
)q0−1
=
u
u+ 1
= (η + 1)q0−1
⇐⇒ a
b
= η.
Since (u + 1)-edges are reversed u-edges, we obtain (6.1b) by switching a and b in the
computation above. To show (6.1d), we replace u by u+ 1 and use the computation above
to obtain
Φa,(u+1)aq0+1
u+1−→ Φb,(u+1)bq0+1 ⇐⇒
(a
b
+ 1
)q0−1
=
u+ 1
u
=
(
1
1 + η
)q0−1
⇐⇒ a
b
=
η
1 + η
.
This proves (6.1c) by switching a and b. For (6.1e):
Φa,uaq0+1
u−→ Φb,(u+1)bq0+1 ⇐⇒ uaq0+1 + (u+ 1)bq0+1 = (a+ b)q0(ua+ (u+ 1)b)
⇐⇒ 0 = (u+ 1)aq0b+ uabq0
⇐⇒
(a
b
)q0−1
=
u
u+ 1
= (η + 1)q0−1
⇐⇒ a
b
= 1 + η.
Finally,
Φa,uaq0+1
u+1−→ Φb,(u+1)bq0+1 ⇐⇒ uaq0+1 + (u+ 1)bq0+1 = (a+ b)q0((u+ 1)a+ ub)
⇐⇒ 0 = aq0+1 + uaq0b+ (u+ 1)abq0 + bq0+1
⇐⇒ 0 =
(a
b
)q0+1
+ u
(a
b
)q0
+ (u+ 1)
(a
b
)
+ 1,
which shows (6.1f).
Our next step is to describe the structure of the neighborhood of the vertex ε in Γu.
For this purpose, we recall the concept of generalized Petersen graphs [3]. Let n and k be
integers with 1 ≤ k < n/2, the vertex set of GPG(n, k) is {c1, . . . , cn, c′1, . . . , c′n} and
the edge set consists of all pairs of the form
cici+1, cic
′
i, cic
′
i+k, i ∈ {1, . . . , n},
where all subscripts are to be read modulo n. In order to describe the automorphism group
of GPG(n, k), define the permutations
ρ : ci 7→ ci+1, c′i 7→ c′i+1,
δ : ci 7→ c−i, c′i 7→ c′−i,
α : ci 7→ c′ki, c′i 7→ cki
for all i ∈ {1, . . . , n}. By [3, Theorem 1 and 2],
⟨ρ, δ⟩ ≤ Aut(GPG(n, k)) ≤ ⟨ρ, δ, α⟩
96 Ars Math. Contemp. 20 (2021) 89–102
provided that n ̸∈ {4, 5, 8, 10, 12, 24}. Moreover, the generators ρ, δ, satisfy the relations
ρn = δ2 = id, δρδ = ρ−1, hence, ⟨ρ, δ⟩ is isomorphic to the dihedral group of order
2n. Also, αδ = δα, α2 ∈ {id, δ}, and most importantly α−1ρα = ρk. This implies the
following lemma:
Lemma 6.2. Let n be an odd integer, n ̸= 5, and 1 ≤ k < n. In Aut(GPG(n, k)), the
following properties hold:
(i) The elements of odd order form a unique cyclic normal subgroup of order n.
(ii) For k ̸= ±1, no involution commutes with the cyclic normal subgroup of order n.
Proposition 6.3. Assume gcd(q − 1, q20 − 1) = 1 and u ∈ Uq,q0 . Then, the neighborhood
Ωu ∪ Ωu+1 of ε in Γu is isomorphic to the generalized Petersen graph GPG(q − 1, k),
where u = (1 + ηq0)/(η + ηq0) and the integer k is defined by 1 + η = ηk+1.
Proof. By the choice of u, η is a primitive element of Fq . Define
ci = Φηi,uηi(q0+1) , c
′
i = Φηi/(1+η),(u+1)(ηi/(1+η))q0+1 .
From Lemma 6.1, cici+1, cic′i are edges and there are no more edges in Ωu and between
Ωu and Ωu+1. In Ωu+1, c′i and c
′
j are connected with an u-edge if and only if
ηj
1 + η
=
ηi
1 + η
· 1 + η
η
⇐⇒ ηj−i+1 = 1 + η = ηk+1 ⇐⇒ j ≡ i+ k (mod q − 1).
This finishes the proof.
Notice that k = ±1 would imply η = 0 or 1 + η + η2 = 0, which is not possible if
gcd(q − 1, q20 − 1) = 1 and η generates F∗q .
Corollary 6.4. Assume gcd(q − 1, q20 − 1) = 1 and u ∈ Uq,q0 . Let A be the permutation
group induced by the stabilizer Aut(Γu)ε on Ωu ∪ Ωu+1. Then A is solvable, its order is
either (q − 1), 2(q − 1) or 4(q − 1), and it has a unique cyclic normal subgroup of odd
order q − 1. Moreover, Aut(Γu)ε either preserves Ωu and Ωu+1, or it interchanges them.
Proof. A contains the cyclic subgroup of order q − 1 that is induced by H on Ωu ∪ Ωu+1.
Proposition 6.3 and Lemma 6.2 apply.
We finish this section with another property of the stabilizer of ε in Aut(Γu).
Lemma 6.5. Assume gcd(q − 1, q20 − 1) = 1 and u ∈ Uq,q0 .
(i) Let A be the centralizer of the commutator subgroup K ′ in Aut(Γu). Then K ≤ A
and |A : K| ≤ 2. Moreover, any element of A \ K interchanges the sets Ωu and
Ωu+1.
(ii) Let α ∈ Aut(Γ) be an involution which centralizes H . Then α fixes Ωu ∪ Ωu+1
point-wise.
G. Korchmáros and G. P. Nagy: Graphical Frobenius representations of non-abelian groups 97
Proof. (i): Obvoiusly, K ≤ A and A is transitive. From the last sentence of Corollary 6.4,
an element α ∈ Aε either preserves Ωu and Ωu+1, or it interchanges them. We show
that if α preserves Ωu then α = id. This will imply |Aε| ≤ 2 and |A| ≤ 2q2. Since α
commutes with K ′ and fixes ε, it fixes all points in the orbit εK
′
= {ε}∪Ω∞. The elements
Φa,uaq0+1 ∈ Ωu and Φ0,d ∈ K ′ satisfy both relations
Φa,uaq0+1
u−→ Φ0,d ⇐⇒ d = 0,
Φa,uaq0+1
u+1−→ Φ0,d ⇐⇒ d = aq0+1.
This means that each element in Ωu is connected with a unique element in Ω∞. Hence, α
fixes all elements in Ωu. As each K ′-orbit contains a unique element in Ωu, we see that
each K ′-orbit is preserved. Once again, α commutes with K ′ and fixes an element in each
K ′-orbit. Therefore, α fixes all points in each K ′-orbit.
(ii): As ε is the unique fixed point of H , εα = ε and α leaves the neighborhood
Ωu∪Ωu+1 of ε invariant. By Lemma 6.2(ii), the restriction of α to Ωu∪Ωu+1 cannot have
order 2, therefore, it must be trivial.
7 Imprimitivity
In this section we show that an appropriate choice of u ∈ Fq ensures that Aut(Γu) cannot
act primitively on the set of vertices of Γu. We recall that a primitive permutation group G is
of affine type if it has an abelian regular normal subgroup, which is necessarily elementary
abelian of order rn for some prime r. In this case G is embedded in the affine group
AGL(n, r) with the socle being the translation subgroup. Its stabiliser of 0 ∈ Fnr is a
subgroup of GL(n, r) which acts irreducibly on Fnr . For our purpuse, a useful tool is the
following result by Guralnick and Saxl.
Proposition 7.1 (Guralnick and Saxl [4]). Let G be a primitive permutation group of degree
2n. Then either G is of affine type, or G has a unique minimal normal subgroup N =
S × · · · × S = St, t ≥ 1, S is a non-abelian simple group, and one of the following holds:
(i) S = Am, m = 2e ≥ 8, n = te, and the 1-point stabilizer in N is N1 = Am−1 ×
· · · ×Am−1, or
(ii) S = PSL(2, p), p = 2e − 1 ≥ 7 is a Mersenne prime, n = te, and the 1-point
stabilizer in N is the direct product of maximal parabolic subgroups each stabilizing
a 1-space.
Lemma 7.2. Let G be a group acting transitively on the set X . For x ∈ X and let H = Gx
be the stabilizer of x in G.
(i) For y ∈ X , choose g ∈ G such that y = xg . Then the subgroup of H , fixing the
H-orbit of y point-wise, coincides with ∩h∈HHgh.
(ii) If G is 2-transitive on X then ∩h∈HHgh is either H or {1}, depending upon whether
g ∈ H or g ̸∈ H .
Proof. If y′ ∈ yH , then y′ = yh = xgh for some h ∈ H . Hence, for the stabilizer we
have Gy′ = Gghx = H
gh. Therefore, the point-wise stabilizer of yH is ∩y′∈yHGy′ =
∩h∈HHgh. This proves (i). Clearly, if g ∈ H then ∩h∈HHgh = H . If g ∈ G \ H
then x ̸= y = xg and ∩h∈HHgh fixes all points in {x} ∪ yH . The latter set is X if G is
2-transitive.
98 Ars Math. Contemp. 20 (2021) 89–102
Lemma 7.3. Assume gcd(q − 1, q20 − 1) = 1 and u ∈ Uq,q0 . If Aut(Γu) acts primitively
on Γu, then its action is of affine type.
Proof. Let us assume on the contrary that Aut(Γu) is not of affine type. Let N be its unique
minimal normal subgroup. With the notation in Proposition 7.1, we have N = St where
either S = Am, m ≥ 8, or S = PSL(2, p), with a Mersenne prime p = m − 1 ≥ 7. In
both cases, S has a 2-transitive action on m points. Moreover, if B is the 1-point stabilizer
in S, then the point stabilizer of ε = Φ0,0 in N is Nε = Bt. For (g1, . . . , gt) ∈ St take
a generic vertex y = ε(g1,...,gt) of Γu. Let Y be the Bt-orbit of y. By Lemma 7.2(i) the
point-wise stabilizer of Y is
(∩b∈BBg1b)× · · · × (∩b∈BBgtb).
By Lemma 7.2(ii), each factor is either {1} or B, depending upon whether gi ∈ B or not.
Thus, the point-wise stabilizer of Y in Bt is Bt0 , where 0 ≤ t0 ≤ t, and t0 = t occurs
if and only if Y = {ε}. Therefore, the Bt induces a permutation group on Y which is
isomorphic to Bt1 , where t1 = t− t0. Furthermore, t1 = 0 if and only if Y = {ε}.
The stabilizer Nε acts on Ωu ∪ Ωu+1. Let Y be a nontrivial Nε-orbit contained in
Ωu ∪ Ωu+1. If S = Am, then Nε induces a nonsolvable group of automorphisms of
Ωu ∪ Ωu+1. If S = PSL(2, p), then |B| = p(p− 1)/2, and Nε induces a noncyclic group
of odd order on Ωu ∪ Ωu+1. Both possibilities are inconsistent with Corollary 6.4.
We are now able to prove the imprimitivity of Aut(Γu).
Proposition 7.4. Assume gcd(q − 1, q20 − 1) = 1 and u ∈ Uq,q0 . Then, Aut(Γu) acts
imprimitively on Γu.
Proof. As before, G is identified with its permutation action on Γu. In particular, we
consider H , K as subgroups of Aut(Γu). At the same time, K is the set of vertices of Γu.
Assume on the contrary that Aut(Γu) is primitive, hence of affine type by Lemma 7.3.
Let N be the unique minimal normal subgroup of Aut(Γu). Then N is a regular elementary
abelian 2-group. Since H has odd order, N decomposes into the direct product of H-
invariant subgroups. For any 1 ̸= h ∈ H and 1 ̸= n ∈ N , h has a unique fixed point,
while n has no fixed point. Hence nh ̸= hn. Therefore N = A1 × A2 where Ai is an
elementary abelian group of order q and H acts regularly on Ai \ {1}, i = 1, 2. Consider
the subgroup M = NNK(K). Since NK is nilpotent, we have K ⪇ M and K ′ ◁ M . The
latter implies K ′ ∩ Z(M) ̸= {1}. Since both K ′ and Z(M) are H-invariant while H acts
regularly on K ′ \ {1}, we have K ′ ≤ Z(M). By Lemma 6.5, |M : K| = 2. On the one
hand, M = (M ∩ N)K. On the other hand, N ∩ K is an H-submodule of M ∩ N . By
Maschke’s Theorem [1, (10.8)] applied to M ∩N , viewed as a F2-vector space, there is an
H-invariant subgroup B in M ∩N such that M ∩N = B × (N ∩K). Therefore,
B ∼= (M ∩N)/(N ∩K) ∼= (M ∩N)K/K ∼= M/K ∼= F2,
that is, the nontrivial element of B ≤ N commutes with H , a contradiction.
8 Proof of the main result Theorem 3.1(ii)
In this section, we complete the proof of Theorem 3.1. As before, G is identified with
its permutation action on Γu. From Proposition 7.4, we know that A = Aut(Γu) acts
G. Korchmáros and G. P. Nagy: Graphical Frobenius representations of non-abelian groups 99
imprimitively on Γu. We claim that the only nontrivial blocks of imprimitivity of A are
the cosets of the commutator subgroup K ′ of K. Or equivalently, K ′ is the only nontrivial
block containing ε. Let B be an arbitrary nontrivial block of imprimitivity of A which
contains ε. Then the stabilizer of the set B in G is a subgroup GB of G, lying properly
between H and G. By Lemma 4.2(v), GB = HK ′ and B = K ′, which proves the claim.
The next two lemmas describe the point-wise stabilizer of K ′ in A.
Lemma 8.1. Let E be the point-wise stabilizer of K ′ = {ε} ∪Ω∞ in Aut(Γu). Then E is
either trivial or it is an elementary abelian 2-group which fixes all pairs {Φa,c, Φ−1a,c}.
Proof. The observations made prior to Lemma 6.1 show
Φa,c
u−→ Φ0,d ⇐⇒ d = c+ uaq0+1
for all a, c, d ∈ Fq . Thus, any vertex Φa,c, a ̸= 0, is u-connected to a unique element Φ0,d1
of K ′ and (u+1)-connected to a unique element Φ0,d2 of K
′, where d1 = c+ uaq0+1 and
d2 = c + (u + 1)a
q0+1. If d1 and d2 are distinct nonzero elements, then Φ0,d1 , Φ0,d2 are
distinct vertices in Ω∞, whose common neighbors are Φa,c and Φ−1a,c = Φa,c+aq0+1 , where
a = (d1 + d2)
1
q0+1 and c ∈ {d1 + uaq0+1, d2 + uaq0+1}.
This shows that any automorphism of Γu, which fixes Ω∞ point-wise, must leave the pair
{Φa,c, Φ−1a,c} invariant. It follows that E either trivial or has exponent 2 and in the latter
case E is elementary abelian.
Actually, E is trivial by the following lemma.
Lemma 8.2. The only automorphism that fixes {ε} ∪ Ω∞ point-wise is the identity.
Proof. Let E be defined as in Lemma 8.1. Since HK ′ preserves the set of vertices in K ′,
HK ′ normalizes E. Assume on the contrary that E ̸= {1}, then CE(K ′) ̸= {1} is H-
invariant. Since K ′ acts regularly on itself, E ∩ K ′ = {1}. We apply Lemma 6.5(i) to
conclude that |CE(K ′)| = 2. This means that there is a unique involutory automorphism
α ∈ A which centralizes both K ′ and H . Now, Lemma 6.5(ii) implies that α fixes Ωu ∪
Ωu+1 point-wise. Finally, Lemma 6.5(i) yields α ∈ K, a contradiction.
Let us now focus on the point stabilizer Aε of ε in A = Aut(Γu). Clearly, Aε leaves
Ωu∪Ωu+1 invariant. Moreover, by the imprimitivity of A, Aε preserves Ω∞ as well. Since
any element of Ωu is connected with a unique element of Ω∞, each automorphism fixing
all points in {ε} ∪ Ωu ∪ Ωu+1 fixes all points in Ω∞. Hence by Lemma 8.2, the action of
Aε on Ωu ∪ Ωu+1 is faithful and the possibilities for |Aε| are q − 1, 2(q − 1) or 4(q − 1)
by Corollary 6.4.
Let S denote the stabilizer of the set K ′ in A. On the one hand, HK ′ ≤ S, hence
S is transitive on K ′. On the other hand, Aε ≤ S since K ′ is a block of imprimitivity.
Therefore, Aε = Sε, and
|S| = q|Aε| ∈ {q(q − 1), 2q(q − 1), 4q(q − 1)}.
This implies that S induces a 2-transitive solvable permutation group S̄ on K ′. Since the
order of K ′ is a power of 2, Huppert’s Theorem [6, Theorem XII.7.3] yields that S̄ is
similar to a subgroup of the group AΓL(1, q) of all semilinear mappings
z 7→ azα + b, a, b ∈ Fq, a ̸= 0, α ∈ Aut(Fq)
100 Ars Math. Contemp. 20 (2021) 89–102
on Fq . Here, |AΓL(1, q)| = fq(q − 1) for q = 2f . Since gcd(q − 1, q20 − 1) = 1, f is
odd, and the only possibility for the cardinality of S̄ is q(q−1). We apply Lemma 8.2 once
more to conclude that |S| = q(q − 1), which implies Aε = H and A = HK = G. This
finishes the proof of Theorem 3.1(ii).
Remark 8.3. Since the proof of Lemma 8.2 depends on Huppert’s classification of solvable
2-transitive groups of degree 2h + 1, a natural question is whether the possibility |S| ∈
{2q(q − 1), 4q(q − 1)} can be ruled out by purely combinatorial arguments based on the
structure of the graph Γu rather than by the use of Huppert’s classification. We are likely
to think that the answer is negative. In fact, the action of an extra-automorphism in case
|S| ∈ {2q(q − 1), 4q(q − 1)} does not seem to produce further useful constraint on the
structure of the graph Gu in the case where the kernel K is nonabelian and the complement
H has odd order. Actually, as Spiga himself pointed out in [10, Section 1.1], this case is by
far the hardest in the GFR problem.
9 Spiga’s bound
To state Spiga’s bound we need some notation consistent with that used in [9]. For a
Frobenius group G = N ⋊H with kernel N and complement H , let
d11(|N |, |H|) = 1 +
|N | − 1
|H|
−
(√
|N |
4
− (log2 |N |)2
)
;
d12(|N |, |H|) = 1 +
|N | − 1
|H|
−
(√
|N | − 1− |H|
|H|(1 + 2|H|)
log2
4
3 − 2 log2
√
|N |+ 1
)
;
d1(|N |, |H|) = (log2 |N |)2;
f1(|N |, |H|) = 2d1(|N |,|H|)( 12 |N | − 1)max{2
d11(|N |,|H|), 2d12(|N |,|H|)};
c2(|N |, |H|) =
3
4
|N |
|H|
− 1
2|H|
+
1
2
+
√
|N | |H| − 1
2|H|
+ (log2 |N |)2;
f2(|N |, |H|) = 2c2(|(N |,|H|);
d31(|N |, |H|) =
(
2|N |
|H|
) 2
3
− 2|H|
√
|N |;
d32(|N |, |H|) =
√
N ;
d33(|N |, |H|) = 4
√
|N | − 2|H| 4
√
|N |;
F3(|N |, |H|) =
1
4|H|
min{d31(|N |, |H|), d32(|N |, |H|), d33(|N |, |H|)};
c3(|N |, |H|) =
3
2
+
|N |
|H|
+
1
2|H|
− F3(|N |, |H|) + (log2 |N |)2;
f3(|N |, |H|) = 2c3(|N |,|H|).
Theorem 9.1 (Spiga’s Bound). If
21+(|N |−1)/|H| > f1(|N |, |H|) + f2(|N |, |H|) + f3(|N |, |H|)
then G admits a GFR.
G. Korchmáros and G. P. Nagy: Graphical Frobenius representations of non-abelian groups 101
As stated in [9, Theorem 2], Spiga’s bound implies that when |N | is large compared
to |H|, then a random H-invariant subset S of N gives rise to GFR on Cay(N,S). Spiga
also claimed on his strategy that “Theoretically this strategy is sound, in practice, even for
relatively small groups H , the lower bound on |N | is so large that it seems hopeless and
infeasible to study the small groups N with a computer.”
Proposition 9.2. For a 2-power q ≥ 2, let G be a Frobenius group of order q2(q− 1) with
nucleus of order q2 and complement of order q − 1. Then Spiga’s bound does not hold
for G.
Proof. In our case, in the exponent on the left hand side in Spiga’s bound we have q + 2.
On the other hand, for q = 2m,
c2(|N |, |H|) >
3
4
q2
q − 1
+
q(q − 2)
2(q − 1)
+ 2m ≥ q(5q − 4)
4(q − 1)
+ 2.
A straightforward computation shows that the last number is always bigger than q+2 when
the claim follows.
ORCID iDs
Gábor Korchmáros https://orcid.org/0000-0002-2776-5754
Gábor P. Nagy https://orcid.org/0000-0002-9558-4197
References
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 103–127
https://doi.org/10.26493/1855-3974.2101.b76
(Also available at http://amc-journal.eu)
On few-class Q-polynomial association schemes:
feasible parameters and nonexistence results
Alexander L. Gavrilyuk *
Center for Math Research and Education, Pusan National University,
2, Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan, 46241, Republic of Korea
Janoš Vidali †
Faculty of Mathematics and Physics, University of Ljubljana,
Jadranska ulica 21, 1000 Ljubljana, Slovenia, and
Institute of Mathematics, Physics and Mechanics,
Jadranska ulica 19, 1000 Ljubljana, Slovenia
Jason S. Williford ‡
Department of Mathematics and Statistics, University of Wyoming,
1000 E. University Ave., Laramie, WY 82071, United States of America
Received 28 August 2019, accepted 28 August 2020, published online 19 August 2021
Abstract
We present the tables of feasible parameters of primitive 3-class Q-polynomial associ-
ation schemes and 4- and 5-class Q-bipartite association schemes (on up to 2800, 10000,
and 50000 vertices, respectively), accompanied by a number of nonexistence results for
such schemes obtained by analysing triple intersection numbers of putative open cases.
Keywords: Association scheme, Q-polynomial, feasible parameters, distance-regular graph.
Math. Subj. Class. (2020): 05E30
*The author is supported by BK21plus Center for Math Research and Education at Pusan National University,
by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the
Ministry of Education (grant number NRF-2018R1D1A1B07047427) and by the Slovenian Research Agency
(Slovenia-Russia bilateral grant number BI-RU/19-20-007).
†The author is supported by the Slovenian Research Agency (research program P1-0285, research projects
J1-8130, J1-1691, J1-1692 and Slovenia-Russia bilateral grant (number BI-RU/19-20-007)).
‡The author was supported by National Science Foundation (NSF) grant DMS-1400281.
E-mail addresses: gavrilyuk@riko.shimane-u.ac.jp (Alexander L. Gavrilyuk), janos.vidali@fmf.uni-lj.si
(Janoš Vidali), jwillif1@uwyo.edu (Jason S. Williford)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
104 Ars Math. Contemp. 20 (2021) 103–127
1 Introduction
Much attention in literature on association schemes has been paid to distance-regular graphs,
in particular to those of diameter 2, also known as strongly regular graphs – however, their
complete classification is still a widely open problem. The tables of their feasible parame-
ters, maintained by A. E. Brouwer [4, 5], are very helpful for the algebraic combinatorics
community, in particular when one wants to check whether a certain example has already
been proven (not) to exist, to be unique, etc. Compiling such a table can be a challeng-
ing problem, as, for example, some feasibility conditions require calculating roots of high
degree polynomials.
The goal of this work is to present the tables of feasible parameters of Q-polynomial
association schemes, compiled by the third author, and accompanied by a number of nonex-
istence results obtained by the first two authors.
Recall that Q-polynomial association schemes can be seen as a counterpart of distance-
regular graphs, which, however, remains much less explored, although they have received
considerable attention in the last few years [11, 25, 27, 28] due to their connection with
some objects in quantum information theory such as equiangular lines and real mutually
unbiased bases [24].
More precisely, let A0, . . . , AD and E0, . . . , ED denote the adjacency matrices and the
primitive idempotents of an association scheme, respectively. An association scheme is
P -polynomial (or metric) if, after suitably reordering the relations, there exist polynomials
vi of degree i such that Ai = vi(A1) (0 ≤ i ≤ D). If this is the case, the matrix Ai can be
seen as the distance-i adjacency matrix of a distance-regular graph and vice-versa. Simi-
larly, an association scheme is Q-polynomial (or cometric) if, after suitably reordering the
eigenspaces, there exist polynomials v∗j of degree j such that Ej = v
∗
j (E1) (0 ≤ j ≤ D),
where the matrix multiplication is entrywise. These notions are due to Delsarte [15], who
introduced the P -polynomial property as an algebraic definition of association schemes
generated by distance-regular graphs, and then defined Q-polynomial association schemes
as the dual concept to P -polynomial association schemes.
Many important examples of P -polynomial association schemes, which arise from clas-
sical algebraic objects such as dual polar spaces and forms over finite fields, also possess
the Q-polynomial property. Bannai and Ito [1] posed the following conjecture.
Conjecture 1.1. For D large enough, a primitive association scheme of D classes is P -
polynomial if and only if it is Q-polynomial.
We are not aware of any progress towards its proof. The discovery of a feasible set of
parameters of hypothetical counter-examples (see [30]) casts some doubt on the conjecture,
and in the very least shows that this will likely be difficult to prove (see the next section
for the definition of feasible parameter sets). Moreover, the problem of classification of
association schemes which are both P - and Q-polynomial (i.e., Q-polynomial distance-
regular graphs) is still open. We refer the reader to [13] for its current state.
Recall that, for a P -polynomial association scheme defined on a set X , its intersection
numbers pkij satisfy the triangle inequality: p
k
ij = 0 if |i − j| > k or i + j < k, which
naturally gives rise to a graph structure on X . Perhaps, due to the lack of such an intu-
itive combinatorial characterization, much less is known about Q-polynomial association
schemes when the P -polynomial property is absent (which also indicates that there should
be much more left to discover). To date, only few examples of Q-polynomial schemes are
known which are neither P -polynomial nor duals of P -polynomial schemes [28] – most
A. L. Gavrilyuk et al.: On few-class Q-polynomial association schemes: feasible . . . 105
of them are imprimitive and related to combinatorial designs. The first infinite family of
primitive Q-polynomial schemes that are not also P -polynomial was recently constructed
in [31]. Due to Conjecture 1.1, it seems that the most promising area for constructing new
examples of Q-polynomial association schemes which are not P -polynomial includes those
with few classes, say, in the range 3 ≤ D ≤ 6. The tables of feasible parameters of prim-
itive 3-class Q-polynomial association schemes and 4- and 5-class Q-bipartite association
schemes presented in Section 3 may serve as a source for new constructions.
We note that imprimitive Q-polynomial 3-class schemes are either Taylor graphs (see
[5, pp. 4–6]) or linked systems of symmetric designs (see [27]). For current research on Q-
antipodal 4- and 5-class association schemes, see [24, 25] and [11]. Due to this recent work
on Q-antipodal schemes, the third author has focused only on the less studied primitive
and Q-bipartite cases in his tables. We note the primitive case is far more computationally
demanding than the Q-bipartite case, and this is the reason the class number in the tables
does not go to 4 or 5.
The parameters of P -polynomial association schemes are restricted by a number of
conditions implied by the triangle inequality. On the other hand, the Q-polynomial prop-
erty allows us to consider triple intersection numbers with respect to some triples of ver-
tices, which can be thought of as a generalization of intersection numbers to triples of
starting vertices instead of pairs. This technique has been previously used by various re-
searchers [8, 10, 17, 21, 22, 23, 36, 37], mostly to prove nonexistence of some strongly reg-
ular and distance-regular graphs with equality in the so-called Krein conditions, in which
case combining the restrictions implied by the triangle inequality with triple intersection
numbers seems the most fruitful. Yet, while calculating triple intersection numbers when
the P -polynomial property is absent is harder, we managed to rule out a number of open
cases from the tables. This includes a putative Q-polynomial association scheme on 91
vertices whose existence has been open since 1999 [12].
The paper is organized as follows. In Section 2, we recall the basic theory of association
schemes and their triple intersection numbers. In Section 3, we comment on the tables of
feasible parameters of Q-polynomial association schemes and how they were generated. In
Section 4, we explain in detail the analysis of triple intersection numbers of Q-polynomial
association schemes and prove nonexistence for many open cases from the tables. Finally,
in Section 5, we discuss the generalization of triple intersection numbers to quadruples of
vertices.
2 Preliminaries
In this section we prepare the notions needed in subsequent sections.
2.1 Association schemes
Let X be a finite set of vertices and {R0, R1, . . . , RD} be a set of non-empty subsets of
X ×X . Let Ai denote the adjacency matrix of the (di-)graph (X,Ri) (0 ≤ i ≤ D). The
pair (X, {Ri}Di=0) is called a (symmetric) association scheme of D classes (or a D-class
scheme for short) if the following conditions hold:
(1) A0 = I|X|, which is the identity matrix of size |X|,
(2)
∑D
i=0 Ai = J|X|, which is the square all-one matrix of size |X|,
(3) A⊤i = Ai (1 ≤ i ≤ D),
106 Ars Math. Contemp. 20 (2021) 103–127
(4) AiAj =
∑D
k=0 p
k
ijAk, where p
k
ij are nonnegative integers (0 ≤ i, j ≤ D).
The nonnegative integers pkij are called intersection numbers: for a pair of vertices x, y ∈ X
with (x, y) ∈ Rk and integers i, j (0 ≤ i, j, k ≤ D), pkij equals the number of vertices
z ∈ X such that (x, z) ∈ Ri, (y, z) ∈ Rj .
The vector space A over R spanned by the matrices Ai forms an algebra. Since A is
commutative and semisimple, there exists a unique basis of A consisting of primitive idem-
potents E0 = 1|X|J|X|, E1, . . . , ED (i.e., projectors onto the maximal common eigenspaces
of A0, . . . , AD). Since the algebra A is closed under the entry-wise multiplication denoted
by ◦, we define the Krein parameters qkij (0 ≤ i, j, k ≤ D) by
Ei ◦ Ej =
1
|X|
D∑
k=0
qkijEk. (2.1)
It is known that the Krein parameters are nonnegative real numbers (see [15, Lem-
ma 2.4]). Since both {A0, A1, . . . , AD} and {E0, E1, . . . , ED} form bases of A, there
exists matrices P = (Pij)Di,j=0 and Q = (Qij)
D
i,j=0 defined by
Ai =
D∑
j=0
PjiEj and Ei =
1
|X|
D∑
j=0
QjiAj . (2.2)
The matrices P and Q are called the first and second eigenmatrix of (X, {Ri}Di=0).
Let ni, 0 ≤ i ≤ D, denote the valency of the graph (X,Ri), and mj , 0 ≤ j ≤ D,
denote the multiplicity of the eigenspace of A0, . . . , AD corresponding to Ej . Note that
ni = p
0
ii, while mj = q
0
jj .
For an association scheme (X, {Ri}Di=0), an ordering of A1, . . . , AD such that for each
i (0 ≤ i ≤ D), there exists a polynomial vi(x) of degree i with Pji = vi(Pj1) (0 ≤ j ≤ D),
is called a P -polynomial ordering of relations. An association scheme is said to be P -
polynomial if it admits a P -polynomial ordering of relations. The notion of an association
scheme together with a P -polynomial ordering of relations is equivalent to the notion of
a distance-regular graph – such a graph has adjacency matrix A1, and Ai (0 ≤ i ≤ D)
is the adjacency matrix of its distance-i graph (i.e., (x, y) ∈ Ri precisely when x and y
are at distance i in the graph), and the number of classes equals the diameter of the graph.
It is also known that an ordering of relations is P -polynomial if and only if the matrix of
intersection numbers L1, where Li := (pkij)
D
k,j=0 (0 ≤ i ≤ D), is a tridiagonal matrix with
nonzero superdiagonal and subdiagonal [1, p. 189] – then pkij = 0 holds whenever the triple
(i, j, k) does not satisfy the triangle inequality (i.e., when |i − j| < k or i + j > k). For
a P -polynomial ordering of relations of an association scheme, set ai = pi1,i, bi = p
i
1,i+1,
and ci = pi1,i−1. These intersection numbers are usually gathered in the intersection array
{b0, b1, . . . , bD−1; c1, c2, . . . , cD}, as the remaining intersection numbers can be computed
from them (in particular, ai = b0−bi−ci for all i, where bD = c0 = 0). For an association
scheme with a P -polynomial ordering of relations, the ordering E1, . . . , ED is called the
natural ordering of eigenspaces if (Pi1)Di=0 is a decreasing sequence.
Dually, for an association scheme (X, {Ri}Di=0), an ordering of E1, . . . , ED such that
for each i (0 ≤ i ≤ D), there exists a polynomial v∗i (x) of degree i with Qji = v∗i (Qj1)
(0 ≤ j ≤ D), is called a Q-polynomial ordering of eigenspaces. An association scheme
A. L. Gavrilyuk et al.: On few-class Q-polynomial association schemes: feasible . . . 107
is said to be Q-polynomial if it admits a Q-polynomial ordering of eigenspaces. Similarly
as before, it is known that an ordering of eigenspaces is Q-polynomial if and only if the
matrix of Krein parameters L∗1, where L
∗
i := (q
k
ij)
D
k,j=0 (0 ≤ i ≤ D), is a tridiagonal ma-
trix with nonzero superdiagonal and subdiagonal [1, p. 193] – then qkij = 0 holds whenever
the triple (i, j, k) does not satisfy the triangle inequality. For a Q-polynomial ordering of
eigenspaces, set a∗i = q
i
1,i, b
∗
i = q
i
1,i+1, and c
∗
i = q
i
1,i−1. Again, these Krein parameters
are usually gathered in the Krein array {b∗0, b∗1, . . . , b∗D−1; c∗1, c∗2, . . . , c∗D} containing all
the information needed to compute the remaining Krein parameters (in particular, we have
a∗i = b
∗
0 − b∗i − c∗i for all i, where b∗D = c∗0 = 0). For an association scheme with a Q-
polynomial ordering of eigenspaces, the ordering A1, . . . , AD is called the natural ordering
of relations if (Qi1)Di=0 is a decreasing sequence. Unlike for the P -polynomial association
schemes, there is no known general combinatorial characterization of Q-polynomial asso-
ciation schemes.
An association scheme is called primitive if all of A1, . . . , AD are adjacency matrices
of connected graphs. It is known that a distance-regular graph is imprimitive precisely
when it is a cycle of composite length, an antipodal graph, or a bipartite graph (possibly
more than one of these), see [5, Thm. 4.2.1]. The last two properties can be recognised from
the intersection array as bi = cD−i (0 ≤ i ≤ D, i ̸= ⌊D/2⌋) and ai = 0 (0 ≤ i ≤ D),
respectively. We may define dual properties for a Q-polynomial association scheme – we
say that it is Q-antipodal if b∗i = c
∗
D−i (0 ≤ i ≤ D, i ̸= ⌊D/2⌋), and Q-bipartite if
a∗i = 0 (0 ≤ i ≤ D). All imprimitive Q-polynomial association schemes are schemes of
cycles of composite length, Q-antipodal or Q-bipartite (again, possibly more than one of
these). The original classification theorem by Suzuki [34] allowed two more cases, which
have however been ruled out later [9, 35]. An association scheme that is both P - and Q-
polynomial is Q-antipodal if and only if it is bipartite, and is Q-bipartite if and only if it is
antipodal.
A formal dual of an association scheme with first and second eigenmatrices P and Q
is an association scheme such that, for some orderings of its relations and eigenspaces, its
first and second eigenmatrices are Q and P , respectively. Note that this duality occurs on
the level of parameters – an association scheme might have several formal duals, or none
at all (we can speak of duality when there exists a regular abelian group of automorphisms,
see [5, §2.10B]). An association scheme with P = Q for some orderings of its relations and
eigenspaces is called formally self-dual. For such orderings, pkij = q
k
ij (0 ≤ i, j, k ≤ D)
holds – in particular, a formally self-dual association scheme is P -polynomial if and only
if it is Q-polynomial, and then its intersection array matches its Krein array.
Any imprimitive association scheme with two classes is both P - and Q-polynomial for
either of the two orderings of relations and eigenspaces. The graph with adjacency matrix
A1 of such a scheme is said to be strongly regular (an SRG for short) with parameters
(n, k, λ, µ), where n = |X| is the number of vertices, k = p011 is the valency of each
vertex, and each two distinct vertices have precisely λ = p111 common neighbours if they
are adjacent, and µ = p211 common neighbours if they are not adjacent. In the sequel, we
will identify P -polynomial association schemes with their corresponding strongly regular
or distance-regular graphs.
By a parameter set of an association scheme, we mean the full set of pkij , q
k
ij , Pij
and Qij described in this section, which are real numbers satisfying the identities in [5,
Lemma 2.2.1, Lemma 2.3.1]. We say that a parameter set for an association scheme is
feasible if it passes all known conditions for the existence of a corresponding association
108 Ars Math. Contemp. 20 (2021) 103–127
scheme. For distance-regular graphs, there are many known feasibility conditions, see [5,
13, 37]. For Q-polynomial association schemes, much less is known – see Section 3 for
the feasibility conditions we have used.
2.2 Triple intersection numbers
For a triple of vertices x, y, z ∈ X and integers i, j, k (0 ≤ i, j, k ≤ D) we denote by[
x y z
i j k
]
(or simply [i j k] when it is clear which triple (x, y, z) we have in mind) the
number of vertices w ∈ X such that (x,w) ∈ Ri, (y, w) ∈ Rj and (z, w) ∈ Rk. We call
these numbers triple intersection numbers.
Unlike the intersection numbers, the triple intersection numbers depend, in general, on
the particular choice of (x, y, z). Nevertheless, for a fixed triple (x, y, z), we may write
down a system of 3D2 linear Diophantine equations with D3 triple intersection numbers as
variables taking nonnegative values, thus relating them to the intersection numbers, cf. [22]:
D∑
ℓ=0
[ℓ j k] = ptjk,
D∑
ℓ=0
[i ℓ k] = psik,
D∑
ℓ=0
[i j ℓ] = prij , (1 ≤ i, j, k ≤ D) (2.3)
where (x, y) ∈ Rr, (x, z) ∈ Rs, (y, z) ∈ Rt, and
[0 j k] = δjrδks, [i 0 k] = δirδkt, [i j 0] = δisδjt (0 ≤ i, j, k ≤ D)
are constants. Note that the equations (2.3) are not all linearly independent, so the system
is underdetermined in general when D ≥ 3. Moreover, the following theorem sometimes
gives additional equations.
Theorem 2.1 ([10, Theorem 3], cf. [7], [5, Theorem 2.3.2]). Let (X, {Ri}Di=0) be an
association scheme of D classes with second eigenmatrix Q and Krein parameters qtrs
(0 ≤ r, s, t ≤ D). Then,
qtrs = 0 ⇐⇒
D∑
i,j,k=0
QirQjsQkt
[
x y z
i j k
]
= 0 for all x, y, z ∈ X.
Note that in a Q-polynomial association scheme, many Krein parameters are zero, and
we can use Theorem 2.1 to obtain an equation for each of them.
3 Tables of feasible parameters for Q-polynomial association schemes
In this section we will describe the tables of feasible parameter sets for primitive 3-class
Q-polynomial schemes and 4- and 5-class Q-bipartite schemes.
These tables were all completed using the MAGMA programming language (see [2]).
Any parameter set meeting the following conditions was included in the table:
(1) The parameters satisfy the Q-polynomial condition.
(2) All pkij are nonnegative integers, all valencies p
0
jj are positive.
(3) For each j > 0 we have np0jj is even (the handshaking lemma applied to the graph
(X,Rj)).
A. L. Gavrilyuk et al.: On few-class Q-polynomial association schemes: feasible . . . 109
(4) For each j, k > 0 we have p0jjp
j
jk is even (the handshaking lemma applied to the
subconstituent (Y, {(y, z) ∈ Y × Y | (y, z) ∈ Rk}), where x ∈ X and Y = {y ∈
X | (x, y) ∈ Rj}).
(5) For each j > 0 we have np0jjp
j
jj is divisible by 6 (the number of triangles in each
graph (X,Rj) is integral).
(6) All qkij are nonnegative and for each j the multiplicity q
0
jj (i.e., the dimension of the
Ej-eigenspace) is a positive integer (see [5, Proposition 2.2.2]).
(7) For all i, j we have
∑
qkij ̸=0
mk ≤ mimj if i ̸= j and
∑
qkii ̸=0
mk ≤ mi(mi−1)2 (the
absolute bound, see [5, Theorem 2.3.3] and the references therein).
(8) The splitting field is at most a degree 2 extension of the rationals (see [29]).
We note that there are many other conditions known for the special case of distance-
regular graphs. It was decided to apply these conditions after the construction of the table,
and those not meeting these extra conditions were labelled as nonexistent with a note as
to the condition not met. We leave as an open question whether if any of these conditions
could be generalized to any cases beyond distance-regular graphs; this (perhaps faint) hope
is the main reason that they are included in the table.
We begin with the tables for Q-bipartite schemes, since this case is somewhat sim-
pler than the primitive case. Schemes which are Q-bipartite are formally dual to bipartite
distance-regular graphs. As a consequence, the formal dual to [5, Theorem 4.2.2(i)] gives
the Krein array for the quotient scheme of a Q-bipartite scheme (see [27]). Namely, if the
scheme has Krein array {b∗0, b∗1, . . . , b∗D−1; c∗1, . . . , c∗D} and q211 = µ∗, then the Krein array
of the quotient is{
b∗0b
∗
1
µ∗
,
b∗2b
∗
3
µ∗
, . . . ,
b∗2t−2b
∗
2t−1
µ∗
;
c∗1c
∗
2
µ∗
,
c∗3c
∗
4
µ∗
, . . . ,
c∗2t−1c
∗
2t
µ∗
}
,
where t = ⌊D2 ⌋. Note that the quotient scheme has multiplicities 1,m2,m4, . . . ,m2t,
from which it follows that the condition
∑t
i=0 m2i =
∑D−t
i=1 m2i−1 must be satisfied for a
D-class Q-bipartite scheme.
When D = 4, 5 we obtain t = 2, so the quotient structure is a strongly regular graph.
A database of strongly regular graph parameters up to 5000 vertices can be generated very
quickly. From there, we can use the above condition on the multiplicities. The following
proposition shows that the multiplicities determine all the parameters of the scheme.
Proposition 3.1. A D-class Q-bipartite Q-polynomial association scheme with D ∈ {4, 5}
and multiplicities 1,m1,m2, . . . ,mD has the Krein array{
m1,m1 − 1,
m1(m2 −m1 + 1)
m2
,
m1(m3 −m2 +m1 − 1)
m3
;
1,
m1(m1 − 1)
m2
,
m1(m2 −m1 + 1)
m3
,m1
} (D = 4)
or {
m1,m1−1,
m1(m2−m1+1)
m2
,
m1(m3−m2+m1−1)
m3
,
m1(m4−m3+m2−m1+1)
m4
;
1,
m1(m1−1)
m2
,
m1(m2−m1+1)
m3
,
m1(m3−m2+m1−1)
m4
,m1
}
(D = 5).
110 Ars Math. Contemp. 20 (2021) 103–127
Proof. Follows easily from the identities of [5, Lemma 2.3.1].
In the 4-class case, the parameters are entirely determined by the quotient’s multiplici-
ties (with a chosen Q-polynomial ordering) and m1. To search, we take a strongly regular
graph parameter set, choose one of two possible orderings for its multiplicities, calling its
multiplicities m0 = 1, m2, m4. From the absolute bound, we have 1 +m2 ≤ m1(m1+1)2 ,
and from the positivity of c∗2 we have
(m2−m1+1)m1
m2
≥ 0. We then search over all√
2(1 +m2) − 12 ≤ m1 ≤ m2, checking the conditions above. Given that we are iter-
ating over SRG parameters together with two orderings and one integer, this search is very
fast. The limitation of the table to 10000 vertices is mainly readability and practicality. The
third author has unpublished tables (without comments or details) to 100000 vertices.
We note that Q-bipartite schemes with 5 classes are very similar, except we must iterate
over both m1 and m3. Again, this is a very quick search, but the relative scarcity of 5-class
parameter sets makes listing up to 50000 vertices, with annotation, manageable. The table
actually goes slightly higher, to 50520 vertices, because of the existence of an example on
that number of vertices.
The trickiest search was the primitive 3-class Q-polynomial parameter sets. In this
case, there is no non-trivial quotient scheme to build on.
We use the following observation.
Theorem 3.2. A primitive Q-polynomial association scheme of 3 classes must have a ma-
trix Li with 4 distinct eigenvalues.
Proof. Assume not. If a matrix Ai has only two distinct eigenvalues, it is either complete,
contradicting the fact that it is a 3-class scheme, or a disjoint union of more than one
complete graph, contradicting the fact the scheme is primitive. Therefore, the only case left
to consider is when A1, A2, A3 all have three distinct eigenvalues, meaning the graphs are
all strongly regular. A 3-class scheme where every non-trivial relation is strongly-regular
is amorphic, see [19] and [14] for a definition and details on amorphic schemes. It was
shown in [20] that amorphic schemes are formally self-dual. This implies that no column
of Q has 4 distinct entries. Therefore, the second eigenmatrix Q cannot be generated by
one column via polynomials, thus the scheme cannot be Q-polynomial.
We note that, in fact, all Q-polynomial D-class schemes must have a relation with D+1
distinct eigenvalues. However, the above theorem and its proof is sufficient for our needs.
From this we conclude that each 3-class primitive Q-polynomial scheme has an ad-
jacency matrix, which we label A1, which has four distinct eigenvalues. Then the corre-
sponding 4 × 4 intersection matrix L1 has four distinct eigenvalues. From this matrix, all
of the other parameters may be determined. In particular, from [5, Proposition 2.2.2], the
left-eigenvectors of L1, normalized so their leftmost entry is 1, must be the rows of P .
The rest of the parameters can be derived from the equations:
Lj = P
−1 diag(P0j , P1j , . . . , PDj)P,
L∗j = Q
−1 diag(Q0j , Q1j , . . . , QDj)Q.
However, checking the Q-polynomial condition is done before the computation of all
parameters. We use the following theorem, a proof of which can be found in [30].
A. L. Gavrilyuk et al.: On few-class Q-polynomial association schemes: feasible . . . 111
Theorem 3.3. Let Li be an intersection matrix of a D-class association scheme, where Li
has exactly D + 1 distinct eigenvalues. Then the scheme is Q-polynomial if and only if
there is a Vandermonde matrix U such that U−1LiU = T where T is upper triangular.
It is not hard to show that, without loss of generality, we can take T01 to be 0, implying
that the first column of U is an eigenvector of L1. We only then need to iterate over the
three (nontrivial) eigenvectors of L1 to check this condition. If the Q-polynomial condition
is met, the rest of the parameters are computed and checked for the above conditions.
The schemes are then split into types depending on whether there is a strongly regular
graph as a relation, and whether the splitting field is rational or not. These are split in this
manner to aid in computation (following the list of types we give details on how these were
used):
(1) Diameter 3 distance-regular graphs (DRG for short).
(2) No diameter 3 DRG, there is a strongly regular graph as a relation, the splitting field
is the rational field.
(3) No diameter 3 DRG, there is a strongly regular graph as a relation, the splitting field
is a degree-2 extension of the rational field.
(4) No diameter 3 DRG, there is no strongly regular graph as a relation, the splitting field
is the rational field.
(5) No diameter 3 DRG, there is no strongly regular graph as a relation, the splitting field
is a degree-2 extension of the rational field.
We note that we do not have any examples of primitive 3-class Q-polynomial schemes
with an irrational splitting field, but there are open parameter sets of such (for example, see
entry ⟨216, 20⟩ in the third author’s primitive 3-class table at [39]). It would be interesting
to determine if these exist. We also point out that all the feasible parameter sets known to
us have rational Krein parameters.
Type 1. For DRG’s, we iterated over the number of vertices, intersection array and valen-
cies. The order was n, b0 = n1, b1, n2 (noting n2 is a divisor of n1b1), then b2 (noting b2
must be a multiple of n3gcd(n2,n3) , where n3 = n− n1 − n2), from which the rest could be
determined.
When there is no DRG, it is tempting to try to formally dualize the above process.
However, the Krein parameters of a scheme do not have to be integral, or even rational.
For this reason, it seemed more advantageous to iterate over parameters that needed to be
integral, namely the parameters pkij . All arithmetic was done in MAGMA using the rational
field, or a splitting field of a degree two irreducible polynomial over the rationals. Floating
point arithmetic was avoided to minimize numerical errors.
For the rest of the types, L1 and the valencies were iterated over. In particular, the
parameters a = p112, b = p
1
13 and c = p
2
13, together with n, n1, n2 determine the rest of L1,
noting that a + b ≤ n1 − 1 and c ≤ n1 − n1an2 . Any matrix without 4 distinct eigenvalues
or with an irreducible cubic factor in its characteristic polynomial was discarded.
Types 2 and 3. For these types, we iterate over strongly regular graphs first, with param-
eters (n, k, λ, µ). We choose A3 to be the adjacency matrix of the strongly regular graph
relation, and L1, L2 to be fissions of the complement. Given this, the choice of n1 will
112 Ars Math. Contemp. 20 (2021) 103–127
determine n2. The possibilities for n1 can be narrowed by observing that p133 = µ, n3 = k
and p133n1 = p
3
13n3, implying that n1 is divisible by
n3
gcd(n3,µ)
.
Using similar identities, we find b is divisible by n3gcd(n1,n3) , a is divisible by
n2
gcd(n1,n2)
,
and c = n1(n3−b−µ)n2 . After choosing these parameters all of L1 follows.
Types 4 and 5. For these types, we know L1, L2 and L3 all have 4 distinct eigenvalues.
Therefore, we can assume n1 is the smallest valency, and that a ≤ b. Using a is divisible
by n2gcd(n1,n2) , b is divisible by
n3
gcd(n1,n3)
, and n2 divides an1, we choose n1, a, n2, b, c,
from which the rest is determined. This is the slowest part of the search, and the reason the
primitive table goes to 2800 vertices.
We close this section with some comments on the irrational splitting field types. The
2-class primitive Q-polynomial association schemes are equivalent to complementary pairs
of primitive strongly regular graphs. The only case where strongly regular graphs have an
irrational splitting field is the so-called “half-case”, when the graph has valency n−12 . Such
graphs do exist, for example the Paley graphs for non-square prime powers q with q ≡ 1
(mod 4). We note that no primitive Q-polynomial schemes with more than 2 classes and a
quadratic splitting field are known. All feasible parameter sets we know of are 3-class and
have a strongly regular graph relation (type 3). The corresponding strongly regular graphs
are also all unknown (see [4]). We have no feasible parameter set for type 5. However, one
type 5 parameter set satisfied all criteria except the handshaking lemma. Given this, we
expect feasible parameter sets for type 5 to exist, but may be quite large. This parameter
set is listed below (including L∗1, so it can be seen it is Q-polynomial, but not including the
other L∗i matrices), though this set is not included in the online table:
P =

1 285 285 405
1 19+8
√
19 −38+1
√
19 18−9
√
19
1 −3 5 −3
1 19−8
√
19 −38−1
√
19 18+9
√
19
, Q=

1 60 855 60
1 76+32
√
19
19
−9 76−32
√
19
19
1 −152+4
√
19
19
15 −152−4
√
19
19
1 8−4
√
19
3
−19
3
8+4
√
19
3
,
L1=

0 285 0 0
1 116 60 108
0 60 90 135
0 76 95 114
 , L2=

0 0 285 0
0 60 90 135
1 90 59 135
0 95 95 95
 , L3=

0 0 0 405
0 108 135 162
0 135 135 135
1 114 95 195
 ,
L∗1 =

0 60 0 0
1 400+32
√
19
61
3199−32
√
19
61
0
0 12796−128
√
19
3477
181184+128
√
19
3477
80
19
0 0 60 0
 .
While feasible parameters may exist, the complete lack of examples elicits the follow-
ing question:
Question 3.4. Do all 3-class primitive Q-polynomial schemes have a rational splitting
field?
This is a special case of the so-called “Sensible Caveman” conjecture of William J. Mar-
tin:
Conjecture 3.5. For Q-polynomial schemes of 3 or more classes that is not a polygon, if
the scheme is primitive then its splitting field is rational.
A. L. Gavrilyuk et al.: On few-class Q-polynomial association schemes: feasible . . . 113
4 Nonexistence results
We derived our nonexistence results by analyzing triple intersection numbers of Q-poly-
nomial association schemes. For some choice of relations Rr, Rs, Rt, the system of Dio-
phantine equations derived from (2.3) and Theorem 2.1 may have multiple nonnegative
solutions, each giving the possible values of the triple intersection numbers with respect to
a triple (x, y, z) with (x, y) ∈ Rr, (x, z) ∈ Rs and (y, z) ∈ Rt. However, in certain cases,
there might be no nonnegative solutions – in this case, we may conclude that an association
scheme with the given parameters does not exist.
Even when there are solutions for all choices of Rr, Rs, Rt such that ptrs ̸= 0, some-
times nonexistence can be derived by other means. We may, for example, employ double
counting.
Proposition 4.1. Let x and y be vertices of an association scheme with (x, y) ∈ Rr.
Suppose that α1, α2, . . . , αm are distinct integers such that there are precisely κℓ vertices
z with (x, z) ∈ Rs, (y, z) ∈ Rt and
[
x y z
i j k
]
= αℓ (1 ≤ ℓ ≤ m,
∑m
ℓ=1 κℓ = p
r
st),
and β1, β2, . . . , βn are distinct integers such that there are precisely λℓ vertices w with
(w, x) ∈ Ri, (w, y) ∈ Rj and
[
w x y
k s t
]
= βℓ (1 ≤ ℓ ≤ n,
∑n
ℓ=1 λℓ = p
r
ij). Then,
m∑
ℓ=1
κℓαℓ =
n∑
ℓ=1
λℓβℓ.
Proof. Count the number of pairs (w, z) with (x, z) ∈ Rs, (y, z) ∈ Rt, (w, x) ∈ Ri,
(w, y) ∈ Rj and (w, z) ∈ Rk.
We consider the special case of Proposition 4.1 when a triple intersection number is
zero for all triples of vertices in some given relations.
Corollary 4.2. Suppose that for all vertices x, y, z of an association scheme with (x, y) ∈
Rr, (x, z) ∈ Rs, (y, z) ∈ Rt,
[
x y z
i j k
]
= 0 holds. Then,
[
w x y
k s t
]
= 0 holds for all
vertices w, x, y with (w, x) ∈ Ri, (w, y) ∈ Rj and (x, y) ∈ Rr.
Proof. Apply Proposition 4.1 to all (x, y) ∈ Rr, with m ≤ 1 and α1 = 0. Since βℓ and λℓ
(1 ≤ ℓ ≤ n) must be nonnegative, it follows that n ≤ 1 and β1 = 0.
4.1 Computer search
The sage-drg package [38, 37] by the second author for the SageMath computer al-
gebra system [32] has been used to perform computations of triple intersection numbers
of Q-polynomial association schemes with Krein arrays that were marked as open in the
tables of feasible parameter sets by the third author [39], see Section 3. The package
was originally developed for the purposes of feasibility checking for intersection arrays of
distance-regular graphs and included a routine to find general solutions to the system of
equations for computing triple intersection numbers.
For the purposes of the current research, the package has been extended to support
parameters of general association schemes, in particular, given as Krein arrays of Q-poly-
nomial association schemes. Additionally, the package now supports generating integral
solutions for systems of equations with constraints on the solutions (e.g., nonnegativity of
114 Ars Math. Contemp. 20 (2021) 103–127
triple intersection numbers) – these can also be added on-the-fly. The routine uses Sage-
Math’s mixed integer linear programming facilities, which support multiple solvers. We
have used SageMath’s default GLPK solver [26] and the CBC solver [16] in our computa-
tions – however, other solvers can also be used if they are available.
We have thus been able to implement an algorithm which tries to narrow down the
possible solutions of the systems of equations for determining triple intersection numbers
of an association scheme such that they satisfy Corollary 4.2, and conclude inequality if
any of the systems of equations has no such feasible solutions.
(1) For each triple of relations (Rr, Rs, Rt) such that ptrs > 0, initialize an empty
set of solutions, obtain a general (i.e., parametric) solution to the system of equa-
tions derived from (2.3) and Theorem 2.1, and initialize a generator of solutions
with the constraint that the intersection numbers be integral and nonnegative. All
generators (r, s, t) are initially marked as active, and all triple intersection num-
bers (r, s, t; i, j, k) (representing
[
x y z
i j k
]
with (x, y) ∈ Rr, (x, z) ∈ Rs and
(y, z) ∈ Rt) are initially marked as unknown.
(2) For each active generator, generate one solution and add it to the corresponding set
of solutions. If a generator does not return a new solution (i.e., it has exhausted all
of them), then mark it as inactive.
(3) For each inactive generator, verify that the corresponding set of solutions is non-
empty – otherwise, terminate and conclude nonexistence.
(4) Initialize an empty set Z.
(5) For each unknown triple intersection number (r, s, t; i, j, k), mark it as nonzero if a
solution has been found in which its value is not zero. If such a solution has not been
found yet, make a copy of the generator (r, s, t) with the constraint that (r, s, t; i, j, k)
be nonzero, and generate one solution. If such a solution exists, add it to the set of
solutions and mark (r, s, t; i, j, k) as nonzero, otherwise mark (r, s, t; i, j, k) as zero
and add it to Z.
(6) If Z is empty, terminate without concluding nonexistence.
(7) For each triple intersection number (r, s, t; i, j, k) ∈ Z and for each nonzero (a, b, c;
d, e, f) ∈ {(r, i, j; s, t, k), (s, i, k; r, t, j), (t, j, k; r, s, i)}, remove all solutions from
the corresponding set in which the value of the latter is nonzero, mark (a, b, c; d, e, f)
as zero, mark all nonzero (a, b, c; ℓ,m, n) with (ℓ,m, n) ̸= (d, e, f) as unknown, and
add a constraint that (a, b, c; d, e, f) be zero to the generator (a, b, c) if it is active.
(8) Go to (2).
Note that generators and triple intersection numbers are considered equivalent under
permutation of vertices, i.e., under actions (r, s, t) 7→ (r, s, t)π and (r, s, t; i, j, k) 7→
((r, s, t)π; (i, j, k)π
(1 3)
) for π ∈ S3.
The above algorithm is available as the check_quadruples method of sage-drg’s
ASParameters class. We ran it for all open cases in the tables from Section 3, and ob-
tained 29 nonexistence results for primitive 3-class schemes, 92 nonexistence results for
Q-bipartite 4-class schemes, and 11 nonexistence results for Q-bipartite 5-class schemes.
The results are summarized in the following theorem and in the tables in Appendix A.
A. L. Gavrilyuk et al.: On few-class Q-polynomial association schemes: feasible . . . 115
Theorem 4.3. A Q-polynomial association scheme with Krein array listed in one of Ta-
bles 1, 2 and 3 does not exist.
Proof. In all but two cases, it suffices to observe that for some triple of relations Rr, Rs, Rt,
the system of equations derived from (2.3) and Theorem 2.1 has no integral nonnegative
solutions – Tables 1 and 2 list the triple (r, s, t), while for all examples in Table 3, this is
true for (r, s, t) = (1, 1, 1). Note that the natural ordering of the relations is used.
Let us now consider the cases ⟨225, 24⟩ and ⟨1470, 104⟩ from Table 1. In the first case,
the Krein array is {24, 20, 36/11; 1, 30/11, 24}. Such an association scheme has two Q-
polynomial orderings, so we can augment the system of equations (2.3) with six equations
derived from Theorem 2.1. Let w, x, y, z be vertices such that (x, z), (y, z) ∈ R1 and
(w, x), (w, y), (x, y) ∈ R3. Since p311 = 22 and p333 = 3, such vertices must exist. We
first compute the triple intersection numbers with respect to x, y, z. There are two integral
nonnegative solutions, both having [3 3 1] = 0. On the other hand, there is a single solution
for the triple intersection numbers with respect to w, x, y, giving [1 1 1] = 3. However, this
contradicts Corollary 4.2, so such an association scheme does not exist.
In the second case, the Krein array is {104, 70, 25; 1, 7, 80}. Let w, x, y, z be vertices
such that (x, y), (x, z) ∈ R1, (w, y), (y, z) ∈ R2 and (w, x) ∈ R3. Since p112 = 70 and
p132 = 250, such vertices must exist. There is a single solution for the triple intersection
numbers with respect to x, y, z, giving [3 2 3] = 0. On the other hand, there are four
solutions for the triple intersection numbers with respect to w, x, y, from which we obtain
[3 1 2] ∈ {15, 16, 17, 18}. Again, this contradicts Corollary 4.2, so such an association
scheme does not exist. This completes the proof.
Remark 4.4. The sage-drg package repository provides two Jupyter notebooks con-
taining the computation details in the proofs of nonexistence of two cases from Table 1:
• QPoly-24-20-36_11-1-30_11-24.ipynb for the case ⟨225, 24⟩, and
• DRG-104-70-25-1-7-80.ipynb for the case ⟨1470, 104⟩.
Remark 4.5. The parameter set ⟨91, 12⟩ from Table 1 was listed by Van Dam [12] as the
smallest feasible Q-polynomial parameter set for which no scheme is known. The next
such open case is now the Krein array {14, 108/11, 15/4; 1, 24/11, 45/4} for a primitive
3-class Q-polynomial association scheme with 99 vertices, which was also listed by Van
Dam.
Since some of the parameters from Table 1 also admit a P -polynomial ordering, we
can derive nonexistence of distance-regular graphs with certain intersection arrays. We
have also found an intersection array for a primitive Q-polynomial distance-regular graph
of diameter 4, which is listed in [5] and [3], and for which, to the best of our knowledge,
nonexistence has not been previously known.
Theorem 4.6. There is no distance-regular graph with intersection array
{83, 54, 21; 1, 6, 63},
{104, 70, 25; 1, 7, 80},
{195, 160, 28; 1, 20, 168},
{125, 108, 24; 1, 9, 75},
{126, 90, 10; 1, 6, 105}, or
{203, 160, 34; 1, 16, 170}.
116 Ars Math. Contemp. 20 (2021) 103–127
Proof. The cases ⟨1080, 83⟩, ⟨1470, 104⟩, ⟨2016, 195⟩ and ⟨2640, 203⟩ from Table 1 are
formally self-dual for the natural ordering of relations, while ⟨2197, 126⟩ is formally self-
dual with ordering of relations A2, A3, A1 relative to the natural ordering. In each case,
the corresponding association scheme is P -polynomial with intersection array equal to
the Krein array. The case ⟨2106, 65⟩ is not formally self-dual, yet the natural ordering
of relations is P -polynomial with intersection array {125, 108, 24; 1, 9, 75}. In all of the
above cases, Theorem 4.3 implies nonexistence of the corresponding association scheme,
so a distance-regular graph with such an intersection array does not exist.
Theorem 4.7. There is no distance-regular graph with intersection array
{53, 40, 28, 16; 1, 4, 10, 28}.
Proof. Consider a distance-regular graph with intersection array {53, 40, 28, 16; 1, 4, 10,
28}. Such a graph is formally self-dual for the natural ordering of eigenspaces and there-
fore also Q-polynomial. Augmenting the system of equations (2.3) with twelve equations
derived from Theorem 2.1 gives a two parameter solution for triple intersection numbers
with respect to three vertices mutually at distances 1, 3, 3. However, it turns out that there
is no integral solution, leading to nonexistence of the graph.
Remark 4.8. The non-existence of a distance-regular graph with intersection array {53, 40,
28, 16; 1, 4, 10, 28} also follows by applying the Terwilliger polynomial [17]. Recall that
this polynomial, say TΓ(x), which depends only on the intersection numbers of a Q-
polynomial distance-regular graph Γ and its Q-polynomial ordering, satisfies:
TΓ(η) ≥ 0, (4.1)
where η is any non-principal eigenvalue of the local graph of an arbitrary vertex x of Γ.
Furthermore, by [5, Theorem 4.4.3(i)], η satisfies
−1− b1
θ1 + 1
≤ η ≤ −1− b1
θD + 1
, (4.2)
where b0 = θ0 > θ1 > · · · > θD are the D + 1 distinct eigenvalues of Γ.
For the above-mentioned intersection array, TΓ(x) is a polynomial of degree 4 with a
negative leading term and the following roots: − 73 (= −1−
b1
θ1+1
), 9−
√
249
4 ≈ −1.695,
17
3
(= −1− b1θD+1 ),
9+
√
249
9 ≈ 6.195.
Thus, combining (4.1) and (4.2), we obtain
−7
3
≤ η ≤ 9−
√
249
4
or η =
17
3
,
and one can finally obtain a contradiction as in [18, Claim 4.3].
4.2 Infinite families
The data from Tables 1, 2 and 3 allows us to look for infinite families of Krein arrays for
which we can show nonexistence of corresponding Q-polynomial association schemes. We
find three families, one for each number of classes.
A. L. Gavrilyuk et al.: On few-class Q-polynomial association schemes: feasible . . . 117
The first family of Krein arrays is given by
{2r2 − 1, 2r2 − 2, r2 + 1; 1, 2, r2 − 1}. (4.3)
This Krein array is feasible for all integers r ≥ 2. A Q-polynomial association scheme
with Krein array (4.3) has 3 classes and 4r4 vertices. Examples exist when r is a power of
2 – they are realized by duals of Kasami codes with minimum distance 5, see [5, §11.2].
Theorem 4.9. A Q-polynomial association scheme with Krein array (4.3) and r odd does
not exist.
Proof. Consider a Q-polynomial association scheme with Krein array (4.3). Besides the
Krein parameters failing the triangle inequality, q111 is also zero. Therefore, in order to
compute triple intersection numbers, the system of equations (2.3) can be augmented with
four equations derived from Theorem 2.1. We compute triple intersection numbers with
respect to vertices x, y, z such that (x, y), (x, z) ∈ R1 and (y, z) ∈ R2. Since p211 =
r(r+2)(r2 − 1)/4 > 0, such vertices must exist. We obtain a four parameter solution (see
the notebook QPoly-d3-1param-odd.ipynb on the sage-drg package repository
for computation details). Then we may express
[1 2 3] = −r
4
2
+ 2r2 + [1 3 1] + 3 · [2 3 3]− [3 1 1] + 4 · [3 3 3].
Clearly, the above triple intersection number can only be integral when r is even. Therefore,
we conclude that a Q-polynomial association scheme with Krein array (4.3) and r odd does
not exist.
The next family is a two parameter family of Krein arrays
{m,m− 1,m(r2 − 1)/r2,m− r2 + 1; 1,m/r2, r2 − 1,m}. (4.4)
This Krein array is feasible for all integers m and r such that 0 < 2(r2 − 1) ≤ m ≤
r(r − 1)(r + 2) and m(r + 1) is even. A Q-polynomial association scheme with Krein
array (4.4) is Q-bipartite with 4 classes and 2m2 vertices. One may take the Q-bipartite
quotient of such a scheme (i.e., identify vertices in relation R4) to obtain a strongly regular
graph with parameters (n, k, λ, µ) = (m2, (m − 1)r2,m + r2(r2 − 3), r2(r2 − 1)), i.e.,
a pseudo-Latin square graph. Therefore, we say that a scheme with Krein array (4.4) is of
Latin square type.
There are several examples of Q-polynomial association schemes with Krein array (4.4)
for some r and m. For (r,m) = (2, 6) and (r,m) = (3, 16), this Krein array is realized
by the schemes of shortest vectors of the E6 lattice and an overlattice of the Barnes-Wall
lattice in R16 [28], respectively. For (r,m) = (2ij , 2i(2j+1)), there are examples arising
from duals of extended Kasami codes [5, §11.2] for each choice of positive integers i and
j. In particular, the Krein array obtained by setting i = j = 1 uniquely determines the
halved 8-cube.
In the case when r is a prime power and m = r3, the formal dual of this parameter
set (i.e., a distance-regular graph with the corresponding intersection array) is realized by
a Pasechnik graph [6].
Theorem 4.10. A Q-polynomial association scheme with Krein array (4.4) and m odd
does not exist.
118 Ars Math. Contemp. 20 (2021) 103–127
Proof. Consider a Q-polynomial association scheme with Krein array (4.4). Since the
scheme is Q-bipartite, we have qkij = 0 whenever i+ j+k is odd or the triple (i, j, k) does
not satisfy the triangle inequality. This allows us to augment the system of equations (2.3)
with many equations derived from Theorem 2.1. We compute triple intersection numbers
with respect to vertices x, y, z such that (x, y), (x, z) ∈ R1 and (y, z) ∈ R2. Since p211 =
r2(r2 − 1)/2 > 0, such vertices must exist. We obtain a one parameter solution (see
the notebook QPoly-d4-LS-odd.ipynb on the sage-drg package repository for
computation details) which allows us to express
[1 1 3] = r +
r2(1− r)
2
− m
2
+ [1 1 1].
Clearly, the above triple intersection number can only be integral when m is even. There-
fore, we conclude that a Q-polynomial association scheme with Krein array (4.4) and m
odd does not exist.
The last family is given by the Krein array{
r2 + 1
2
,
r2 − 1
2
,
(r2 + 1)2
2r(r + 1)
,
(r − 1)(r2 + 1)
4r
,
r2 + 1
2r
;
1,
(r − 1)(r2 + 1)
2r(r + 1)
,
(r + 1)(r2 + 1)
4r
,
(r − 1)(r2 + 1)
2r
,
r2 + 1
2
}
.
(4.5)
This Krein array is feasible for all odd r ≥ 5. A Q-polynomial association scheme with
Krein array (4.5) is Q-bipartite with 5 classes and 2(r+1)(r2 +1) vertices. One may take
the Q-bipartite quotient of such a scheme to obtain a strongly regular graph with parameters
(n, k, λ, µ) = ((r+1)(r2 +1), r(r+1), r− 1, r+1) – these are precisely the parameters
of collinearity graphs of generalized quadrangles GQ(r, r). The scheme also has a second
Q-polynomial ordering of eigenspaces, namely the ordering E5, E2, E3, E4, E1 relative to
the ordering implied by the Krein array. For r ≡ 1 (mod 4) a prime power, the Krein
array (4.5) is realized by a scheme derived by Moorhouse and Williford [30] from a double
cover of the C2(r) dual polar graph.
Theorem 4.11. A Q-polynomial association scheme with Krein array (4.5) and r ≡ 3
(mod 4) does not exist.
Proof. Consider a Q-polynomial association scheme with Krein array (4.5). Since the
scheme is Q-bipartite, we have qkij = 0 whenever i + j + k is odd or the triple (i, j, k)
does not satisfy the triangle inequality. This allows us to augment the system of equa-
tions (2.3) with many equations derived from Theorem 2.1. We compute triple inter-
section numbers with respect to vertices x, y, z that are mutually in relation R1. Since
p111 = (r − 1)/2 > 0, such vertices must exist. We obtain a single solution (see the note-
book QPoly-d5-1param-3mod4.ipynb on the sage-drg package repository for
computation details) with
[1 1 1] =
r − 5
4
.
Clearly, the above triple intersection number can only be integral when r ≡ 1 (mod 4).
Therefore, we conclude that a Q-polynomial association scheme with Krein array (4.5) and
r ≡ 3 (mod 4) does not exist.
A. L. Gavrilyuk et al.: On few-class Q-polynomial association schemes: feasible . . . 119
5 Quadruple intersection numbers
The argument of the proof of Theorem 2.1 ([5, Theorem 2.3.2]) can be further extended
to s-tuples of vertices (see Remark (iii) in [5, §2.3]; cf. [34, Lemma 4(2)]). In particular,
we may consider quadruple intersection numbers with respect to a quadruple of vertices
w, x, y, z ∈ X . For integers h, i, j, k (0 ≤ h, i, j, k ≤ D), denote by
[
w x y z
h i j k
]
(or
simply [h i j k] when it is clear which quadruple (w, x, y, z) we have in mind) the number
of vertices u ∈ X such that (u,w) ∈ Rh, (u, x) ∈ Ri, (u, y) ∈ Rj , and (u, z) ∈ Rk.
For a fixed quadruple (w, x, y, z), one can obtain a system of linear Diophantine equa-
tions with quadruple intersection numbers as variables which relates them to the intersec-
tion numbers (or to the triple intersection numbers).
The following analogue of Theorem 2.1 allows us to obtain some additional equations.
Theorem 5.1. Let (X, {Ri}Di=0) be an association scheme of D classes with second eigen-
matrix Q and Krein parameters qkij (0 ≤ i, j, k ≤ D). Then, for fixed indices ι1, ι2, ι3, ι4
(0 ≤ ι1, ι2, ι3, ι4 ≤ D) and any permutation p, r, s, t of ι1, ι2, ι3, ι4,
D∑
ℓ=0
qℓprq
ℓ
st = 0 ⇐⇒
D∑
h,i,j,k=0
QhpQirQjsQkt
[
w x y z
h i j k
]
= 0 for all w, x, y, z ∈ X .
Proof. Since Ei is a symmetric idempotent matrix, one has∑
w∈X
Ei(u,w)Ei(v, w) = Ei(u, v). (5.1)
Let Σ(M) denote the sum of all entries of a matrix M . Then, by (5.1),
Σ(Ep ◦ Er ◦ Es ◦ Et) =
∑
u,v∈X
Ep(u, v)Er(u, v)Es(u, v)Et(u, v)
=
∑
w,x,y,z∈X
(∑
u∈X
Ep(u,w)Er(u, x)Es(u, y)Et(u, z)
)
·
(∑
v∈X
Ep(v, w)Er(v, x)Es(v, y)Et(v, z)
)
=
∑
w,x,y,z∈X
σ(w, x, y, z)2 ≥ 0, (5.2)
where σ(w, x, y, z) =
∑
u∈X Ep(u,w)Er(u, x)Es(u, y)Et(u, z).
On the other hand, by (2.1),
|X|2 Σ(Ep ◦ Er ◦ Es ◦ Et) = |X|2 Tr((Ep ◦ Er) · (Es ◦ Et))
= Tr
((
D∑
ℓ=0
qℓprEℓ
)
·
(
D∑
ℓ=0
qℓstEℓ
))
=
D∑
ℓ=0
mℓq
ℓ
prq
ℓ
st, (5.3)
120 Ars Math. Contemp. 20 (2021) 103–127
where mℓ is the rank of Eℓ (i.e., the multiplicity of the corresponding eigenspace), and by
(2.2),
|X|3 Σ(Ep ◦ Er ◦ Es ◦ Et) =
1
|X|
D∑
ℓ=0
QℓpQℓrQℓsQℓtΣ(Aℓ)
=
D∑
ℓ=0
nℓQℓpQℓrQℓsQℓt, (5.4)
where nℓ is the valency of (X,Rℓ).
Since the multiplicities mℓ are positive numbers and the Krein parameters are non-
negative numbers, by (5.2), (5.3), (5.4), we have Σ(Ep ◦ Er ◦ Es ◦ Et) = 0 if and only if
qℓprq
ℓ
st = 0 (with fixed p, r, s, t) for all ℓ = 0, . . . , D. In this case, we have σ(w, x, y, z) = 0
for all quadruples (w, x, y, z), which implies
0 = |X|4 σ(w, x, y, z) = |X|4
∑
u∈X
Ep(u,w)Er(u, x)Es(u, y)Et(u, z)
=
D∑
h,i,j,k=0
QhpQirQjsQkt
[
w x y z
h i j k
]
,
which completes the proof.
The condition of Theorem 5.1 is satisfied when, for example, an association scheme is
Q-bipartite, i.e., qkij = 0 whenever i+ j+k is odd (take p+ r and s+ t of different parity).
Suda [33] lists several families of association schemes which are known to be triply reg-
ular, i.e., their triple intersection numbers
[
x y z
i j k
]
only depend on i, j, k and the mutual
distances between x, y, z, and not on the choices of the vertices themselves:
• strongly regular graphs with q111 = 0 (cf. [8]),
• Taylor graphs (antipodal Q-bipartite schemes of 3 classes),
• linked systems of symmetric designs (certain Q-antipodal schemes of 3 classes) with
a∗1 = 0,
• tight spherical 7-designs (certain Q-bipartite schemes of 4 classes), and
• collections of real mutually unbiased bases (Q-antipodal Q-bipartite schemes of 4
classes).
Schemes belonging to the above families seem natural candidates for the computations of
their quadruple intersection numbers. However, the condition of Theorem 5.1 is never sat-
isfied for primitive strongly regular graphs, while for Taylor graphs the obtained equations
do not give any information that could not be obtained through relating the quadruple inter-
section numbers to the triple intersection numbers. This was also the case for the examples
of triply regular linked systems of symmetric designs that we have checked. However, in
the cases of tight spherical 7-designs and mutually unbiased bases, we do get new restric-
tions on quadruple intersection numbers. So far, we have not succeeded in using this new
information for either new constructions or proofs of nonexistence.
A. L. Gavrilyuk et al.: On few-class Q-polynomial association schemes: feasible . . . 121
ORCID iDs
Alexander L. Gavrilyuk https://orcid.org/0000-0001-9296-0313
Janoš Vidali https://orcid.org/0000-0001-8061-9169
Jason S. Williford https://orcid.org/0000-0002-8697-5997
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124 Ars Math. Contemp. 20 (2021) 103–127
Appendix A Tables of nonexistence results
Here, we give the tables of nonexistence results obtained by running the algorithm from
Subsection 4.1 on the open cases in the tables from Section 3. Tables 1, 2 and 3 give
nonexistence results for Q-polynomial schemes which are primitive of 3 classes, and Q-
bipartite (but not Q-antipodal) of 4 and 5 classes, respectively.
Label Krein array DRG Nonexistence Family
⟨91, 12⟩ {12, 338
35
, 39
25
; 1, 312
175
, 39
5
} (3, 3, 3)
⟨225, 24⟩ {24, 20, 36
11
; 1, 30
11
, 24} (3, 1, 1; 3, 3, 1)
⟨324, 17⟩ {17, 16, 10; 1, 2, 8} (1, 1, 2) (4.3)
⟨324, 19⟩ {19, 128
9
, 10; 1, 16
9
, 10} (1, 1, 3)
⟨441, 20⟩ {20, 378
25
, 12; 1, 42
25
, 9} (1, 1, 3)
⟨540, 33⟩ {33, 20, 63
5
; 1, 12
5
, 15} (1, 1, 3)
⟨540, 35⟩ {35, 243
10
, 27
2
; 1, 27
10
, 45
2
} (1, 1, 3)
⟨576, 23⟩ {23, 432
25
, 15; 1, 48
25
, 9} (1, 1, 3)
⟨729, 26⟩ {26, 486
25
, 18; 1, 54
25
, 9} (1, 1, 3)
⟨1000, 37⟩ {37, 24, 14; 1, 2, 12} (1, 1, 3)
⟨1015, 28⟩ {28, 2523
130
, 4263
338
; 1, 1218
845
, 203
26
} (1, 1, 3)
⟨1080, 83⟩ {83, 54, 21; 1, 6, 63} FSD (1, 1, 2)
⟨1134, 49⟩ {49, 48, 644
75
; 1, 196
75
, 42} (1, 1, 1)
⟨1189, 40⟩ {40, 5043
203
, 123
7
; 1, 615
406
, 164
7
} (1, 1, 2)
⟨1470, 104⟩ {104, 70, 25; 1, 7, 80} FSD (1, 1, 2; 3, 2, 3)
⟨1548, 35⟩ {35, 2187
86
, 45
2
; 1, 135
86
, 27
2
} (1, 1, 3)
⟨1680, 69⟩a {69, 42, 7; 1, 2, 63} (1, 1, 2)
⟨1702, 45⟩ {45, 4761
148
, 115
4
; 1, 345
148
, 69
4
} (1, 1, 2)
⟨1944, 29⟩ {29, 22, 25; 1, 2, 5} (1, 1, 2)
⟨2016, 195⟩ {195, 160, 28; 1, 20, 168} FSD (1, 2, 2)
⟨2106, 65⟩ {65, 64, 676
25
; 1, 104
25
, 26} {125, 108, 24; 1, 9, 75} (1, 1, 1)
⟨2185, 114⟩ {114, 4761
65
, 58121
1521
; 1, 11799
1690
, 6118
117
} (1, 1, 3)
⟨2197, 36⟩ {36, 45
2
, 45
2
; 1, 3
2
, 15
2
} (1, 1, 3)
⟨2197, 126⟩ {126, 90, 10; 1, 6, 105} FSD (0231) (2, 2, 3)
⟨2304, 47⟩ {47, 135
4
, 33; 1, 9
4
, 15} (1, 1, 3)
⟨2376, 95⟩ {95, 63, 12; 1, 3, 84} (1, 1, 3)
⟨2401, 48⟩ {48, 30, 29; 1, 3
2
, 20} (1, 1, 2)
⟨2500, 49⟩a {49, 48, 26; 1, 2, 24} (1, 1, 2) (4.3)
⟨2640, 203⟩ {203, 160, 34; 1, 16, 170} FSD (1, 2, 2)
Table 1: Nonexistence results for feasible Krein arrays of primitive 3-class Q-polynomial
association schemes on up to 2800 vertices. For P -polynomial parameters (for the natural
ordering of relations, unless otherwise indicated), the DRG column indicates whether the
parameters are formally self-dual (FSD), or the intersection array is given. The Nonex-
istence column gives either the triple of relation indices for which there is no solution
for triple intersection numbers, or the 6-tuple of relation indices (r, s, t; i, j, k) for which
Corollary 4.2 is not satisfied. The Family column specifies the infinite family from Subsec-
tion 4.2 that the parameter set is part of.
A. L. Gavrilyuk et al.: On few-class Q-polynomial association schemes: feasible . . . 125
Label Krein array Nonexistence Family
⟨200, 12⟩ {12, 11, 256
25
, 36
11
; 1, 44
25
, 96
11
, 12} (1, 1, 2)
⟨462, 21⟩ {21, 20, 196
11
, 49
5
; 1, 35
11
, 56
5
, 21} (1, 1, 2)
⟨486, 45⟩ {45, 44, 36, 5; 1, 9, 40, 45} (1, 1, 2)
⟨578, 17⟩ {17, 16, 136
9
, 9; 1, 17
9
, 8, 17} (1, 1, 2) (4.4)
⟨686, 28⟩ {28, 27, 25, 8; 1, 3, 20, 28} (1, 2, 2)
⟨702, 36⟩ {36, 35, 405
13
, 72
7
; 1, 63
13
, 180
7
, 36} (1, 2, 2)
⟨722, 19⟩ {19, 18, 152
9
, 11; 1, 19
9
, 8, 19} (1, 1, 2) (4.4)
⟨882, 21⟩ {21, 20, 56
3
, 13; 1, 7
3
, 8, 21} (1, 1, 2) (4.4)
⟨990, 66⟩ {66, 65, 847
15
, 88
13
; 1, 143
15
, 770
13
, 66} (1, 2, 2)
⟨1014, 78⟩ {78, 77, 65, 8; 1, 13, 70, 78} (1, 2, 2)
⟨1058, 23⟩ {23, 22, 184
9
, 15; 1, 23
9
, 8, 23} (1, 1, 2) (4.4)
⟨1250, 25⟩ {25, 24, 200
9
, 17; 1, 25
9
, 8, 25} (1, 1, 2) (4.4)
⟨1458, 27⟩ {27, 26, 24, 19; 1, 3, 8, 27} (1, 1, 2) (4.4)
⟨1458, 36⟩ {36, 35, 33, 16; 1, 3, 20, 36} (1, 2, 2)
⟨1482, 38⟩ {38, 37, 12635
351
, 76
37
; 1, 703
351
, 1330
37
, 38} (1, 2, 2)
⟨1674, 45⟩ {45, 44, 1296
31
, 135
11
; 1, 99
31
, 360
11
, 45} (1, 1, 2)
⟨1682, 29⟩ {29, 28, 232
9
, 21; 1, 29
9
, 8, 29} (1, 1, 2) (4.4)
⟨1694, 55⟩ {55, 54, 352
7
, 15; 1, 33
7
, 40, 55} (1, 1, 2)
⟨1862, 21⟩ {21, 20, 364
19
, 81
5
; 1, 35
19
, 24
5
, 21} (1, 1, 2)
⟨2058, 49⟩ {49, 48, 686
15
, 77
5
; 1, 49
15
, 168
5
, 49} (1, 1, 2)
⟨2060, 50⟩ {50, 49, 4800
103
, 110
7
; 1, 350
103
, 240
7
, 50} (1, 1, 2)
⟨2394, 27⟩ {27, 26, 3240
133
, 279
13
; 1, 351
133
, 72
13
, 27} (1, 1, 2)
⟨2466, 36⟩ {36, 35, 4617
137
, 144
7
; 1, 315
137
, 108
7
, 36} (1, 2, 2)
⟨2550, 85⟩ {85, 84, 1156
15
, 187
7
; 1, 119
15
, 408
7
, 85} (1, 1, 2)
⟨2662, 121⟩ {121, 120, 5324
49
, 77
5
; 1, 605
49
, 528
5
, 121} (1, 1, 2)
⟨2706, 66⟩ {66, 65, 2541
41
, 44
3
; 1, 165
41
, 154
3
, 66} (1, 2, 2)
⟨2730, 78⟩ {78, 77, 507
7
, 52
3
; 1, 39
7
, 182
3
, 78} (1, 2, 2)
⟨2750, 25⟩ {25, 24, 250
11
, 185
9
; 1, 25
11
, 40
9
, 25} (1, 1, 2)
⟨2862, 53⟩ {53, 52, 11236
225
, 265
13
; 1, 689
225
, 424
13
, 53} (1, 1, 2)
⟨2890, 153⟩ {153, 152, 136, 9; 1, 17, 144, 153} (1, 1, 2)
⟨2926, 171⟩ {171, 170, 11552
77
, 171
17
; 1, 1615
77
, 2736
17
, 171} (1, 1, 2)
⟨2970, 54⟩ {54, 53, 567
11
, 12; 1, 27
11
, 42, 54} (1, 2, 2)
⟨3042, 65⟩ {65, 64, 182
3
, 25; 1, 13
3
, 40, 65} (1, 1, 2)
⟨3074, 106⟩ {106, 105, 2809
29
, 212
9
; 1, 265
29
, 742
9
, 106} (1, 2, 2)
⟨3174, 184⟩ {184, 183, 161, 16; 1, 23, 168, 184} (1, 2, 2)
⟨3250, 50⟩ {50, 49, 625
13
, 100
9
; 1, 25
13
, 350
9
, 50} (1, 2, 2)
⟨3402, 126⟩ {126, 125, 343
3
, 28; 1, 35
3
, 98, 126} (1, 2, 2)
⟨3498, 77⟩ {77, 76, 3872
53
, 231
19
; 1, 209
53
, 1232
19
, 77} (1, 1, 2)
⟨3610, 133⟩ {133, 132, 608
5
, 21; 1, 57
5
, 112, 133} (1, 1, 2)
⟨3726, 36⟩ {36, 35, 783
23
, 24; 1, 45
23
, 12, 36} (1, 2, 2)
⟨4070, 55⟩ {55, 54, 1936
37
, 77
3
; 1, 99
37
, 88
3
, 55} (1, 1, 2)
⟨4250, 119⟩ {119, 118, 13872
125
, 1309
59
; 1, 1003
125
, 5712
59
, 119} (1, 1, 2)
⟨4370, 190⟩ {190, 189, 3971
23
, 76
7
; 1, 399
23
, 1254
7
, 190} (1, 2, 2)
⟨4410, 210⟩ {210, 209, 189, 12; 1, 21, 198, 210} (1, 2, 2)
⟨4464, 24⟩ {24, 23, 2048
93
, 488
23
; 1, 184
93
, 64
23
, 24} (1, 1, 2)
⟨4526, 73⟩ {73, 72, 10658
155
, 511
15
; 1, 657
155
, 584
15
, 73} (1, 1, 2)
⟨4558, 86⟩ {86, 85, 12943
159
, 1376
51
; 1, 731
159
, 3010
51
, 86} (1, 2, 2)
⟨4590, 75⟩ {75, 74, 1200
17
, 35; 1, 75
17
, 40, 75} (1, 1, 2)
(Continued on next page.)
126 Ars Math. Contemp. 20 (2021) 103–127
(Continued.)
Label Krein array Nonexistence Family
⟨4758, 117⟩ {117, 116, 6760
61
, 273
29
; 1, 377
61
, 3120
29
, 117} (1, 1, 2)
⟨4802, 49⟩ {49, 48, 1176
25
, 25; 1, 49
25
, 24, 49} (1, 1, 2) (4.4)
⟨5046, 261⟩ {261, 260, 232, 21; 1, 29, 240, 261} (1, 1, 2)
⟨5202, 51⟩ {51, 50, 1224
25
, 27; 1, 51
25
, 24, 51} (1, 1, 2) (4.4)
⟨5480, 100⟩ {100, 99, 12800
137
, 140
3
; 1, 900
137
, 160
3
, 100} (1, 1, 2)
⟨5566, 66⟩ {66, 65, 1463
23
, 24; 1, 55
23
, 42, 66} (1, 2, 2)
⟨5590, 78⟩ {78, 77, 3211
43
, 312
11
; 1, 143
43
, 546
11
, 78} (1, 2, 2)
⟨5618, 53⟩ {53, 52, 1272
25
, 29; 1, 53
25
, 24, 53} (1, 1, 2) (4.4)
⟨5618, 106⟩ {106, 105, 901
9
, 36; 1, 53
9
, 70, 106} (1, 2, 2)
⟨5642, 91⟩ {91, 90, 2704
31
, 65
3
; 1, 117
31
, 208
3
, 91} (1, 1, 2)
⟨5670, 105⟩ {105, 104, 98, 49; 1, 7, 56, 105} (1, 1, 2)
⟨5670, 105⟩a {105, 104, 100, 25; 1, 5, 80, 105} (1, 1, 2)
⟨6050, 55⟩ {55, 54, 264
5
, 31; 1, 11
5
, 24, 55} (1, 1, 2) (4.4)
⟨6278, 73⟩ {73, 72, 21316
301
, 365
21
; 1, 657
301
, 1168
21
, 73} (1, 1, 2)
⟨6358, 85⟩ {85, 84, 884
11
, 45; 1, 51
11
, 40, 85} (1, 1, 2)
⟨6422, 91⟩ {91, 90, 1664
19
, 119
5
; 1, 65
19
, 336
5
, 91} (1, 1, 2)
⟨6426, 147⟩ {147, 146, 2352
17
, 35; 1, 147
17
, 112, 147} (1, 1, 2)
⟨6450, 105⟩ {105, 104, 4320
43
, 357
13
; 1, 195
43
, 1008
13
, 105} (1, 1, 2)
⟨6498, 57⟩ {57, 56, 1368
25
, 33; 1, 57
25
, 24, 57} (1, 1, 2) (4.4)
⟨6962, 59⟩ {59, 58, 1416
25
, 35; 1, 59
25
, 24, 59} (1, 1, 2) (4.4)
⟨7210, 103⟩ {103, 102, 84872
875
, 927
17
; 1, 5253
875
, 824
17
, 103} (1, 1, 2)
⟨7442, 61⟩ {61, 60, 1464
25
, 37; 1, 61
25
, 24, 61} (1, 1, 2) (4.4)
⟨7854, 66⟩ {66, 65, 1089
17
, 88
3
; 1, 33
17
, 110
3
, 66} (1, 2, 2)
⟨7878, 78⟩ {78, 77, 7605
101
, 104
3
; 1, 273
101
, 130
3
, 78} (1, 2, 2)
⟨7906, 134⟩ {134, 133, 22445
177
, 2948
57
; 1, 1273
177
, 4690
57
, 134} (1, 2, 2)
⟨7938, 63⟩ {63, 62, 1512
25
, 39; 1, 63
25
, 24, 63} (1, 1, 2) (4.4)
⟨8120, 100⟩ {100, 99, 19200
203
, 620
11
; 1, 1100
203
, 480
11
, 100} (1, 1, 2)
⟨8190, 90⟩ {90, 89, 1125
13
, 40; 1, 45
13
, 50, 90} (1, 2, 2)
⟨8246, 217⟩ {217, 216, 3844
19
, 155
3
; 1, 279
19
, 496
3
, 217} (1, 1, 2)
⟨8450, 65⟩ {65, 64, 312
5
, 41; 1, 13
5
, 24, 65} (1, 1, 2) (4.4)
⟨8450, 78⟩ {78, 77, 377
5
, 36; 1, 13
5
, 42, 78} (1, 2, 2)
⟨8470, 88⟩ {88, 87, 429
5
, 16; 1, 11
5
, 72, 88} (1, 2, 2)
⟨8478, 27⟩ {27, 26, 3888
157
, 327
13
; 1, 351
157
, 24
13
, 27} (1, 1, 2)
⟨8750, 325⟩ {325, 324, 300, 13; 1, 25, 312, 325} (1, 1, 2)
⟨8758, 232⟩ {232, 231, 32799
151
, 464
11
; 1, 2233
151
, 2088
11
, 232} (1, 2, 2)
⟨8798, 106⟩ {106, 105, 8427
83
, 424
9
; 1, 371
83
, 530
9
, 106} (1, 2, 2)
⟨8802, 351⟩ {351, 350, 52488
163
, 351
25
; 1, 4725
163
, 8424
25
, 351} (1, 1, 2)
⟨8978, 67⟩ {67, 66, 1608
25
, 43; 1, 67
25
, 24, 67} (1, 1, 2) (4.4)
⟨9310, 105⟩ {105, 104, 17500
171
, 165
13
; 1, 455
171
, 1200
13
, 105} (1, 1, 2)
⟨9350, 153⟩ {153, 152, 8092
55
, 459
19
; 1, 323
55
, 2448
19
, 153} (1, 1, 2)
⟨9386, 171⟩ {171, 170, 2128
13
, 27; 1, 95
13
, 144, 171} (1, 1, 2)
⟨9522, 69⟩ {69, 68, 1656
25
, 45; 1, 69
25
, 24, 69} (1, 1, 2) (4.4)
⟨9522, 161⟩ {161, 160, 460
3
, 49; 1, 23
3
, 112, 161} (1, 1, 2)
⟨9702, 126⟩ {126, 125, 1323
11
, 56; 1, 63
11
, 70, 126} (1, 2, 2)
Table 2: Nonexistence results for feasible Krein arrays of Q-bipartite (but not Q-antipodal)
4-class Q-polynomial association schemes on up to 10000 vertices. The Nonexistence
column gives either the triple of relation indices for which there is no solution for triple
intersection numbers. The Family column specifies the infinite family from Subsection 4.2
that the parameter set is part of.
A. L. Gavrilyuk et al.: On few-class Q-polynomial association schemes: feasible . . . 127
Label Krein array Family
⟨576, 21⟩ {21, 20, 18, 21
2
, 27
7
; 1, 3, 21
2
, 120
7
, 21}
⟨800, 25⟩ {25, 24, 625
28
, 75
7
, 25
7
; 1, 75
28
, 100
7
, 150
7
, 25} (4.5)
⟨2000, 25⟩ {25, 24, 625
27
, 50
3
, 25
9
; 1, 50
27
, 25
3
, 200
9
, 25}
⟨2400, 22⟩ {22, 21, 20, 88
5
, 32
11
; 1, 2, 22
5
, 210
11
, 22}
⟨2928, 61⟩ {61, 60, 3721
66
, 305
11
, 61
11
; 1, 305
66
, 366
11
, 610
11
, 61} (4.5)
⟨7232, 113⟩ {113, 112, 12769
120
, 791
15
, 113
15
; 1, 791
120
, 904
15
, 1582
15
, 113} (4.5)
⟨14480, 181⟩ {181, 180, 32761
190
, 1629
19
, 181
19
; 1, 1629
190
, 1810
19
, 3258
19
, 181} (4.5)
⟨25440, 265⟩ {265, 264, 70225
276
, 2915
23
, 265
23
; 1, 2915
276
, 3180
23
, 5830
23
, 265} (4.5)
⟨37752, 121⟩ {121, 120, 14641
125
, 484
5
, 121
25
; 1, 484
125
, 121
5
, 2904
25
, 121}
⟨40880, 365⟩ {365, 364, 133225
378
, 4745
27
, 365
27
; 1, 4745
378
, 5110
27
, 9490
27
, 365} (4.5)
⟨47040, 116⟩ {116, 115, 112, 696
7
, 144
29
; 1, 4, 116
7
, 3220
29
, 116}
Table 3: Nonexistence results for feasible Krein arrays of Q-bipartite (but not Q-antipodal)
5-class Q-polynomial association schemes on up to 50000 vertices. In all cases, there is no
solution for triple intersection numbers for a triple of vertices mutually in relation R1. The
Family column specifies the infinite family from Subsection 4.2 that the parameter set is
part of.

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 129–142
https://doi.org/10.26493/1855-3974.2227.e1a
(Also available at http://amc-journal.eu)
The enclaveless competition game
Michael A. Henning *
Department of Mathematics and Applied Mathematics, University of Johannesburg,
Auckland Park, 2006 South Africa
Douglas F. Rall
Professor Emeritus of Mathematics
Furman University, Greenville, SC, USA
Received 24 January 2020, accepted 11 November 2020, published online 19 August 2021
Abstract
For a subset S of vertices in a graph G, a vertex v ∈ S is an enclave of S if v and
all of its neighbors are in S, where a neighbor of v is a vertex adjacent to v. A set S is
enclaveless if it does not contain any enclaves. The enclaveless number Ψ(G) of G is the
maximum cardinality of an enclaveless set in G. As first observed in 1997 by Slater, if G is
a graph with n vertices, then γ(G)+Ψ(G) = n where γ(G) is the well-studied domination
number of G. In this paper, we continue the study of the competition-enclaveless game
introduced in 2001 by Philips and Slater and defined as follows. Two players take turns in
constructing a maximal enclaveless set S, where one player, Maximizer, tries to maximize
|S| and one player, Minimizer, tries to minimize |S|. The competition-enclaveless game
number Ψ+g (G) of G is the number of vertices played when Maximizer starts the game and
both players play optimally. We study among other problems the conjecture that if G is
an isolate-free graph of order n, then Ψ+g (G) ≥ 12n. We prove this conjecture for regular
graphs and for claw-free graphs.
Keywords: Competition-enclaveless game, domination game.
Math. Subj. Class. (2020): 05C65, 05C69
1 Introduction
In 2010 Brešar, Klavžar, and Rall [2] published the seminal paper on the domination game
which belongs to the growing family of competitive optimization graph games. Domi-
nation games played on graphs are now very well studied in the literature. Indeed, the
*Research supported in part by the University of Johannesburg.
E-mail addresses: mahenning@uj.ac.za (Michael A. Henning), doug.rall@furman.edu (Douglas F. Rall)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
130 Ars Math. Contemp. 20 (2021) 129–142
subsequent rapid growth by the scientific community of research on domination games
played on graphs resulted in several dozen papers to date on the domination-type games
(see, for example, [3, 4, 10, 11, 12, 14, 17]). A recent book entitled “Domination games
played on graphs” by Brešar, Henning, Klavžar, and Rall [1] presents the state of the art
results to date, and shows that the area is rich for further research. In this paper, we con-
tinue the study of domination games played on graphs, and investigate in more depth the
competition-enclaveless game birthed by Philips and Slater [15, 16].
A neighbor of a vertex v in G is a vertex that is adjacent to v. A vertex dominates itself
and its neighbors. A dominating set of a graph G is a set S of vertices of G such that every
vertex in G is dominated by a vertex in S. The domination number of G, denoted γ(G),
is the minimum cardinality of a dominating set in G, while the upper domination number
of G, denoted Γ(G), is the maximum cardinality of a minimal dominating set in G. A
minimal dominating set of cardinality Γ(G) we call a Γ-set of G.
The open neighborhood of a vertex v in G is the set of neighbors of v, denoted NG(v).
Thus, NG(v) = {u ∈ V | uv ∈ E(G)}. The closed neighborhood of v is the set NG[v] =
{v} ∪NG(v). If the graph G is clear from context, we simply write N(v) and N [v] rather
than NG(v) and NG[v], respectively.
As defined by Alan Goldman and introduced by Slater in [19], for a subset S of vertices
in a graph G, a vertex v ∈ S is an enclave of S if it and all of its neighbors are also in S;
that is, if N [v] ⊆ S. A set S is enclaveless if it does not contain any enclaves. We note that
a set S is a dominating set of a graph G if and only if the set V (G) \ S is enclaveless. The
enclaveless number of G, denoted Ψ(G), is the maximum cardinality of an enclaveless set
inG, and the lower enclaveless number ofG, denoted by ψ(G), is the minimum cardinality
of a maximal enclaveless set. The domination and enclaveless numbers of a graph G are
related by the following equations.
Observation 1.1. If G is a graph of order n, then γ(G) + Ψ(G) = n = Γ(G) + ψ(G).
The domination game on a graph G consists of two players, Dominator and Staller,
who take turns choosing a vertex from G. Each vertex chosen must dominate at least one
vertex not dominated by the vertices previously chosen. Upon completion of the game, the
set of chosen (played) vertices is a dominating set in G. The goal of Dominator is to end
the game with a minimum number of vertices chosen, while Staller has the opposite goal
and wishes to end the game with as many vertices chosen as possible.
The Dominator-start domination game and the Staller-start domination game is the
domination game when Dominator and Staller, respectively, choose the first vertex. We
refer to these simply as the D-game and S-game, respectively. The game domination num-
ber, γg(G), of G is the number of moves in a D-game when both players play optimal
strategies consistent with their goals. The Staller-start game domination number, γ′g(G),
of G is defined analogously for the S-game.
Philips and Slater [15, 16] introduced what they called the competition-enclaveless
game. The game is played by two players, Maximizer and Minimizer, on some graph
G. They take turns in constructing a maximal enclaveless set S of G. That is, in each turn
a player plays a vertex v that is not in the set S of the vertices already chosen and such that
S ∪ {v} does not contain an enclave, until there is no such vertex. We call such a vertex a
playable vertex. The goal of Maximizer is to make the final set S as large as possible and
for Minimizer to make the final set S as small as possible.
The competition-enclaveless game number, or simply the enclaveless game number,
M. A. Henning and D. F. Rall: The enclaveless competition game 131
Ψ+g (G), of G is the number of vertices chosen when Maximizer starts the game and both
players play an optimal strategy according to the rules. The Minimizer-start competi-
tion-enclaveless game number, or simply the Minimizer-start enclaveless game number,
Ψ−g (G), of G is the number of vertices chosen when Minimizer starts the game and both
players play an optimal strategy according to the rules. The competition-enclaveless game,
which has been studied for example in [8, 9, 15, 16, 18], has not yet been explored in as
much depth as the domination game. In this paper we continue the study of the competition-
enclaveless game. Our main motivation for our study are the following conjectures that
have yet to be settled, where an isolate-free graph is a graph that does not contain an iso-
lated vertex.
Conjecture 1.2. If G is an isolate-free graph of order n, then Ψ+g (G) ≥ 12n.
Conjecture 1.2 was first posed as a question by Peter Slater to the 2nd author on 8th May
2015, and subsequently posed as a conjecture in [9]. We refer to Conjecture 1.2 for general
isolate-free graphs as the 12 -Enclaveless Game Conjecture. We also pose the following
conjecture for the Minimizer-start enclaveless game, where δ(G) denotes the minimum
degree of the graph G.
Conjecture 1.3. If G is a graph of order n with δ(G) ≥ 2, then Ψ−g (G) ≥ 12n.
We proceed as follows. By Observation 1.1, if the domination number of a graph is
known, then we immediately know the enclaveless number, and vice versa. In contrast,
we show in Section 2 that the game domination number and the enclaveless game number
are very different and are not related in the same way that the domination and enclaveless
numbers are related. Indeed, knowledge of the game domination number gives no informa-
tion of the enclaveless game number, and vice versa. We show that the domination game
and the enclaveless game are intrinsically different. In Section 3, we present fundamental
bounds on the enclaveless game number and the Minimizer-start enclaveless game number.
In Sections 4 and 5, we show that the 12 -Enclaveless Game Conjecture holds for regular
graphs and claw-free graphs, respectively. We use the standard notation [k] = {1, . . . , k}.
2 Game domination versus enclaveless game
Although the domination and enclaveless numbers of a graph G of order n are related by
the equation γ(G) +Ψ(G) = n (see Observation 1.1), as remarked in [9] the competition-
enclaveless game is very different from the domination game. To illustrate this, we present
two simple examples showing that the sum γg(G) + Ψ+g (G) on the class of graphs of a
fixed order can differ greatly even when restricted to trees.
For the first example, for k ≥ 3, let G be a tree with exactly two non-leaf vertices
both of which have k leaf neighbors, that is, G is a double star S(k, k). In this case,
Ψ+g (G) = Ψ
−
g (G) = k + 1 and γg(G) = 3 and γ
′
g(G) = 4. Thus if G is a double star of
order n, then γg(G) + Ψ+g (G) =
1
2n+ 3.
For the second example, we consider the class of paths; Pn denotes the path on n
vertices. For n ≥ 1, Košmrlj [14] showed that γ′g(Pn) =
⌈
n
2
⌉
and that γg(Pn) =
⌈
n
2
⌉
− 1
if n ≡ 3 (mod 4) and γg(Pn) =
⌈
n
2
⌉
, otherwise. For n ≥ 2, Phillips and Slater [16]
showed that Ψ+g (Pn) = ⌊ 3n+15 ⌋ and Ψ
−
g (Pn) = ⌊ 3n5 ⌋. Thus if G is a path Pn, then
γg(G) + Ψ
+
g (G) ≈ n+ 110n.
132 Ars Math. Contemp. 20 (2021) 129–142
The most important general fact in the domination game is the so-called Continuation
Principle, which provides a much-used monotonicity property of the game domination
number and allows us to assume that each optimal move of Dominator (and of Staller) is
taken from a restricted subset of the unchosen vertices. Due to its importance in the dom-
ination game, we recall this well-studied Continuation Principle. A partially dominated
graph is a graph together with a declaration that some vertices are already dominated and
need not be dominated, but can be played, in the rest of the game. Given a graph G and a
subset S of vertices of G, we denote by G|S the partially dominated graph with S as the
set of declared vertices already dominated. We use γg(G|S) (resp. γ′g(G|S)) to denote the
number of moves remaining in the game onG|S under optimal play when Dominator (resp.
Staller) has the next move. We are now in a position to state the Continuation Principle
presented by Kinnersley, West, and Zamani in [12, Lemma 2.1].
Lemma 2.1 (Continuation Principle). If G is a graph and A,B ⊆ V (G) with B ⊆ A,
then γg(G|A) ≤ γg(G|B) and γ′g(G|A) ≤ γ′g(G|B).
The Continuation Principle is one of the most important proof techniques to obtain
results on the domination game and its variants. It yields, for example, the following fun-
damental monotonicity property of the domination game; see [2, Theorem 6] and [12,
Corollary 4.1].
Theorem 2.2. The Dominator-start game domination number and the Staller-start game
domination number can differ by at most 1, that is, for any graph G, we have
|γg(G)− γ′g(G)| ≤ 1.
The most significant difference between the competition-enclaveless game and the
domination game is that the Continuation Principle holds for the domination game but
does not hold for the competition-enclaveless game. If the Continuation Principle were to
hold for the competition-enclaveless game, then this would imply that the Maximizer-start
enclaveless game number and the Minimizer-start enclaveless game number can differ by
at most 1. However, this is not the case, and these two game numbers can differ signifi-
cantly. For example, if n ≥ 1 and G is a star K1,n, then Ψ+g (G) = n while Ψ−g (G) = 1.
Thus, the numbers Ψ+g (G) and Ψ
−
g (G) can vary greatly.
Without the powerful proof method of the Continuation Principle at our disposal, the
competition-enclaveless game is raised to a greater level of difficulty than other domination
games played on graphs. Indeed, this suggests that there may exist a graph G (or infinite
classes of graphs) for which Ψ−g (G) > Ψ
+
g (G) is possible, and such that the difference
Ψ−g (G) − Ψ+g (G) can possibly be made arbitrarily large. However, we are unable at this
time to construct such examples, if they exist. Moreover, we are also unable to prove at this
time that Ψ−g (G) ≤ Ψ+g (G) is always true.
Another significant difference between the domination game and the competition-en-
claveless game is that upon completion of the domination game, the set of played vertices
is a dominating set although not necessarily a minimal dominating set, while upon com-
pletion of the competition-enclaveless game, the set of played vertices is always a maximal
enclaveless set. Thus, the enclaveless game numbers of a graph G are always squeezed
between the lower enclaveless number ψ(G) of G and the enclaveless number Ψ(G) of G.
We state this formally as follows.
M. A. Henning and D. F. Rall: The enclaveless competition game 133
Observation 2.3. If G is a graph, then
ψ(G) ≤ Ψ−g (G) ≤ Ψ(G) and ψ(G) ≤ Ψ+g (G) ≤ Ψ(G).
A graph G is well-dominated if all the minimal dominating sets of G have the same
cardinality. Examples of well-dominated graphs include, for example, the complete graph
Kn, C7, P4, the corona of any graph, and the graph formed from two vertex disjoint cycles
of order 5 joined by a single edge. Finbow, Hartnell and Nowakowski [7] characterized
the well-dominated graphs having no 3-cycle nor 4-cycle. As observed earlier, upon com-
pletion of the enclaveless game, the set of played vertices is always a maximal enclaveless
set. Hence, any sequence of legal moves by Maximizer and Minimizer (regardless of strat-
egy) in the enclaveless game played in a well-dominated graph of order n will always lead
to the game ending in n − γ(G) moves. Thus as a consequence of Observation 2.3, we
have the following interesting connection between the enclaveless game and the class of
well-dominated graphs.
Observation 2.4. If G is a well-dominated graph of order n, then
Ψ−g (G) = Ψ
+
g (G) = n− γ(G).
It is well-known that ifG is an isolate-free graph of order n, then γ(G) ≤ 12n, implying
by Observation 1.1 that Ψ(G) = n − γ(G) ≥ 12n. Hence one might think that γg(G) ≤
Ψ+g (G) for such a graph G with no isolated vertex. We now provide an infinite class of
graphs to show that the ratio γg/Ψ+g of these two graphical invariants can be strictly larger
than, and bounded away from, 1. The corona cor(G) of a graph G, also denoted G ◦K1 in
the literature, is the graph obtained from G by adding for each vertex v of G a new vertex
v′ and the edge vv′ (and so, the vertex v′ has degree 1 in cor(G)). The edge vv′ is called a
pendant edge. We shall need the following 2014 result due to Košmrlj [13].
Theorem 2.5 ([13, Theorem 4.1]). For k ≥ 1, if G = cor(Pk), then γg(G) = k + ⌈k−710 ⌉.
Let G be the (infinite) family of coronas of paths Pk where k ≥ 8 and kmod10 ∈
{8, 9}, that is,
G = {cor(Pk) : k mod 10 ∈ {8, 9}}.
As a consequence of Theorem 2.5, we have the following result.
Theorem 2.6. For every graph G ∈ G, we have
γg(G)
Ψ+g (G)
>
11
10
.
Proof. Let G ∈ G, and so G = cor(Pk) for some positive integer k where kmod10 ∈
{8, 9}. Every minimal dominating set of G has cardinality k, which implies by Observa-
tion 1.1 that every maximal enclaveless set of G also has cardinality k; that is, ψ(G) =
Ψ(G) = k where we recall that ψ(G) denotes the cardinality of the smallest maximal en-
claveless set in G and Ψ(G) is the cardinality of a largest enclaveless set in G. Hence by
Observation 2.3, Ψ+g (G) = k. Consequently, by Theorem 2.5 and since kmod10 ∈ {8, 9}
we have
γg(G)
Ψ+g (G)
=
k + ⌈k−710 ⌉
k
>
11
10
.
Hence, by Theorem 2.6, the difference γg(G) − Ψ+g (G) can be made arbitrarily large
for an infinite family of graphs.
134 Ars Math. Contemp. 20 (2021) 129–142
3 Fundamental bounds
In this section, we establish some fundamental bounds on the (Maximizer-start) enclaveless
game number and the Minimizer-start enclaveless game number. We establish next a lower
and upper bound on the enclaveless number of a graph in terms of the maximum degree
and order of the graph.
Proposition 3.1. If G is an isolate-free graph of order nwith maximum degree ∆(G) = ∆,
then (
1
∆ + 1
)
n ≤ ψ(G) ≤ Ψ(G) ≤
(
∆
∆+ 1
)
n.
Proof. If G is any graph of order n and maximum degree ∆, then γ(G) ≥ n∆+1 . Hence,
by Observation 1.1,
Ψ(G) = n− γ(G) ≤ n− n
∆+ 1
=
(
∆
∆+ 1
)
n.
On the other hand, let D be a minimal dominating set of maximum cardinality, and
so |D| = Γ(G). Let D = V (G) \ D, and so |D| = n − |D|. Let ℓ be the number of
edges between D and D. Since G is an isolate-free graph and D is a minimal dominating
set, every vertex in D has at least one neighbor in D, and so ℓ ≥ |D|. Since G has
maximum degree ∆, every vertex inD has at most ∆ neighbors inD, and so ℓ ≤ ∆ · |D| =
∆(n− |D|). Hence, |D| ≤ ∆(n− |D|), implying that Γ(G) = |D| ≤ ∆n/(∆+ 1). Thus
by Observation 1.1,
ψ(G) = n− Γ(G) ≥ n−
(
∆
∆+ 1
)
n =
(
1
∆ + 1
)
n.
This completes the proof of Proposition 3.1.
By Observation 2.3, the set of played vertices in either the Maximizer-start enclaveless
game or the Minimizer-start enclaveless game is a maximal enclaveless set of G. Thus as
an immediate consequence of Proposition 3.1, we have the following result.
Corollary 3.2. If G is an isolate-free graph of order n with maximum degree ∆(G) = ∆,
then(
1
∆ + 1
)
n ≤ Ψ−g (G) ≤
(
∆
∆+ 1
)
n and
(
1
∆ + 1
)
n ≤ Ψ+g (G) ≤
(
∆
∆+ 1
)
n.
We show that the upper bounds in Corollary 3.2 are realized for infinitely many con-
nected graphs.
Proposition 3.3. There exist infinitely many positive integers n along with a connected
graph G of order n satisfying
Ψ−g (G) = Ψ
+
g (G) =
(
∆(G)
∆(G) + 1
)
n.
Proof. Let r be an integer such that r ≥ 4 and let m be any positive integer. For each
i ∈ [m], let Hi be a graph obtained from a complete graph of order r + 1 by removing
M. A. Henning and D. F. Rall: The enclaveless competition game 135
the edge xiyi for two distinguished vertices xi and yi. The graph Fm is obtained from
the disjoint union of H1, . . . ,Hm by adding the edges yixi+1 for each i ∈ [m] where the
subscripts are computed modulo m. The vertices xi and yi are called connectors in Fm,
and each of the r − 1 vertices in the set V (Hi) \ {xi, yi} is called a hidden vertex of Hi.
Note that Fm is r-regular and has order n = m(r + 1).
We first show that Ψ−g (Fm) = (
r
r+1 )n. Suppose the Minimizer-start enclaveless game
is played on Fm. We provide a strategy for Maximizer that forces exactly rm vertices to be
played. Maximizer’s strategy is to make sure that all the connector vertices in the graph are
played. If he can accomplish this, then exactly rm vertices will be played when the game
ends because of the structure of Fm. Suppose that at some point in the game Minimizer
plays a vertex from some Hj . If one of the connector vertices, say xj , is playable, then
Maximizer responds by playing xj . If both connector vertices have already been played
and some hidden vertex, say w, in Hj is playable, then Maximizer plays w. If no vertex
of Hj is playable, then Maximizer plays a connector vertex from Hi for some i ̸= j if one
is playable and otherwise plays any playable vertex. Since Hk contains at least 3 hidden
vertices for each k ∈ [m], it follows that Maximizer can guarantee that all the connector
vertices are played by following this strategy. This implies that for each i ∈ [m], exactly
one hidden vertex of Hi is not played during the course of the game. That is, the set of
played vertices has cardinality
rm =
(
r
r + 1
)
m(r + 1) =
(
∆(Fm)
∆(Fm) + 1
)
n ,
where we recall that ∆(Fm) = r. Thus,
Ψ−g (Fm) ≥
(
∆(Fm)
∆(Fm) + 1
)
n.
By Corollary 3.2,
Ψ−g ((Fm)) ≤
(
∆(Fm)
∆(Fm) + 1
)
n.
Consequently, Ψ−g (Fm) =
(
∆(Fm)
∆(Fm)+1
)
n.
If the Maximizer-start enclaveless game is played on Fm, then the same strategy as
above for Maximizer forces rm vertices to be played (even with the relaxed condition that
r be an integer larger than 2). Thus as before, Ψ+g (Fm) =
(
∆(Fm)
∆(Fm)+1
)
n.
The lower bound in Corollary 3.2 on Ψ−g (G) is achieved, for example, by taking G =
K1,∆ for any given ∆ ≥ 1 in which case Ψ−g (G) = 1 = ( 1∆+1 )n where n = n(G) =
∆+ 1.
The lower bound in Corollary 3.2 on Ψ+g (G) is trivially achieved when ∆ = 1, in which
case G is the disjoint union of copies of K2. However, as remarked in the introductory
section, the main open problem in the competition-enclaveless game is the 12 -Enclaveless
Game Conjecture (stated formally in Conjecture 1.2) that claims that if G is an isolate-free
graph of order n, then Ψ+g (G) ≥ 12n. If the conjecture is true, then, from our earlier exam-
ples such as the double star, the bound is achieved for isolate-free graphs with arbitrarily
large maximum degree ∆.
136 Ars Math. Contemp. 20 (2021) 129–142
4 Regular graphs
In this section, we show that 12 -Enclaveless Game Conjecture (see Conjecture 1.2) holds
for the class of regular graphs, as does Conjecture 1.3 for the Minimizer-start enclaveless
game. For a set S ⊂ V (G) of vertices in a graph G and a vertex v ∈ S, we define the
S-external private neighborhood of a vertex v, abbreviated epnG(v, S), as the set of all
vertices outside S that are adjacent to v but to no other vertex of S; that is,
epnG(v, S) = {w ∈ V (G) \ S : NG(w) ∩ S = {v}}.
As remarked in the introduction, if the graph G is clear from the context, we omit the
subscript G in the above definitions. We define an S-external private neighbor of v to be a
vertex in epn(v, S).
We establish next a 12 -lower bound on Ψ
+
g (G) and Ψ
−
g (G) by forbidding induced stars
of a certain size. We remark that the proof of the following result uses similar counting
techniques to those employed by Southey and Henning in [20].
Proposition 4.1. If G is a graph with order n, minimum degree δ and with no induced
K1,δ+1, then ψ(G) ≥ 12n.
Proof. LetD be an arbitrary minimal dominating set ofG. Denote byD1 the set of vertices
in D that have a D-external private neighbor. That is, D1 = {x ∈ D : epn(x,D) ̸= ∅}.
In addition, let D2 = D \D1. Since D is a minimal dominating set, the set D2 consists of
those vertices in D that are isolated in the subgraph G[D] of G induced by D. Let
C1 =
⋃
x∈D1
epn(x,D) and C2 = V (G) \ (D ∪ C1).
We note that by definition, there are no edges in G joining a vertex in D2 and a vertex
in C1. That is, each vertex in D2 has at least δ neighbors in C2. Since the set D2 is
independent and G has no induced K1,δ+1, each vertex of C2 has at most δ neighbors in
D2. Denote by ℓ the number of edges of the form uv where u ∈ D2 and v ∈ C2. It now
follows that δ|D2| ≤ ℓ ≤ δ|C2|, that is, |D2| ≤ |C2|. Now,
|D| = |D1|+ |D2| ≤ |C1|+ |C2| = n− |D| ,
which shows that Γ(G) ≤ 12n. Hence by Observation 1.1, we have ψ(G) ≥
1
2n.
Observation 2.3 and Proposition 4.1 now yield the following result.
Corollary 4.2. If G is a graph with order n, minimum degree δ and with no induced
K1,δ+1, then Ψ+g (G) ≥ 12n and Ψ
−
g (G) ≥ 12n.
As a special case of Corollary 4.2, we have the desired 12 -lower bound on Ψ
+
g (G) and
Ψ−g (G) for regular graphs G without isolated vertices.
Corollary 4.3. If G is a regular graph of order n without isolated vertices, then Ψ+g (G) ≥
1
2n and Ψ
−
g (G) ≥ 12n.
We remark that if G is a graph of order n that is a disjoint union of copies of K2, then
Ψ+g (G) =
1
2n and Ψ
−
g (G) =
1
2n. The same conclusion holds if G is a disjoint union of
copies of C4. Hence, for k ∈ {1, 2} there are k-regular graphs G that achieve equality in
the lower bound in Corollary 4.3. However, it remains an open problem to characterize the
graphs achieving equality in Corollary 4.3 for each value of k ≥ 3.
M. A. Henning and D. F. Rall: The enclaveless competition game 137
5 Claw-free graphs
A graph is claw-free if it does not contain the star K1,3 as an induced subgraph. In this
section, we show that 12 -Enclaveless Game Conjecture (see Conjecture 1.2) holds for the
class of claw-free graphs with no isolated vertex, as does Conjecture 1.3 for the Minimizer-
start enclaveless game. For this purpose, we recall the definition of an irredundant set. For
a set S of vertices in a graph G and a vertex v ∈ S, the S-private neighborhood of v is the
set
pnG[v, S] = {w ∈ V : N [w] ∩ S = {v}}.
If the graph G is clear from context, we simply write pn[v, S] rather than pnG[v, S].
We note that epn(v, S) ⊆ pn[v, S] ⊆ epn(v, S) ∪ {v} and v ∈ pn[v, S] if and only if
v is isolated in G[S]. A vertex in the set pn[v, S] is called an S-private neighbor of v.
The set S is an irredundant set if every vertex of S has an S-private neighbor. The upper
irredundance number IR(G) is the maximum cardinality of an irredundant set in G.
The independence number α(G) of G is the maximal cardinality of an independent set
of vertices in G. An independent set of vertices of G of cardinality α(G) is called an α-set
of G. Every maximum independent set in a graph is a minimal dominating set, and every
minimal dominating set is an irredundant set. Hence we have the following inequality
chain.
Observation 5.1 ([5]). For every graph G, we have α(G) ≤ Γ(G) ≤ IR(G).
The inequality chain in Observation 5.1 is part of the canonical domination chain which
was first observed by Cockayne, Hedetniemi, and Miller [5] in 1978.
In 2004, Favaron [6] established the following upper bound on the upper irredundance
number of a claw-free graph.
Theorem 5.2 ([6]). If G is a connected, claw-free graph of order n, then IR(G) ≤
1
2 (n+ 1). Moreover, if IR(G) =
1
2 (n+ 1), then α(G) = Γ(G) = IR(G).
In addition, she proved the following stronger upper bound for the upper irredundance
number of claw-free graphs with minimum degree at least 2.
Corollary 5.3 ([6]). If G is a connected, claw-free graph of order n and minimum degree
at least 2, then IR(G) ≤ 12n.
Suppose that G is a claw-free graph of order n and minimum degree δ ≥ 2. By Corol-
lary 5.3, IR(G) ≤ 12n, and thus by Observations 1.1 and 5.1, we have
ψ(G) = n− Γ(G) ≥ n− IR(G) ≥ n− 1
2
n =
1
2
n.
By Observation 2.3, we therefore have the following result.
Theorem 5.4. If G is a claw-free graph of order n and δ(G) ≥ 2, then
Ψ+g (G) ≥
1
2
n and Ψ−g (G) ≥
1
2
n.
By Theorem 5.4, we note that Conjecture 1.3 holds for connected claw-free graphs.
In order to prove that Conjecture 1.2 holds for connected claw-free graphs, we need the
138 Ars Math. Contemp. 20 (2021) 129–142
characterization due to Favaron [6] of the graphs achieving equality in the bound of The-
orem 5.2. For this purpose, we recall that a vertex v of a graph G is a simplicial vertex if
its open neighborhood N(v) induces a complete subgraph of G. A clique of a graph G is a
maximal complete subgraph of G. The clique graph of G has the set of cliques of G as its
vertex set, and two vertices in the clique graph are adjacent if and only if they intersect as
cliques of G. A non-trivial tree is a tree of order at least 2.
Favaron [6] defined the family F of claw-free graphs G as follows. Let T1, . . . , Tq be
q ≥ 1 non-trivial trees. Let Li be the line graph of the corona cor(Ti) of the tree Ti for
i ∈ [q]. If q = 1, let G = L1. If q ≥ 2, let G be the graph constructed from the line graphs
L1, L2, . . . , Lq by choosing q − 1 pairs {xij , xji} such that the following holds.
• xij and xji are simplicial vertices of Li and Lj , respectively, where i ̸= j.
• The 2(q − 1) vertices from the q − 1 pairs {xij , xji} are all distinct vertices.
• Contracting each pair of vertices xij and xji into one common vertex cij results in a
graph whose clique graph is a tree.
To illustrate the above construction of a graph G in the family F consider, for example,
such a construction when q = 3 and the trees T1, T2, T3 are given in Figure 1.
c12 c23G
x12 x21 x23 x32L1 L2 L3⇓
cor(T1) cor(T2) cor(T3)⇓
T1 T2 T3⇓
Figure 1: An illustration of the construction of a graph G in the family F .
We note that if G is an arbitrary graph of order n in the family F , then n ≥ 3 is odd
and the vertex set of G can be partitioned into two sets A and B such that the following
holds.
• |A| = 12 (n− 1) and |B| =
1
2 (n+ 1).
• The set B is an independent set.
M. A. Henning and D. F. Rall: The enclaveless competition game 139
• Each vertex in A has exactly two neighbors in B.
We refer to the partition (A,B) as the partition associated withG. For the graphG ∈ F
illustrated in Figure 1, the set A consists of the darkened vertices and the set B consists of
the white vertices.
We are now in a position to state the characterization due to Favaron [6] of the graphs
achieving equality in the bound of Theorem 5.2.
Theorem 5.5 ([6]). If G is a connected, claw-free graph of order n ≥ 3, then IR(G) ≤
1
2 (n+ 1), with equality if and only if G ∈ F .
We prove next the following property of graphs in the family F .
Lemma 5.6. If G ∈ F and (A,B) is the partition associated with G, then the set B is the
unique IR-set of G.
Proof. We proceed by induction on the order n ≥ 3 of G ∈ F . If n = 3, then G = P3.
In this case, the set B consists of the two leaves of G, and the desired result is immediate.
This establishes the base case. Suppose that n ≥ 5 and that the result holds for all graphs
G′ ∈ F of order n′, where 3 ≤ n′ < n. Let Q be an IR-set of G.
By construction of the graph G, the set B contains at least two vertices of degree 1 in
G. Let v be an arbitrary vertex inB of degree 1 inG, and let u be its neighbor. We note that
u ∈ A. LetG′ = G−{u, v} and letG′ have order n′, and so n′ = n−2. LetA′ = A\{u}
and B′ = B \ {v}. By construction of the graph G and our choice of the vertex v, we note
that G′ ∈ F and that (A′, B′) is the partition associated with G′. Applying the inductive
hypothesis to G′, the set B′ is the unique IR-set of G′. Let w be the second neighbor of u
in G that belongs to the set B, and so N(u) ∩ B = {v, w}. By the structure of the graph
G ∈ F , we note thatN [w] ⊂ N [u] and that the subgraph ofG induced byN [w] is a clique.
Suppose, to the contrary, that Q ̸= B. Let Q′ be the restriction of Q to the graph
G′, and so Q′ = Q ∩ V (G′). Suppose that u ∈ Q. Since Q is an irredundant set, this
implies that v /∈ Q. If w ∈ Q, then pn[w,Q] = ∅, contradicting the fact that Q is an
irredundant set. Hence, w /∈ Q, and so Q′ ̸= B′. By the inductive hypothesis, the set Q′
is therefore not an IR-set of G′, and so |Q′| < IR(G′). Thus, IR(G) = |Q| = |Q′|+ 1 ≤
(IR(G′) − 1) + 1 = 12 (n
′ + 1) = 12 (n − 1) < IR(G), a contradiction. Hence, u /∈ Q. In
this case, IR(G) = |Q| ≤ |Q′|+ 1 ≤ IR(G′) + 1 = 12 (n
′ + 1) + 1 = 12 (n+ 1) = IR(G).
Hence, we must have equality throughout this inequality chain. This implies that v ∈ Q
and |Q′| = IR(G′). By the inductive hypothesis, we therefore have Q′ = B′. Hence,
Q = Q′ ∪ {v} = B′ ∪ {v} = B. Thus, the set B is the unique IR-set of G.
Corollary 5.7. If G ∈ F and (A,B) is the partition associated with G, then the set B is
the unique α-set of G and the unique Γ-set of G.
Proof. By Theorem 5.2, α(G) = Γ(G) = IR(G) = 12 (n + 1). By Lemma 5.6, the set B
is the unique IR-set of G. Since every α-set of G is an IR-set of G and α(G) = IR(G),
this implies that B is the unique α-set of G. Since every Γ-set of G is an IR-set of G and
Γ(G) = IR(G), this implies that B is the unique Γ-set of G.
We show next that Conjecture 1.2 holds for connected claw-free graphs.
Theorem 5.8. If G is a connected, claw-free graph of order n ≥ 2, then the following
holds.
140 Ars Math. Contemp. 20 (2021) 129–142
(a) Ψ+g (G) ≥ 12n.
(b) If G ̸= P3, then Ψ−g (G) ≥ 12n.
Proof. Let G be a connected, claw-free graph of order n ≥ 2. If IR(G) ≤ 12n, then
min{Ψ+g (G),Ψ−g (G)} ≥ ψ(G) ≥ n− IR(G) ≥
1
2
n.
Therefore, by Theorem 5.5, we can assume that IR(G) = 12 (n+1) andG ∈ F . Let (A,B)
be the partition associated with G. We show in this case we have min{Ψ+g (G),Ψ−g (G)} ≥
ψ(G) + 1.
By Observation 1.1, Γ(G) + ψ(G) = n. Moreover, the complement of every Γ-set
of G is a maximal enclaveless set, and the complement of every ψ-set of G is a minimal
dominating set. By Corollary 5.7, the set B is the unique Γ-set of G. These observations
imply that the complement of the set B, namely the set A, is the unique ψ-set of G. Thus
every maximal enclaveless set of G of cardinality ψ(G) is precisely the set A.
In the Maximizer-start enclaveless game played on G, Maximizer plays as his first
move any vertex from the set B. In the Minimizer-start enclaveless game played on G,
by supposition we have G ̸= P3, implying that there are at least two vertices in the set B
at distance at least 3 apart in G. Thus, whatever the first move is played by Minimizer,
Maximizer can always respond by playing as his first move a vertex chosen from the set
B. Hence, no matter who starts the game, Maximizer can play a vertex in B as his first
move. Thus if S denotes the set of all vertices played when the game ends (in either game),
then the set S is a maximal enclaveless set in G. By our earlier observations, such a set
S contains a vertex of B and is therefore different from the set A. Since the set A is the
unique ψ-set of G, this implies that |S| > ψ(G). Therefore,
|S| ≥ ψ(G) + 1 = (n− Γ(G)) + 1 = n− 1
2
(n+ 1) + 1 =
1
2
(n+ 1).
Since the first move of Maximizer from the set B may not be an optimal move, we
have that Ψ+g (G) ≥ ψ(G) + 1 if we are playing the Maximizer-start enclaveless game and
Ψ−g (G) ≥ ψ(G) + 1 if we are playing the Minimizer-start enclaveless game. Thus, in both
games Maximizer has a strategy to finish the game in at least 12 (n+1) moves. Hence, if we
assume that IR(G) = 12 (n+ 1), then Ψ
+
g (G) ≥ min{Ψ+g (G),Ψ−g (G)} ≥ 12 (n+ 1).
By Theorem 5.8(a), we note that Conjecture 1.2 holds for connected claw-free graphs.
Moreover by Theorem 5.8(b), we note that Conjecture 1.3 holds for connected claw-free
graphs even if we relax the minimum degree two condition and replace it with the require-
ment that the graph is isolate-free and different from the path P3.
6 Open problems and conjectures
In this paper, we have shown that the 12 -Enclaveless Game Conjecture (see, Conjecture 1.2)
is true for special classes of graphs, such as regular graphs and claw-free graphs. However,
the conjecture has yet to be solved in general. It would be very interesting to prove or dis-
prove the conjecture, or at least prove the conjecture for certain other important classes of
graphs. We have also shown that the related conjecture for the Minimizer-start enclaveless
game (see, Conjecture 1.3) holds for special classes. Again, it would be interesting to make
M. A. Henning and D. F. Rall: The enclaveless competition game 141
further inroads into the conjecture. We close with the following two questions that we have
yet to settle.
Question 6.1. Do there exist graphs G such that Ψ−g (G) > Ψ+g (G)? If so, how large can
the difference Ψ−g (G)−Ψ+g (G) be made? Or is Ψ−g (G) ≤ Ψ+g (G) always true?
Question 6.2. Is it possible to characterize graphs G such that Ψ−g (G) = Ψ+g (G)?
ORCID iDs
Michael A. Henning https://orcid.org/0000-0001-8185-067X
Douglas F. Rall https://orcid.org/0000-0002-5482-756X
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 143–149
https://doi.org/10.26493/1855-3974.2194.eab
(Also available at http://amc-journal.eu)
Strongly involutive self-dual polyhedra
Javier Bracho *
Instituto de Matemáticas, UNAM, Mexico
Luis Montejano †
Instituto de Matemáticas, UNAM-Campus Juriquilla, Mexico
Eric Pauli Pérez ‡
Instituto de Matemáticas, UNAM-Campus Juriquilla and IMAG, Univ. Montpellier,
CNRS, Montpellier, France
Jorge Luis Ramı́rez Alfonsı́n §
UMI2924 - Jean-Christophe Yoccoz, CNRS-IMPA and IMAG, Univ. Montpellier,
CNRS, Montpellier, France
Received 10 December 2019, accepted 11 October 2020, published online 2 September 2021
Abstract
A polyhedron is a graph which is simple, planar and 3-connected. In this note, we
classify the family of strongly involutive self-dual polyhedra. The latter is done by using
a well-known result due to Tutte characterizing 3-connected graphs. We also show that
in this special class of polyhedra self-duality behaves topologically as the antipodal map-
ping. These self-dual polyhedra are related with several problems in convex and discrete
geometry including the Vázsonyi problem.
Keywords: Polyhedra, graphs, duality, self-dual, antipodal.
Math. Subj. Class. (2020): 05C15, 05C10
*Supported by PAPIIT-UNAM under project IN109218.
†Supported by CONACyT under project 166306 and support from PAPIIT-UNAM under project IN112614.
‡Supported by CONACyT Grant 268597.
§Supported by MATHAMSUD 18-MATH-01, Project FLaNASAGraTA and by PICS07848 CNRS.
E-mail addresses: jbracho@im.unam.mx (Javier Bracho), luis@im.unam.mx (Luis Montejano),
eric@im.unam.mx (Eric Pauli Pérez), jorge.ramirez-alfonsin@umontpellier.fr (Jorge Luis Ramı́rez Alfonsı́n)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
144 Ars Math. Contemp. 20 (2021) 143–149
1 Introduction
A planar and 3-connected graph G = (V,E) can be drawn in essentially one way on the
sphere or the plane. This fundamental fact is a result of the work of Whitney [14]. It
tells us that we not only have the sets V and E defined, but that the set F of faces is
also determined, and furthermore the dual graph G∗ is well defined. The dual graph G∗ is
the graph whose vertex set V ∗ is the set of faces F of G, and two new vertices in G∗ are
connected by an edge if and only if the faces that define them are adjacent in G.
In this class of graphs, each face f is determined by its boundary walk, that is, a cycli-
cally ordered sequence (v1, v2, . . . , vk) consisting of the vertices (and the edges) that are
in the closure of the region defining the face f (see [4]). In this sense we can say that u
incides on f , if it is any of the elements v1, v2, . . . , vk of the cycle defining the face f . We
denote this situation simply by u ∈ f . From Steinitz’s theorem [12] we know that it is
the same to talk about polyhedra in the sense of convex polytopes and to talk about these
graphs, so we will refer to them as polyhedra. A polyhedron is a graph G that is simple
(without loops and multiple edges), planar and 3-connected.
A polyhedron P is said to be self-dual if there exists an isomorphism of graphs τ : P →
P∗. This isomorphism is called a duality isomorphism. There may be several of these
duality isomorphisms and each of them is a bijection between vertices and faces of P ,
such that adjacent vertices correspond to adjacent faces. We are interested in such an
isomorphism that satisfies two more properties:
(1) For each pair u, v of vertices, u ∈ τ(v) if and only if v ∈ τ(u).
(2) For every vertex v, we have that v /∈ τ(v).
Such an isomorphism will be called a strong involution. If P is a self-dual polyhedron
admitting a strong involution τ , we will say that P is a strongly involutive polyhedron.
Strongly involutive self-dual polyhedra are very common, like for example wheels on
n-cycles with n odd and hyperwheels on n-cycles with n-even (see [11]). In fact the
relevance of strongly involutive self-dual polyhedra is partially related with the famous
Vázsonyi problem. Let T be a finite set of points of diameter h in Euclidean d-space.
Characterize those sets T for which the diameter is attained a maximal number of times
as a segment of length h with both endpoints in T . Y. S. Kupitz et al. [5], call these sets
extremal configurations. For d = 3, if T is an extremal configuration and V is the inter-
section of balls of radius h with centers in points of T , a facial structure can be defined on
the boundary of V that is strongly self-dual in the sense that it admits an duality isomor-
phism that is involution and is fixed-point free when acting as an automorphism of the first
barycentric subdivision of the boundary cell complex of V (see [5]). Indeed, this unusual
connection between discrete and convex geometry attracted the attention of several mathe-
maticians to this and other related problems. See, for example L. Lóvasz [6], L. Montejano
and E. Roldán-Pensado [9], L. Montejano et al. [8] and the work of Bezdek et al. [2]. For
more about the Vázsonyi problem see [7].
In order to have a good understanding of strongly involutive self-dual polyhedra, we
will use a result due to Tutte [13] establishing that every 3-connected graph is either a
wheel (a cycle where every vertex is also connected with a central vertex o) or it can be
obtained from a wheel by a finite sequence of two operations: adding an edge between any
pair of vertices and splitting a given vertex v, with degree δ(v) ≥ 4, into two new adjacent
vertices v′ and v′′ in such a way that the new graph obtained is still 3-connected.
J. Bracho et al.: Strongly involutive self-dual polyhedra 145
In the following section we briefly summarize the notions and notation in relation with
the above Tutte’s result restricted to the case of simple and planar graphs. In [1], Grünbaum
and Barnette used this idea for giving two proofs of Steinitz’s Theorem. In Section 3, we
show our main results that classify the strongly involutive self-dual polyhedra. Finally, in
Section 4, we give a geometric interpretation of strong involutions by proving that such a
duality is topologically equivalent to the antipodal mapping on the sphere.
2 Tutte’s theorem for polyhedra
In this section we summarize the main ideas and terminology of a recursive classification
of spherical polyhedra. These results are deduced from Tutte’s work and the details can be
found in [10]. Let G be a polyhedron and e = (uv) any edge of G. We write G\e for the
graph obtained from G by deleting e. We write G/e for the graph obtained from G\e by
identifying its endpoints u and v in a single vertex uv. In the same way, given any subset
X of V , we write G\X for the graph obtained from G by ommiting the elements of X and
any edge such that one of its endpoints is an element of X . We will say that e = (uv)
can be deleted if G\e is a polyhedron and we say that e = (uv) can be contracted if G/e
is a polyhedron. We will say that X is an n-cutting set if it has n vertices and G\X is
not connected. According to Tutte’s terminology, we will say that an edge e is essential if
neither G\e nor G/e are polyhedra. In other words, e is essential if it cannot be deleted and
it cannot be contracted.
Theorem 2.1 ([10]). The following statements are equivalent.
(1) G is a wheel.
(2) Every edge is essential.
(3) Every edge is on a triangular face and has one of its endpoints of degree 3.
This result can be rephrased as follows.
Remark 2.2. Every polyhedron is either a wheel or it can be obtained by a wheel by adding
new edges within faces of the polyhedron or its dual’s. Equivalently: if a polyhedron is not
a wheel there is always a not essential edge, this means, an edge we can delete or contract
in order to obtain a new polyhedron with one fewer edge.
In this way we can reduce any polyhedron by a finite sequence of this operations until
we obtain a wheel. It happens that one can obtain different wheels from a given polyhedron
by selecting different sequences of non essential edges.
3 Strongly involutive polyhedra
Throughout this section, we let P = (V,E, F, τ) be a strongly involutive self-dual poly-
hedron and (ab) ∈ E any edge of P . By definition τ(a) and τ(b) are adjacent faces of P ,
thus there must be an edge (xy) ∈ E such that τ(a) ∩ τ(b) = (xy) and condition (1) of
strong involution implies that τ(x) ∩ τ(y) = (ab). We will write τ(ab) for the edge (xy).
We will say that (ab) is a diameter if and only if a ∈ τ(b) (and therefore b ∈ τ(a)).
Lemma 3.1. If (ab) and (xy) are both diameters, then P is the tetrahedron K4.
146 Ars Math. Contemp. 20 (2021) 143–149
Proof. From the hypotheses we deduce a ∈ τ(x)∩τ(y)∩τ(b) and x ∈ τ(a)∩τ(b)∩τ(y),
then {a, x} ⊂ τ(y) ∩ τ(b) but from the 3-connectivity, the intersection of any two faces
must be empty, a single vertex or a single edge, thus (ax) is an edge, otherwise {a, x} is
a 2-cutting set. Analogously, (bx) is an edge. In the same way, (ya) and (yb) are edges.
It follows that the induced graph on these four vertices is K4. In addition, we have the
faces τ(a), τ(b), τ(x) and τ(y) are triangles. Suppose there exist additional vertices. Take
any vertex v ∈ V \ {a, b, x, y} such that v is connected to some vertex in {a, b, x, y} by
an edge. Assume, without loss of generality, that (av) is an edge. This is a contradiction
because in that case face τ(a) should form a cycle with at least four edges.
Lemma 3.2. If (ab) is a diameter and (xy) is not, then {a, b, x} and {a, b, y} are 3-cutting
sets of P .
Proof. From the hypotheses we can deduce that (τ(a)∪τ(b))\(xy) and (τ(x)∪τ(y))\(ab)
are cycles whose intersection is the set {a, b}, then we can observe that (τ(a) ∪ τ(b)) \
(xy)) ∪ (ab) is the union of two cycles γ1, γ2 whose intersection is the edge (ab) and thus
P \ γ1 consists of two connected pieces R1, R2 and also P \ γ2 consists of two connected
pieces S1, S2. Since τ(x)∩τ(y) = (ab) = γ1∩γ2, we may assume τ(x)\ (ab) ⊂ R1∩S1
and τ(y) \ (ab) ⊂ R2 ∩ S2. Let be w ∈ (τ(y) \ (ab)) \ γ1 ⊂ R2 ∩ S2. It exists because
otherwise τ(y) = γ1, and therefore x ∈ τ(y), a contradiction since (xy) is not a diameter.
Analogoulsy, let be u ∈ (τ(x) \ (ab)) \ γ2 ⊂ S1 ∩R1. Then in the graphs P\{a, b, y} and
P \ {a, b, x}, the vertices u and w are disconnected.
Theorem 3.3. If P is not a wheel, then there exists an edge e satisfying the three following
conditions:
(1) e is not on a triangular face,
(2) e is not in a 3-cutting set and
(3) e is not a diameter.
Proof. Since P is not a wheel then, by Theorem 2.1, there is a not essential edge, say e
that can be either deleted or contracted. In fact we ensure, since P is self-dual, there is an
edge that can be contracted, otherwise we can take an edge e that can be deleted and the
corresponding dual edge e∗ can be contracted in P∗ which is isomorphic to P . Without
loss of generality we may assume e can be contracted in P . We may now check that e
verifies the three desired conditions:
(1) e is not in a triangle. Otherwise, if e were contracted then P/e would have parallel
edges (which is not possible since P/e is simple).
(2) e is not in any 3-cutting set. Otherwise, if e were contracted then P/e would have a
2-cutting set (which is not possible since P/e is 3-connected).
(3) e is not a diameter. Indeed, if e were a diameter then we would have that edge τ(e)
cannot be a diameter (otherwise, by Lemma 3.1, P must be a tetrahedron, that is, a
3-wheel, which is not the case) and thus, by Lemma 3.2, e would be in a 3-cutting
set, which is not possible.
J. Bracho et al.: Strongly involutive self-dual polyhedra 147
Theorem 3.4. Let e = (ab) be an edge which is neither on a triangular face nor in a
3-cutting set nor a diameter. Then, the graph [P/(ab)]\τ(ab), denoted by P⋄ = P⋄ab, is a
strongly involutive self-dual polyhedron.
Proof. Since (ab) satisfies the three properties of last theorem, then P/(ab) is a polyhe-
dron, and therefore its dual P\τ(ab) is also a polyhedron. We will show that P⋄ is a
polyhedron. Indeed, it is simple and planar. We need it to be 3-connected. If it were not,
then it would have a 2-cutting set {m,n}. Since τ(a) and τ(b) are the faces such that
τ(a)∩ τ(b) = τ(ab) we may observe that one of the elements in {m,n} is in τ(a) and the
other is in τ(b). Let’s supose m ∈ τ(a) and n ∈ τ(b). Furthermore the vertex a = b, de-
noted by ab must be one of the elements in {m,n}, otherwise {m,n} would be a 2-cutting
set of P\τ(ab), a contradiction. This implies that in P , a ∈ τ(b) (and therefore b ∈ τ(a)),
so (ab) would be a diameter, which is not by hypothesis. Finally, by definition, P⋄ is self-
dual and it is strongly involutive with isomorphism τ⋄(u) = τ(u) for every u /∈ {a, b}
and with τ⋄(a = b) the face obtained by the union of τ(a) and τ(b) when edge (xy) is
deleted.
By the above theorem, we can define the remove-contract operation in any strongly
involutive polyhedron which is not a wheel: there is at least one edge (ab) that we can
contract and at the same time remove the edge τ(ab) in order to obtain a new strongly
involutive polyhedron. We can apply this operation repeatedly in order to finish with a
strongly involutive wheel (with odd number of vertices in the main cycle). Conversely, we
can start with such a wheel and then diagonalizing faces and splitting their corresponding
vertices carefully in order to expand a strongly involutive polyhedron. By diagonalizing
we mean that given a face that is not a triangle, we add a new edge within the face joining
two non-consecutive vertices.
In the above terms, Theorem 3.4 gives the following.
Corollary 3.5. Every strongly involutive self-dual polyhedra is either a wheel or it can be
obtained from an odd wheel by a finite sequence of operations consisting in diagonalizing
faces of the polyhedron and its dual’s simultaneously.
4 Topological interpretation
In this section we are going to consider topological embeddings of a given graph G on the
surface S2. By Whitney’s Theorem we know that if G is simple, planar and 3-connected,
then any two such embeddings are equivalent in the sense that the set of faces (and their
adjacencies) is fully determined only by the graph (they are independent of the embedding).
It is an interesting fact that with these conditions we can choose one of these embeddings
in such a way that any automorphism of the graph of P acts as an isometry of S2. We will
write this important fact as follows.
Lemma 4.1 (Isometric embedding lemma [11, Lemma 1]). There exists an embedding
i : G → S2 such that for every σ ∈ Aut(G) there exists an isometry σ̃ of S2 satisfying
i ◦ σ = σ̃ ◦ i.
Our goal for now is to interpret geometrically the strong involutions. In the rest of the
section G is the underlying graph (simple, planar and 3-connected) of a strongly involutive
self-dual polyhedron P .
148 Ars Math. Contemp. 20 (2021) 143–149
Let us define G□ the graph of squares of G as follows:
V (G□) = V (G) ∪ F (G) ∪ E(G) and
E(G□) = {(ve) : v ∈ V (G), e ∈ E(G), v ∈ e} ∪
{(ec) : e ∈ E(G), f ∈ F (G), e ∈ f}.
It is easy to observe that G□ is a 3-connected simple planar graph and therefore it can
be drawn on the sphere in such a way that any automorphism of G□ is an isometry. We can
suppose G□ is embedded in that way and we will abuse of notation making no distintion
between G□ and its image under the embedding. By definition, the faces of G□ are all
quadrilaterals of the form (vafb), where v ∈ V (G), a, b ∈ E(G) and f ∈ F (G).
Theorem 4.2. Let τ be a strong involution of P . Then τ̃ is the antipodal mapping α : S2 →
S2, α(x) = −x.
Proof. First we can observe that τ is an automorphism of G□ and condition (1) of strong
involution implies τ2 = id. Therefore, τ̃ (given in Lemma 4.1) must be an involution as
isometry. There are three possible involutive isometries of the sphere: a reflection through
a line (a spherical line), a rotation by π2 and the antipodal mapping (a good reference is [3]).
Only the antipodal mapping has no fixed points, so we will show that τ̃ cannot have fixed
points. We will proceed by contradiction, supposing τ̃ has a fixed point and then we will
conclude there exists a vertex v such that v ∈ τ(v).
If τ is a reflection through a plane H , let us consider v ∈ V (G), a, b ∈ E(G) and
f ∈ F (G) such that H intersects the quadrilateral Q = (vafb) in its interior. The only
points of the edges of quadrilateral Q can intersect H are a and b, so H ∩ S2 = l where l
is the spherical line through a and b, thus we must have τ(v) = f that means v ∈ τ(v).
If τ is a rotation in a line PP ′ (P, P ′ antipodal points on the sphere), let Q = (vafb)
be a quadrilateral containing P . If P is the center (the barycenter) of the quadrilateral, then
since τ is a duality, it must send v into f , but then τ(v) = f , which means v ∈ τ(v). If P
is a or b, say P = a, then the edge (va) is sent to an edge (af ′) where f ′ is a face of G,
distinct from f and containing v, but then the quadrilateral Q′ corresponding to v and f ′
we have τ(v) = f ′, which means v ∈ τ(v). This concludes the proof.
As a consequence of Theorem 4.2 we obtain the following.
Corollary 4.3. For a strongly involutive self-dual polyhedron there is only one duality
which is a strong involution.
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 151–170
https://doi.org/10.26493/1855-3974.2201.b65
(Also available at http://amc-journal.eu)
Geometry of the parallelism in polar spine
spaces and their line reducts
Krzysztof Petelczyc , Krzysztof Prażmowski , Mariusz Żynel
Faculty of Mathematics, University of Białystok,
Ciołkowskiego 1 M, 15-245 Białystok, Poland
Received 18 December 2019, accepted 22 November 2020, published online 20 October 2021
Abstract
The concept of the spine geometry over a polar Grassmann space belongs to a wide
family of partial affine line spaces. It is known that the geometry of a spine space over
a projective Grassmann space can be developed in terms of points, so called affine lines,
and their parallelism (in this case the parallelism is not intrinsically definable as it is not
Veblenian). This paper aims to prove an analogous result for the polar spine spaces. As a
by-product we obtain several other results on primitive notions for the geometry of polar
spine spaces.
Keywords: Grassmann space, projective space, polar space, spine space, coplanarity, pencil of lines.
Math. Subj. Class. (2020): 51A15, 51A45
Introduction
Some properties of the polar spine spaces were already established in [8], where the class
of such spaces was originally introduced. Its definition resembles the definition of a spine
space defined within a (projective) Grassmann space (= the space Pk(V) of pencils of
k-subspaces in a fixed vector space V), cf. [12, 13]. In every case, a spine space is a
fragment of a (projective) Grassmannian whose points are subspaces which intersect a fixed
subspace W in a fixed dimension m. In case of polar spine spaces we consider a two-
step construction, in fact: we consider the subspaces of V that are totally isotropic (self
conjugate, singular) under a fixed nondegenerate reflexive bilinear form ξ on V, and then
we restrict this class to the subspaces which touch W in dimension m.
It is a picture which is seen from the view of V. Clearly, W can be extended to a
subspace M of V with codimension 1 and then M yields a hyperplane M of the polar space
E-mail addresses: kryzpet@math.uwb.edu.pl (Krzysztof Petelczyc), krzypraz@math.uwb.edu.pl (Krzysztof
Prażmowski), mariusz@math.uwb.edu.pl (Mariusz Żynel)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
152 Ars Math. Contemp. 20 (2021) 151–170
Q0 determined by ξ in V. In other words, the projective points on W that are points of Q0
yield a subspace W of Q0 extendable to a hyperplane. Recall that situation of this sort was
already investigated in [10]. The isotropic k-subspaces of V are the (k − 1)-dimensional
linear subspaces of Q0, and first-step restriction yields the so called polar Grassmann space
Qk−1 = Pk−1(Q). The points of Qk−1 which touch W in dimension m are – from view
of Q – the elements of Qk−1 which touch W in dimension m− 1. So, a polar spine space
is also the fragment of a polar Grassmannian which consists of subspaces which touch a
fixed subspace extendable to a hyperplane in a fixed dimension. The analogy seems full.
In particular, when W is a hyperplane of V i.e. W is a hyperplane of Q0 then a k-
subspace of Q either is contained in W or it touches it in dimension k − 1. It is seen that
in this case the only reasonable value of m is m = k − 1 and the obtained structure is
the Grassmannian of subspaces of the affine polar space obtained from Q0 by deleting W
(cf. [3, 11]). So, the class of polar spine spaces contain Grassmannians of k-subspaces of
arbitrary polar slit space: of a polar space with a subspace (extendable to a hyperplane)
removed, see [10]. An interesting case appears, in particular, when we assume that W is
isotropic.
1 Generalities
This section is quoted after [8] with slight modifications.
1.1 Point-line spaces and their fragments
A point-line structure B = ⟨S,L⟩, where the elements of S are called points, the elements
of L are called lines, and where L ⊂ 2S , is said to be a partial linear space, or a point-line
space, if two distinct lines share at most one point and every line is of size (cardinality) at
least 2 (cf. [2]).
A subspace of B is any set X ⊆ S with the property that every line which shares with
X two or more points is entirely contained in X . We say that a subspace X of B is strong
if any two points in X are collinear. If S is strong, then B is said to be a linear space.
Let us fix a nonempty subset H ⊂ S and consider the set
L|H :=
{
k ∩H : k ∈ L and |k ∩H| ≥ 2
}
. (1.1)
The structure
M := B ↾ H = ⟨H,L|H⟩
is a fragment of B induced by H and itself it is a partial linear space. The incidence relation
in M is again ∈, inherited from B, but limited to the new point set and line set. Following a
standard convention we call the points of M proper, and the points in S \H improper. The
set S \ H will be called the horizon of M. To every line L ∈ L|H we can assign uniquely
the line L ∈ L, the closure of L, such that L ⊆ L. For a subspace X ⊆ H the closure of X
is the minimal subspace X of B containing X . A line L ∈ L|H is said to be a projective
line if L = L, and it is said to be an affine line if |L \ L| = 1. With every affine line L one
can correlate the point L∞ ∈ S \H by the condition L∞ ∈ L\L. We write A for the class
of affine lines. In what follows we consider sets H which satisfy the following
|L \ H| ≤ 1 or |L ∩H| ≤ 1 for all L ∈ L.
Note that the above holds when H or S \ H is a subspace of B, but the above does not
force H or S \ H to be a subspace of B. In any case, under this assumption every line
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 153
is either projective or affine. In case L|H contains projective or affine lines only, then
M is a semiaffine geometry (for details on terminology and axiom systems see [18]). In
this approach an affine space is a particular case of a semiaffine space. For affine lines
L1, L2 ∈ L|H we can define a parallelism in a natural way:
L1, L2 are parallel (L1 ∥ L2) iff L1 ∩ L2 ∩ (S \ H) ̸= ∅.
In what follows we assume that the notion of ‘a plane’ (= 2-dimensional strong sub-
space) is meaningful in B: e.g. B is an exchange space, or a dimension function is defined
on its strong subspaces. In the article in most parts we consider B such that its planes are,
up to an isomorphism, projective planes. We say that E is a plane in M if E is a plane in B.
Observe that there are two types of planes in M: projective and semiaffine. A semiaffine
plane E arises from E by removing a point or a line. In result we get a punctured plane
or an affine plane respectively. For lines L1, L2 ∈ L|H we say that they are coplanar and
write
L1 π L2 iff there is a plane E such that L1, L2 ⊂ E. (1.2)
Let E be a plane in M and U ∈ E. The set
p(U,E) :=
{
L ∈ L|H : U ∈ L ⊆ E
}
(1.3)
will be called a pencil of lines if U is a proper point, or a parallel pencil otherwise. The
point U is said to be the vertex and the plane E is said to be the base plane of that pencil.
We write
L1 ρ L2 iff there is a pencil p such that L1, L2 ∈ p. (1.4)
1.2 Cliques
Let ϱ be a binary symmetric relation defined on a set X. A subset of X is said to be a
ϱ-clique iff every two elements of this set are ϱ-related.
For any x1, x2, . . . , xs in X we introduce
∆sϱ(x1, x2, . . . , xs) iff ̸= (x1, x2, . . . , xs) and xi ϱ xj for all i, j = 1, . . . , s
and for all y1, y2 ∈ X if y1, y2 ϱ x1, x2, . . . , xs then y1 ϱ y2, (1.5)
cf. analogous definition of ∆ϱs in [9]. For short we will frequently write ∆ϱ instead of ∆
s
ϱ.
Next, we define
[|x1, x2, . . . , xs|]ϱ :=
{
y ∈ X : y ϱ x1, x2, . . . xs
}
. (1.6)
It is evident that if ∆ϱ(x1, . . . , xs) holds (and ϱ is reflexive) then [|x1, . . . , xs|]ϱ is the
(unique) maximal ϱ-clique which contains {x1, . . . , xs}. Finally, for an arbitrary integer
s ≥ 3 we put
Ksϱ =
{
[|x1, x2, . . . , xs|]ϱ : x1, x2, . . . , xs ∈ X and ∆ϱ(x1, x2, . . . , xs)
}
. (1.7)
Then we write
Kϱ :=
∞⋃
s=3
Ksϱ.
In most of the interesting situations there is an integer smax such that Kϱ =
⋃smax
s=3 K
s
ϱ =
K∗(ϱ), where
K∗(ϱ) is the set of maximal ϱ-cliques.
154 Ars Math. Contemp. 20 (2021) 151–170
1.3 Grassmann spaces and spine spaces
We start with some constructions of a general character. Let X be a nonempty set and
let P be a family of subsets of X . Assume that there is a dimension function dim: P →
{0, . . . , n} such that B = ⟨P,⊂,dim⟩ is an incidence geometry, cf. e.g. [1]. Write Pk for
the set of all U ∈ P with dim(U) = k.
Given H ∈ Pk−1 and B ∈ Pk+1 with H ⊂ B, a k-pencil over B is a set of the form
p(H,B) = {U ∈ Pk : H ⊂ U ⊂ B}.
The idea behind this concept is the same as in (1.3), though this definition is more general.
The family of all such k-pencils over B will be denoted by Pk. Then, the structure
Pk(B) = ⟨Pk,Pk⟩
will be called a Grassmann space over B (cf. [5, Section 2.1.3]). It is a partial linear space
for 0 < k < n.
Let us fix W ∈ P and an integer m. We will write
Fk,m(B,W ) := {U ∈ Pk : dim(U ∩W ) = m}.
The fragment
Ak,m(B,W ) := Pk(B) ↾ Fk,m(B,W )
will be called a spine space over B determined by W . It will be convenient to have an
additional symbol for the line set of a spine space, which is
Gk,m(B,W ) := Pk|Fk,m(B,W ).
What follows are more specific examples of the above constructions that we actu-
ally investigate in our paper. Let V be a vector space and let Sub(V) be the set of all
vector subspaces of V. Then Pk(V) is a partial linear space called a projective Grass-
mann space. In particular P1(V) is the projective space over V. It is well known that
Pk(V) ∼= Pk−1(P1(V)).
Let W ∈ Sub(V). The spine space Ak,m(V,W ) was introduced in [12] and developed
in [13, 14, 15, 16]. Note that Ak,m(V,W ) ∼= Ak−1,m−1(P1(V),Sub1(W )). The concept
of a spine space makes a little sense without the assumption that
0, k − n+ w ≤ m ≤ k,w, (1.8)
where w = dim(W ). It is a partial linear space when (1.8) is satisfied.
For possibly maximal values of m we get Ak,k(V,W ) = Pk(W ), where the points are
basically vector subspaces of W , and Ak,w(V,W ) ∼= Pk−w(V/W ), where the points are
those vector subspaces of V which contain W . Therefore, we assume that
m < k,w. (1.9)
Now, let ξ be a nondegenerate reflexive bilinear form of index r on V. For U,W ∈
Sub(V) we write U ⊥ W iff ξ(U,W ) = 0, meaning that ξ(u,w) = 0 for all u ∈ U ,
w ∈ W . Then the set of all totally isotropic subspaces of V w.r.t. ξ is
Q := {U ∈ Sub(V) : U ⊥ U},
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 155
and Qk := Q ∩ Subk(V). The set Qk is nonempty iff
k ≤ r. (1.10)
Provided that 2 ≤ r the structure Q = P1(Q) is a classical polar space embeddable into
the projective space P1(V). It is clear that Q ∼= ⟨Q1,Q2,⊂⟩ and usually polar space is
defined that way.
A polar Grassmann space is the structure Pk(Q). It is a partial linear space whenever
k < r. (1.11)
Note that Pk(Q) ∼= Pk−1(P1(Q)).
Finally,
M := Ak,m(Q,W ),
a polar spine space, the main subject of our paper, arises. Note that we have M ∼=
Ak−1,m−1(P1(Q),Sub(W ) ∩Q1).
Let rW = ind(ξ ↾ W ) be the index of the form ξ restricted to W . If rW < m, then there
is no totally isotropic subspace of V, which meets W in some m-dimensional subspace.
Every U ∈ Q can be extended to an Y ∈ Qr. Assume that dim(Y ∩ W ) > r − k + m
for all Y ∈ Qr. This means that all totally isotropic subspaces of V, which meet W in
some m-dimensional subspace, are at most (k − 1)-dimensional. On the other hand, this
assumption implies rW > r − k +m. Thus
m ≤ rW ≤ r − k +m (1.12)
is a sufficient condition for Fk,m(Q,W ) ̸= ∅.
Warning. The condition (1.12) is – in the context above – only sufficient. As we shall see
there are sets W such that r − k +m < rW but Fk,m(Q,W ) ̸= ∅. Clearly, the condition
m ≤ rW is necessary.
Under (1.12) no point of M is isolated and M is a partial linear space. Now, let us
have a look at the structure of strong subspaces of polar spine spaces. Following [13] they
are called: α-stars, ω-stars, α-tops and ω-tops. For details see Table 2. Actually, this is
an ‘adaptation’ of the classification of strong subspaces of Ak,m(V,W ) (consult [13]) to
the case when we restrict Pk(V) to Pk(Q). With a slight abuse of language all sets of
the type T α and T ω we call tops, and sets of the form Sα and Sω stars. But note that
due to some specific values of r, k,m and dim(Y ∩ W ) with Y ∈ Qr families of some
of these types may be empty. Moreover, stars and tops consist of strong subspaces of M,
but stars or tops of some kind may be not maximal among strong. In general, Sω and T α
consist of projective spaces, while the other consist of proper slit spaces (cf. [4, 18]), but if
Fk−1,m(Q,W ) ∋ H ⊂ Y ∈ Fr,m(Q,W ) then [H,Y ]k ∩ Fk,m(Q,W ) = [H,Y ]k ∈ Sα
is a projective space as well.
Generally, H ∈ Pk−1 determines a star and B ∈ Pk+1 determines a top as follows
S(H) = {U ∈ Pk : H ⊂ U}, T(B) = {U ∈ Pk : U ⊂ B}.
Here, we occasionally make use of this convention in the context of polar spine spaces,
where Pk = Fk,m(Q,W ).
156 Ars Math. Contemp. 20 (2021) 151–170
2 Lines classification and existence problems
In analogy to [13, 17] the lines of M can be of three sorts: affine (in A), α-projective
(in Lα), and ω-projective (in Lω). To be more concrete, comp. Table 1, these are pencils
L = p(H,B) ∩ Fk,m(Q,W ) such that (we consider parameters k,m,Q,W as fixed)
A: H ∈ Fk−1,m(Q,W ), B ∈ Fk+1,m+1(Q,W ); in this case L∞ = H+(W ∩B) =
(H +W ) ∩ B. Note that L∞ ⊂ B ∈ Q and therefore L∞ ∈ Fk,m+1(Q,W ). In
other words, L∞ is a point of Ak,m+1(Q,W ).
Lα: H ∈ Fk−1,m(Q,W ), B ∈ Fk+1,m(Q,W ).
Lω: H ∈ Fk−1,m−1(Q,W ), B ∈ Fk+1,m+1(Q,W )
(cf. Table 1).
Note that if rW < m + 1 (in view of the global assumption rW ≥ m this means
rW = m) then A ∪ Lω = ∅. Looking at [8, Lemma 1.6] we see that in this case M is
disconnected as well or m = w. In the latter case also A ∪ Lω = ∅. Besides, this also
contradicts (1.9). Consequently, for rW < m+1 the horizon Ak,m+1(Q,W ) of M looses
its sense.
The problem whether one of the three above classes of lines is nonempty reduces, in
fact, to the problem whether the corresponding class of ‘possible tops’ of these lines is
nonempty. More precisely, we have the following criterion.
Lemma 2.1.
(i) Let B ∈ Fk+1,m(Q,W ); then T(B) ̸= ∅.
(ii) Let B ∈ Fk+1,m(Q,W ) and U ∈ T(B); then there is an L = [H,B]k ∈ Lα such
that U ∈ L.
So, if Fk+1,m(Q,W ) ̸= ∅ then Lα ̸= ∅.
(iii) Let B ∈ Fk+1,m+1(Q,W ); then T(B) ̸= ∅.
(iv) Let B ∈ Fk+1,m+1(Q,W ) and U ∈ T(B). Then there are:
• an L′ = [H ′, B]k ∈ Lω (provided that m > 0) such that U ∈ L′ and
• an L′′ = [H ′′, B]k ∈ A such that U ∈ L′′.
Consequently, if Fk+1,m+1(Q,W ) ̸= ∅ then A ≠ ∅, and Lω ̸= ∅ when m > 0.
Proof. To justify (i) present B in the form B = (B ∩ W ) ⊕ D, where D ∩ W = Θ
and dim(D) = k + 1 − m. Let D′ be a (k − m)-dimensional subspace of D and put
H := (B ∩W ) +D′. To justify (ii) we simply use (i) with B replaced by U to obtain the
subspace H .
To justify (iii) we present B in the form B = (B ∩W )⊕D (now, dim(D) = k −m)
and proceed analogously to (i): U = D + Z, where Z is an m-dimensional subspace of
B ∩ W . To justify (iv) to get H ′ we apply (iii) with B replaced by U , and to get H ′′ we
apply (i) with B replaced by U .
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 157
Note that, sufficient conditions for the existence of the corresponding subspaces B in
Lemma 2.1, i.e. for Fk+1,m(Q,W ),Fk+1,m+1(Q,W ) ̸= ∅ are
m ≤ rW ≤ r − (k + 1) +m and m+ 1 ≤ rW ≤ r − k +m, respectively.
We say that an U ∈ Fk,m(Q,W ) is an α-point iff each top containing U is of type α,
i.e. each line through U is of type α. Similarly, an U ∈ Fk,m(Q,W ) is an ω-point iff each
top containing U is of type ω, i.e. each line through U is either affine or of type ω.
Lemma 2.2. Let U ∈ Fk,m(Q,W ).
(i) There is B such that U ⊂ B ∈ Fk+1,m(Q,W ) ∪ Fk+1,m+1(Q,W ).
(ii) U is an α-point iff U⊥ ∩W ⊂ U . In this case
w ≤ k +m. (2.1)
Otherwise, if U⊥ ∩W ̸⊂ U then there is a B ∈ Fk+1,m+1(Q,W ) such that U ⊂ B.
(iii) U is an ω-point when U⊥ ⊂ U +W . In this case
w ≥ n+m− 2k. (2.2)
Otherwise, if U⊥ ̸⊂ U +W then there is a B ∈ Fk+1,m(Q,W ) such that U ⊂ B.
Proof. Clearly, U is not maximal isotropic, so there is a B such that U ⊂ B ∈ Qk+1. As
in [12] we obtain m ≤ dim(B ∩W ) ≤ m+ 1. This justifies (i).
To justify (ii) note that every B ∈ Qk+1 containing U belongs to Fk+1,m(Q,W ), and
then U ≺ B ⊂ U⊥ and U ∩W ⊂ B ∩W ⊂ U⊥ ∩W . If we have dim(B ∩W ) = m for
all B, then dim(U⊥ ∩W ) = m and U ∩W = U⊥ ∩W . As U ⊂ U⊥ by definition of U ,
the obtained condition is equivalent to U⊥ ∩W ⊂ U .
In this case we have W = (U ∩W )⊕D, where D is contained in a linear complement
of U⊥. D is at most codim(U⊥) = k-dimensional, so dim(W ) ≤ m+ k.
To justify (iii) note, first, that if U ⊂ B ∈ Qk+1 then B ∈ Fk+1,m+1(Q,W ). So, if
U ≺ B ∈ Q, then B = U ⊕ ⟨y⟩ with y ∈ U⊥ \ U . If U⊥ ⊂ U +W then y = u + w for
some u ∈ U and w ∈ W \ U and then B = U ⊕ ⟨w⟩. So, B ∩W = (U ∩W ) ⊕ ⟨w⟩. If
there is y ∈ U⊥ \ (U +W ), then U + ⟨y⟩ intersects W in U ∩W .
If U is as required above then n− k = dim(U⊥) ≤ dim(U +W ) = w+ k−m. This
gives w ≥ m+ n− 2k.
From Lemmas 2.1 and 2.2(ii), 2.2(iii) we infer the following geometrical fact.
Corollary 2.3.
(i) If w > k + m then through each point of M there passes an ω-line and an affine
line.
(ii) If w < n+m− 2k then through every point of M there passes an α-line.
Combining Lemmas 2.2(i) with 2.1(ii) and 2.1(iv) we obtain the following Corollary, a
weakening of Corollary 2.3 but with more general assumptions.
Corollary 2.4. If U ∈ Fk,m(Q,W ) then there is a line in Gk,m(Q,W ) through U . Con-
sequently, if Fk,m(Q,W ) ̸= ∅, then Gk,m(Q,W ) ̸= ∅.
158 Ars Math. Contemp. 20 (2021) 151–170
Comments to Lemma 2.2.
ad (ii) Condition (2.1) is a necessary condition for the existence of an α-point.
By (1.12) and (2.1) we get rW ≤ r− k+m ≤ r−w (this implies r− rW ≥ w).
This condition is not inconsistent. So, it may happen that M contains both α-
points and ω-tops.
One can note (it is, practically, proved in the proof of Lemma 2.2(ii) that if (2.1)
is satisfied and U ∈ Qk then there is a subspace W such that U is an α-point in
Ak,m(Q,W ) and dim(W ) = w.
ad (iii) Analogously, condition (2.2) is a necessary condition for the existence of an ω-
point.
It is seen that (under suitable assumption, obtained by (1.12) and (2.2): r− rW ≥
k −m ≥ n− k − w) the space M may contain both ω-points and α-tops.
And there do exist W for which associated spine spaces contain an ω-point.
As an immediate consequence of Lemma 2.2(iii) we obtain the following.
Corollary 2.5. Assume that w < n +m + 1 − 2k. Then, for every U ∈ Fk,m+1(Q,W )
there is L ∈ Ak,m(Q,W ) such that U = L∞.
3 Examples, particular cases
Let us examine in some detail polar spine spaces of some, particularly natural classes.
3.1 Grassmannians of affine polar spaces
Assume that W is a hyperplane of P; in turn this is equivalent to say that Sub1(W ) is a
hyperplane in Q. In this case we have
m = k − 1 and (3.1)
dim(W ∩ Y ) =
{
r when Y ⊂ W
r − 1 when Y ̸⊂ W
for every Y ∈ Qr. (3.2)
It is clear that in this case
Fk,m(Q,W ) ̸= ∅; in view of Corollary 2.4, Ak,m(Q,W ) is nontrivial
simply, because it is impossible to have Qk ⊂ Subk(W ). However, this case raises several
degenerations concerning the structure of strong subspaces of M.
Lemma 3.1.
(i) Let B ∈ Subk+1(V). Then either dim(B∩W ) = k+1 = m+2 (and then B ⊂ W )
or dim(B ∩ W ) = k = m + 1. Therefore, there is no strong subspace in T α.
Moreover, by the same reasons, Lα = ∅.
(ii) If B ∈ Fk+1,m+1(Q,W ) then T(B)∩Fk,m(Q,W ) ∈ T ω is a k-dimensional punc-
tured projective space.
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 159
(iii) Let X = [H,Y ]k ∩ Fk,m(Q,W ), H ∈ Subk−1(V), Y ∈ Qr. Assume that
dim(H ∩ W ) = m = k − 1 i.e. H ⊂ W . If Y ⊂ W then, clearly, X = ∅. If
Y ̸⊂ W then X ∈ Sα is a (r − k)-dimensional affine space.
(iv) Let X = [H,Y ]k ∩ Fk,m(Q,W ), H ∈ Subk−1(V), Y ∈ Qr. Assume that
dim(H ∩ W ) = m − 1 = k − 2 i.e. H ̸⊂ W . Then Y ̸⊂ W and, consequently,
dim(Y ∩ W ) = r − 1. In this case X ∈ Sω is a (r − k)-dimensional projective
space.
Corollary 3.2. If 4 ≤ k + 2 ≤ r then every line of M has at least two extensions to a
maximal at least 2-dimensional strong subspace: one to a top, and one to a star.
3.2 Spine spaces with isotropic ‘holes’
Next, let us assume that W ∈ Q i.e. W is isotropic. In this case we have
rW = w. (3.3)
So, let m < w, k; let us take arbitrary D ∈ Subm(W ) and Y ∈ Qr with W ⊂ Y . Then
there is Y0 ∈ Qr such that Y ∩ Y0 = D. Consider any U such that dim(U) = k and
D ⊂ U ⊂ Y0; then U ∈ Fk,m(Q,W ). Thus we have proved that
Fk,m(Q,W ) ̸= ∅; in view of Corollary 2.4, Ak,m(Q,W ) is nontrivial.
Note that if we assume (1.10) then k+w−m ≤ r+ r−m ≤ n−m ≤ n follows, so (1.8)
holds as well.
Next, let us pay attention to the problem of extending lines. Namely, let L = p(H,B) ∈
Lω . So, dim(B ∩ W ) = m + 1. Suppose that r = m + 1; then we obtain contradictory
m < k < r = m+ 1. As above, we extend W to a maximal isotropic Y and find maximal
isotropic Y ′ with Y ∩ Y ′ = B. This proves
Lemma 3.3. If k < r−1, then every line in Lω can be extended to an at least 2-dimensional
star.
4 Binary collinearity
Let us start with a Chow’like result concerning binary collinearity λ of points in a polar
spine space M = Ak,m(Q,W ) defined for some integers k,m and a fixed subspace W of
a vector space V equipped with a suitable form ξ. To this aim standard reasoning similar
to this of [6, 7, 17] can be used:
a line through two distinct points is the intersection of all the maximal
λ-cliques which contain these points.
In the sequel we intensively analyse Table 2. Let U1 λ U2, U1 ̸= U2. Put L = U1, U2.
Evidently, every line L = p(H,B) can be extended to a top T = T(B) ∩ Fk,m(Q,W ),
which is a (k − m)-dimensional (T ∈ T α) or a k-dimensional (T ∈ T ω) slit space. We
have assumed that k > 1. So, when m < k − 1 then T is greater than L. For any triangle
U1, U2, U3 ∈ T we have ∆λ(U1, U2, U3) and T = [|U1, U2, U3|]λ.
If L is an α-projective line or an affine line then it has at least one extension to a star S
in Sα, which are (r − k)-dimensional slit spaces. Consequently, L = T ∩ S. Assume that
k < r − 1, so L ⊊ S.
160 Ars Math. Contemp. 20 (2021) 151–170
In this point we can choose one of the following two ways. Firstly, we notice that there
is a finite system U1, U2, . . . , Ut ∈ S such that ∆λ(U1, U2, . . . , Ut), so S′ := S =
[|U1, U2, . . . , Ut|]λ. Secondly, we can extend U1, U2 to any triangle U1, U2, U3 ∈ S and
note that S′ := [|U1, U2, U3|]λ is the union of all the extensions of the plane spanned by
U1, U2, U3 to a maximal λ-clique. In both cases L = S′ ∩ T and thus L can be defined in
terms of λ.
A problem may arise when L ∈ Lω . In this case each extension of L to a star S
is contained in a segment [H,Y ]k with a maximal totally isotropic extension Y of B ⊃
H and it has dimension dim(W ∩ Y ) − m. So, it may degenerate to the line L when
dim(W ∩ Y ) = m + 1. Is it possible that every such an extension Y intersects W in
dimension m + 1? Recall that the condition L ∈ Lω yields dim(B ∩ W ) = m + 1 and
therefore we obtain W ∩ Y = W ∩ B, for every Qr ∋ Y ⊃ B. So, our problematic case
reduces to the question: for which B ∈ Fk+1,m+1(Q,W ) there is no reasonable extension
Y and: when each such a B has a required extension. Note that to find Y it suffices to find
D such that B ≺ D ∈ Q and dim(D ∩ W ) = m + 2; then Y is an extension of D to a
maximal totally isotropic subspace. On the other hand, the existence of D in question can
be assured by a suitable substitution in Lemma 2.2(ii), which yields a sufficient condition
for the existence of our Y :
w > k +m+ 2. (4.1)
As a consequence we can formulate the following result.
Theorem 4.1 (The Chow Theorem for M). If m < k − 1, k < r − 1, and each line in
Lω can be extended to at least 2-dimensional star (which is assured, e.g. by (4.1)) then the
structures M and ⟨Fk,m(Q,W ),λ⟩ are definitionally equivalent.
In particular, in view of Corollary 3.2 and Lemma 3.3, the Chow theorem holds in M
when W is an isotropic subspace and k < r−1, and it holds in M when W is a hyperplane
and 4 ≤ k + 2 ≤ r.
One can continue these investigations in the fashion of [17] considering graphs of
collinearity with some sorts of lines distinguished (λα, λω , λα∨ω etc.). Observing cri-
teria in Lemma 2.2 and Corollary 2.5 we see that it may be a hard work: α-points and
ω-points may appear, ‘deep’ improper points may appear as well.
5 Maximal cliques of λσ
Let σ be a one of the symbols
α, ω, α ∨ ω, α+, ω+.
The classes Lσ with σ ∈ {α, ω} are already defined (usually, the arguments like k,m,
V, Q, W will be omitted, if unnecessary or fixed). Next, Lσ+ := Lσ ∪ A, and, finally
Lα∨ω = Lα ∪ Lω . It is evident that
Mσ := ⟨Fk,m(Q,W ),Lσk,m(Q,W )⟩ (5.1)
is a partial linear space for every admissible symbol σ as above, but it may be trivial for
particular values of k,m, r, w etc.: it may have a void line set. Let us write λσ for the
binary collinearity of points of Mσ . Let λaf be the binary collinearity in
A = ⟨Fk,m(Q,W ),Ak,m, ∥⟩.
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 161
In the first part of this section we shall determine (maximal) cliques of λσ for particular
values of σ as above. Clearly, each such a clique is a λ-clique. So, it is contained in an
appropriate strong subspace of M.
We begin with some results which state, generally, that the affine lines in many cases
can be ‘eliminated’: they are definable in terms of other projective lines.
Proposition 5.1. Assume that m > 0 or w < r−k. Then for arbitrary triple U1, U2, U3 ∈
Fk,m(Q,W ) we have
there is a line L0 ∈ A s.t. U1, U2, U3 ∈ L0 ⇐⇒
there is a triangle L1, L2, L3 ∈ Lα∨ωk,m s.t. Ui ∈ Li for i = 1, 2, 3
& there is no L ∈ Lα∨ω s.t. Ui, Uj ∈ L for some 1 ≤ i < j ≤ 3. (5.2)
Proof. Let U1, U2, U3 ∈ L0 ∈ A; then we can write L0 = p(H,B) ∩ Fk,m(Q,W )
for suitable H,B. As in the proof of Lemma 6.6 we examine extensions of L0 to maximal
strong subspaces of M. First, let us have a look at T(B)∩Fk,m(Q,W ). It is an affine space
only when m = 0; otherwise it contains a nonaffine semiaffine plane A which contains L0.
The lines on A are all in Lα∨ω except the direction of L0. It suffices to find adequate
triangle on A to justify (⇒:) of (5.2).
Next, assume that m = 0 and take a look at extensions of L0 of the form [H,Y ]k ∩
Fk,m(Q,W ), then B ⊂ Y ∈ Qr. This extension is an affine space when dim(W ∩ Y ) =
r − k. If there is no such Y , which is assured by the condition assumed, our extension
contains a plane A as above and (⇒:) of (5.2) is justified.
To prove (⇐:) it suffices to note that a triangle spans a plane A in M. Since this plane
contains projective lines it is not affine, and since there are non projectively joinable points
on A it contains just one direction of affine lines. The rest is evident.
Thus we have proved the following result.
Proposition 5.2. Under assumptions made in Proposition 5.1 the class Ak,m(Q,W ) is
definable in Mα∨ω . That means: M is definable in Mα∨ω .
Remark 5.3. Analysing the proof of Proposition 5.1 one can note an even more detailed
result:
(i) If m > 0 then A is definable in Mω and therefore then Mω+ is definable in Mω .
(ii) If every affine line L = p(H,B) can be extended to a non-affine star (dim(W∩Y ) ≥
r − k +m − 3 for some maximal isotropic Y containing B) then A is definable in
Mα. So, Mα
+
is definable in Mα.
For an arbitrary set X of points we write
L(X) = {L ∈ Gk,m(Q,W ) : L ⊂ X}.
Let us remind well known and fundamental classification of lines in strong subspaces of
M.
Fact 5.4. Let X be a strong subspace of M and X = L(X).
If X ∈ T α then X ⊂ Lα, if X ∈ Sα then X ⊂ Lα
+
,
if X ∈ T ω then X ⊂ Lω
+
, if X ∈ Sω then X ⊂ Lω.
162 Ars Math. Contemp. 20 (2021) 151–170
Let us note an elementary
Fact 5.5. Let S be a n0-dimensional slit space with a w0-dimensional hole i.e. let S result
from a n0-dimensional projective space by deleting a w0-dimensional subspace D. Let L0
be the class of projective lines of S and λ0 be the binary collinearity determined by L0.
Then
(i) The maximal affine subspaces of S (i.e. maximal strong subspace w.r.t. to the family
of affine lines of S) are w0+1 dimensional affine spaces. Two such subspaces either
coincide or are disjoint.
(ii) The maximal projective subspaces of S are (n0 − w0 − 1)-dimensional projective
spaces. These are linear complements of D and the elements of K∗(λ0).
(iii) Let X be a maximal projective subspace of S; then X ∈ Kn0−w0λ0 .
If w0 ≤ n0 − 3 (i.e. every projective line of S has two distinct extensions to maximal
projective subspaces) then the Chow Theorem holds:
The class L0 is definable in terms of λ0.
Observing Table 2 and Fact 5.5 we conclude with the following.
Corollary 5.6.
(i) The maximal λα-cliques are (k −m)-dimensional projective tops: elements of T α,
and (r +m− k − dim(W ∩ Y ))-dimensional projective spaces of the form
[H,E]k, where H ⊂ E ⊂ Y , E ∩ ((W ∩ Y ) +H) = H
contained in a suitable element [H,Y ]k ∩ Fk,m(Q,W ) of Sα.
(ii) The maximal λω-cliques are (dim(W ∩ Y )−m)-dimensional projective stars: ele-
ments of Sω , and m-dimensional projective spaces of the form
[G,B]k, where G ⊂ B, G ∩ (B ∩W ) = Θ
contained in a suitable element T(B) ∩ Fk,m(Q,W ) of T ω .
(iii) The maximal λα
+
-cliques are elements of T α ∪ Sα, and the maximal λω
+
-cliques
are elements of T ω ∪ Sω.
(iv) K∗(λα∨ω) = K∗(λα) ∪ K∗(λω), so the maximal λα∨ω-cliques are of the form (i)
and of the form (ii) above.
Corollary 5.7. The following variants of the Chow Theorem hold in projective reducts
of M.
(i) If m > 1 then Mω is definable in ⟨Fk,m(Q,W ),λω⟩.
(ii) If every projective line L = p(H,B) ∈ Lα can be extended to a non-affine star
(dim(W ∩ Y ) ≤ r − k +m − 2 for some maximal isotropic Y containing B) then
Mα is definable in ⟨Fk,m(Q,W ),λα⟩.
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 163
6 Parallelism, horizon, projective completion(s)
Let us summarize the following
(i) {L∞ : L ∈ Ak,m} ⊂ Fk,m+1(Q,W ).
(ii) by Lemma 2.2(ii) {L∞ : L ∈ Ak,m} ⊃ {U ∈ Fk,m+1(Q,W ) : U⊥ ∩W ̸⊂ U},
(iii) {L∞ : L ∈ Ak,m} ⊃ Fk,m+1(Q,W ), when w < n+m+1− 2k by Corollary 2.5.
Note 6.1. The set {L∞ : L ∈ Ak,m} will be frequently referred to as the horizon of M.
We warn that, generally it does not coincide with the horizon Qk \Fk,m(Q,W ) as defined
in Section 1.
Note that the inequality in (iii) above is only sufficient. One can compute e.g.
Lemma 6.2. Let W ∈ Q. Then the claim of Corollary 2.5 holds i.e. for every U ∈
Fk,m+1(Q,W ) there is an L ∈ Ak,m(Q,W ) such that U = L∞. Consequently,
{L∞ : L ∈ Ak,m} = Fk,m+1(Q,W ).
Proof. By assumption, dim(U ∩ W ) = m + 1. There are extensions Y1, Y2 ∈ Qr such
that U ⊂ Y1, W ⊂ Y2, and Y1 ∩ Y2 = U ∩ W . Take B ∈ [U, Y1]k+1; then B ∈
Fk+1,m+1(Q,W ) and we are through.
For a subset X of Fk,m(Q,W ) we write
X∞ := {N∞ : Ak,m ∋ N ⊂ X}.
Lemma 6.3. Let L = p(H,B) ∈ Lωk,m+1 ∪ Lαk,m+1.
(i) If L ∈ Lα then there is in M a plane A = [G,B]k ∩ Fk,m(Q,W ) with G ∈
Fk−2,m(Q,W ) such that A∞ = L.
(ii) Assume that w < n + m − 2k. If L ∈ Lω then A = [H,E]k ∩ Fk,m(Q,W ) with
some E ∈ Fk+2,m+2(Q,W ) is a plane in M such that A∞ = L.
Proof. Ad (i): By assumption, B ∈ Fk+1,m+1(Q,W ) and H ∈ Fk−1,m+1(Q,W ). There
is a point U ∈ L, so U ∈ Fk,m+1(Q,W ). By Lemma 2.1(iii) there is an H0 such that
U ≻ H0 ∈ Fk−1,m(Q,W ). Set G = H0 ∩H; clearly, dim(G) = k − 2, so [G,B]k is a
plane in Pk(Q). Taking into account the fact that H,H0 ≻ G we obtain dim(G ∩W ) ∈
{m+1,m} and dim(G∩W ) ∈ {m,m−1}. Thus dim(G∩W ) = m. As L ⊂ [G,B]k and
[G,B]k ⊃ [H0, B]k while [H0, B]k ∩ Fk,m(Q,W ) ∈ Ak,m we get that A ∩ Fk,m(Q,W )
is a plane in M with A∞ = L.
Ad (ii): By assumption, B ∈ Fk+1,m+2(Q,W ) and H ∈ Fk−1,m(Q,W ). As above,
we take any U ∈ L, so U ∈ Fk,m+1(Q,W ). By assumption of (ii) (they yield w < n +
(m+2)−2(k+1)) and Lemma 2.2(iii) there is an E such that B ≺ E ∈ Fk+2,m+2(Q,W ).
Next, there is B0 ∈ Fk+1,m+1(Q,W ) with U ⊂ E: B = U + ⟨b⟩ with a b ∈ W and
E = B + ⟨e⟩ with an e /∈ W ; we take B0 = U + ⟨e⟩. Clearly, E = B + B0 and
[H,B0]k ∩ Fk,m(Q,W ) ∈ Ak,m. As above we argue that A = [H,E]k ∩ Fk,m(Q,W ) is
a plane in M, and L = A∞.
164 Ars Math. Contemp. 20 (2021) 151–170
Roughly speaking, Lemma 6.3 gives sufficient condition under which a (projective) line
L of Ak,m+1(Q,W ) can be considered as a ‘horizon’ – the set of improper points of a plane
in Ak,m(Q,W ). On the other hand, considering classification of planes in Ak,m(V,W )
presented in some details in [14] we easily conclude with the following
Lemma 6.4. Let X ⊂ Subk(V) and A = X ∩ Fk,m(Q,W ) be a plane of M such that
A∞ is a line of Ak,m+1(Q,W ). Then one of the following holds:
(i) X = [G,B]k for some G ∈ Fk−2,m(Q,W ), B ∈ Fk+1,m+1(Q,W ).
(ii) X = [H,E]k for some H ∈ Fk−1,m(Q,W ) and E ∈ Fk+2,m+2(Q,W ).
Conversely, if X is defined by (i) then X ∩ Fk,m+1(Q,W ) =
(
X ∩ Fk,m(Q,W )
)∞ ∈
Lαk,m+1, and if (ii) holds, then X ∩ Fk,m+1(Q,W ) ∈ Lωk,m+1.
So, Lemma 6.4 states that the ‘horizon’ of any (affine) plane of Fk,m(Q,W ) is a (pro-
jective) line of Fk,m+1(Q,W ). As usually, the conditions of Lemma 6.3 are only suffi-
cient. Dealing with concrete cases one should look for suitable extendability more or less
‘by hand’. Let us quote an example:
Lemma 6.5. Let W ∈ Q. If L = p(H,B) ∈ Lωk,m+1 then A = [H,E]k ∩ Fk,m(Q,W )
with some E ∈ Fk+2,m+2(Q,W ) is a plane in M such that A∞ = L.
Hint. With the reasoning as in the proof of Lemma 6.3(ii) we look for an E such that
B ≺ E ∈ Fk+2,m+2(Q,W ). It suffices to find an E such that E ∩ W = B ∩ W just
considering suitable maximal isotropic extensions of B and W .
To accomplish this part of investigations on the parallelism let us check if directions are
‘isolated’: when for an affine line L of M there are other lines parallel to L and coplanar
with L; with the plane in question being affine in M.
Lemma 6.6. Let L = p(H,B) ∈ Ak,m and U = L∞.
(i) Assume that k > m + 1. There is an L0 = p(H0, B) ∈ Lαk,m+1 such that U ∈ L0
and A = [H0∩H,B]k∩Fk,m(Q,W ) is a plane in M such that A∞ = L0. We have
dim((H0 ∩H) ∩W ) = m− 1.
(ii) If B has an extension to a Y ∈ Qr such that dim(W ∩ Y ) ≥ m + 2 (this yields,
necessarily, m+ 2 ≤ rW ) then there exists an L1 = p(H,B1) ∈ Lωk,m+1 such that
U ∈ L1 and A = [H,B + B1]k is a plane in M such that A∞ = L1. We have
dim((B1 +B) ∩W ) = m+ 2.
Proof. Let us begin with a reminder: H ∈ Fk−1,m(Q,W ), B ∈ Fk+1,m+1(Q,W ). Have
a look at the extension of L to a top T = T(B) ∩ Fk,m(Q,W ) (an ω-top in this case).
Since k > m + 1, this is a semiaffine space, and its hole is at least 1-dimensional. Let L0
be any line of Pk(V) contained in this hole and A be the plane spanned by L ∪ L0. That
way we justify (i).
Next, let us look for appropriate extension of L to an α-star S = [H,Y ]k∩Fk,m(Q,W ).
In general, it is a (r−k)-dimensional semiaffine space. Since Qk+1 ̸= ∅ we have k+1 ≤ r.
So, S is at least a line. To assure that the hole of S contains at least a line of Pk(V) we
must assume that dim(W ∩ Y ) ≥ m+ 2. That way we justify (ii).
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 165
Let us remind that for distinct affine lines L1, L2 contained in a strong subspace of
Ak,m(V,W ) their parallelism ∥ can be characterized by the following formula (so called
Veblenian parallelism).
L1 ∥v L2 ⇐⇒ there are lines L′1, L′2 s.t. |L′1 ∩ L′2| = 1,
and L′1 ∩ L′2 ∩ Li = ∅, |L′i ∩ Lj | = 1 for i = 1, 2, (6.1)
and then L1 ∥ L2 iff L1 ∥v L2. It is easy to note that the same formula (6.1) characterizes
parallelism of affine lines contained in a common strong subspace of M.
Let us begin with a special form of connectedness of the space of lines over M:
Lemma 6.7. Let U ∈ Fk,m+1(Q,W ) and L∞1 = U = L∞2 for L1, L2 ∈ Ak,m. Moreover,
assume that k ≤ r − 2. Then there are lines M1, . . . ,Mt ∈ Ak,m (t ≤ r + 1) such that
L1 = M1, L2 = Mt, and M∞i = U , Mi,Mi+1 are in a strong (semiaffine) subspace of
M or Mi = Mi+1, for i = 1, . . . , t− 1.
Proof. Write M1 := L1. We have H1, H2 ⊂ U ⊂ B1, B2, U ∈ Fk,m+1(Q,W ) and Bi ∈
Fk+1,m+1(Q,W ), Hi ∈ Fk−1,m(Q,W ) for i = 1, 2. Put N1 := [H1, B2]k∩Fk,m(Q,W ),
N2 := [H2, B1]k ∩ Fk,m(Q,W ). Then N1, N2 ∈ Ak,m, N∞2 = U = N∞1 .
If L1 = N2 we set M2 := L1. Assume that L1 ̸= N2. Note that L1, N2 ∈ T(B1) ∩
Fk,m(Q,W ) ∈ T ω . So, we set M2 := N2.
Observe that N2, L2 ⊂ [H2, V ]k ∩ Fk,m(Q,W ). So, the problem reduces to find a
required sequence of lines in the projective star S(H2). Let B1 ⊂ Y ′ ∈ Qr, B2 ⊂ Y ′′ ∈
Qr. There is a sequence Y2, . . . , Yt of elements of Qr such that Y ′ = Y2, Y ′′ = Yt, and
U ⊂ Yi, Ei := Yi ∩ Yi+1, dim(Ei) = r − 1 for i = 2, . . . , t − 1, t ≤ r + 1. Then
dim(Ei ∩W ) ≥ m + 1. From our assumption k + 1 ≤ r − 1 = dim(Ei). So, for every
i = 3, . . . , t− 1 one can find Di such that U ≺ Di ⊂ Ei−1 and dim(Di ∩W ) = m+ 1.
With Ni = [H2, Di]k we close our proof.
Corollary 6.8. Under assumptions of Lemma 6.7 the parallelism ∥ in M coincides with
the transitive closure of ∥v. Actually, it is the (r + 1)-th relational power ∥v ◦ · · · ◦ ∥v︸ ︷︷ ︸
(r+1) times
of
∥v, defined by (6.1), and therefore ∥ is definable in the incidence structure M.
As an immediate corollary we conclude with the following theorem.
Theorem 6.9. Assume the following
(1) w < n+m− 2k to assure that every line in Lωk,m+1 can be extended to a nontrivial
α-star of M (cf. Lemma 6.3),
(2) w < n+m+ 1− 2k to assure extendability of each improper point to an affine line
(cf. Corollary 2.5),
(3) m+1 > 0 or w < r−k to assure definability of Ak,m+1 in Ak,m+1(Q,W ) in terms
of its projective lines (cf. Proposition 5.2),
(4) k ≤ r − 2, to assure definability of parallelism in M (cf. Corollary 6.8).
Then Ak,m+1(Q,W ) is definable within Ak,m(Q,W ).
166 Ars Math. Contemp. 20 (2021) 151–170
In analogy to [17] in the fragment of Pk(Q) determined by R := Fk,m(Q,W ) ∪
Fk,m+1(Q,W ) (i.e. the points of M and the points of the “affine horizon” of M) we
distinguish two substructures corresponding to two possible sorts of lines. Let us set
Lτk,m :=
{
[H,B]k : H ∈ Fk−1,m(Q,W ), B ∈ Fk+1,m+1(Q,W )
}
; it is seen that
Lτk,m = {L : L ∈ Ak,m}. Note evident relation:{
L : L is a line of Pk(Q), L ⊂ R
}
= Lαk,m ∪ Lωk,m ∪ Lαk,m+1 ∪ Lωk,m+1 ∪ Lτk,m. (6.2)
We define (write: −α = ω, −ω = α)
Nσ := ⟨R,Lσk,m ∪ Lτk,m ∪ L−σk,m+1⟩ with σ ∈ {α, ω}.
Evidently, Mσ can be embedded into Nσ . Intuitively, while the structure〈
R, {L : L is a line of Pk(Q), L ⊂ R}
〉
can be considered as a projective completion of M and, under specific assumptions, it is
definable in M, Nσ is a projective completion of Mσ .
To close this part it is worth to note the following analogue of Remark 5.3 and, at the
same time, an analogue of [17, Fact 3.1].
Remark 6.10. Assume (2) and (4) from Theorem 6.9.
(i) If m > 0 (cf. Remark 5.3) then the structure Nω is definable in Mω .
(ii) If for each affine line L = p(H,B) there is a maximal isotropic Y such that B ⊂ Y
and dim(W ∩ Y ) ≥ r − k + m − 3 (cf. Remark 5.3) and w < n + m − 2k (cf.
Lemma 6.3), then the structure Nα is definable in Mα.
According to Corollary 2.5 and Lemma 6.3, under condition w < n + m − 2k each
point of Ak,m+1(Q,W ) is a direction of a line in M and each line of Ak,m+1(Q,W ) is a
direction of a plane in M. This observation leads to the following.
Proposition 6.11. If w < n + m − 2k, then the horizon Ak,m+1(Q,W ) of M can be
defined in terms of A.
Finally, the question arises whether the adjacency of M is definable purely in terms
of the geometry of A? Unfortunately, the answer is not straightforward. The reasoning
for spine spaces that justifies [17, Proposition 4.12], based on the fact that two distinct
stars or tops of A share no line on the horizon, cannot be adopted here without significant
alterations. Note that if Lω ∪ Lα = ∅, then practically A = M. Therefore we assume that
Lω ∪ Lα ̸= ∅.
Theorem 6.12. If the ground field of V is of odd characteristic, then the structure M can
be defined in terms of A.
Proof. The proof is divided into several steps. For distinct points U1, U2 of M we define
U1 ∼+ U2 :⇐⇒ U1, U2 ⊂ B for some B ∈ Fk+1,m+1(Q,W ),
U1 ∼− U2 :⇐⇒ H ⊂ U1, U2 for some H ∈ Fk−1,m(Q,W ),
U1 ∼ U2 :⇐⇒ U1 ∼+ U2 or U1 ∼− U2.
Note that U1 ∼+ U2 yields that either U1 λaf U2 or U1 λω U2, while U1 ∼− U2 yields that
either U1 λaf U2, U1 λα U2, or U1, U2 are not collinear in M.
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 167
Step 1. The following conditions are equivalent.
(i) U1 ∼+ U2 or U1 ∼− U2.
(ii) There is a plane Π1 through U1 parallel to a plane Π2 through U2 in A.
Proof of Step 1. (i) =⇒ (ii): Assume that U1, U2 ⊂ B ∈ Fk+1,m+1(Q,W ), then T(B) is
a semiaffine space (of the form T ω) and one easily finds Π1,Π2 in it.
Next, assume that U1, U2 ⊃ H ∈ Fk−1,m(Q,W ). Set B := U1 + U2. If B ∈ Q
then L = p(H,B) is a line of M. Applying analogous reasoning we find Π1,Π2 in an
extension [H,Y ]k of the type Sα. If B /∈ Q then, in any case L is a line of the surrounding
Ak,m(V,W ). Let us restrict to the subspaces around H; they form a spine space in the
projective space P1(V/H) with the quadric Q(ξ/H) distinguished. Projective reasoning
proves that required planes Π1,Π2 exist.
(ii) =⇒ (i): Let Πi be parallel planes of A with Ui ∈ Πi, i = 1, 2. Let L0 = Π∞1 = Π∞2
be the improper line of Πi. Then L0 ∈ Lαk,m+1 or L0 ∈ Lωk,m+1. In the first case L0, U1, U2
are contained in the (unique) extension to a top T(B) with B ∈ Fk+1,m+1(Q,W ) and
therefore U1 ∼+ U2. In the second case extensions of Πi to maximal strong subspaces have
form [H,Yi]k (they have L0 in common), where H ∈ Fk−1,m(Q,W ). So, U1 ∼− U2. ♢
Let us write
M0 := Ak,m(V,W ) ↾ Fk,m(Q,W )
for the surrounding spine space with point set restricted to totally isotropic subspaces. Note
that the distinction between M and M0 consists in the range of their line sets. More pre-
cisely, for a line L = p(H,B) of M0 its base B needs not to be totally isotropic and
L is a line of M iff |L| ≥ 3.
Step 2. Let U1, U2 ∈ Fk,m(Q,W ) and U1 ̸= U2. The following conditions are equivalent.
(i) U1 ∼ U2.
(ii) U1, U2 are collinear in M0 with exception when the line L of M0 which joins them
has form L = p(H,B) where H ∈ Fk−1,m−1(Q,W ), B ∈ Fk+1,m+1(V,W ), and
B /∈ Q (i.e. L is an ω-line in Ak,m(V,W )).
Proof of Step 2. (i) =⇒ (ii): It is clear that U1, U2 are collinear in the surrounding Grass-
mann space. If U1 ∼+ U2, then they lie on an affine or ω-line in M by Table 1, while if
U1 ∼− U2, then they lie on an affine or α-line in M0.
(ii) =⇒ (i): Now, let U1, U2 be collinear in M0. Hence U1, U2 ∈ p(H,B) for suitable
H,B. If dim(B∩W ) = m, then dim(H∩W ) = m and thus U1 ∼− U2. If dim(B∩W ) =
m+1, then two cases arise: dim(H ∩W ) = m,m− 1. In the former we have U1 ∼− U2.
In the later H ∈ Fk−1,m−1(Q,W ) and B ∈ Fk+1,m+1(V,W ). If B ∈ Q, then U1 ∼+ U2,
otherwise we get the excluded case. ♢
Step 3. A set X of points of A is a maximal at least 3-element ∼-clique iff X has one of
the following forms:
(a) X = T(B) for some B ∈ Fk+1,m+1(Q,W ),
(b) X = T(B) for some B ∈ Fk+1,m(Q,W ),
168 Ars Math. Contemp. 20 (2021) 151–170
(c) X = S(H) for some H ∈ Fk−1,m(Q,W ), or
(d) X = [H,Y ] ∩ Fk,m(Q,W ) for some H ∈ Fk−1,m−1(Q,W ) and H ⊂ Y ∈ Qr.
Proof of Step 3. It is easy to verify that sets defined in (a) – (d) are maximal ∼-cliques.
Now, let X be a maximal at least 3-element ∼-clique. In view of Step 2, X is a subset of
a clique in M0. So, we need general tops T0(B) = [Θ, B]k for B ∈ Subk+1(V) and stars
S0(H) = [H,V ]k for H ∈ Subk−1(V). Let us examine the following four cases:
X ⊆ T0(B), B ∈ Fk+1,m+1(V,W )
If B ∈ Q, then any two points of M0 in T(B) are ∼+-adjacent and thus X =
T(B) ∩ Fk,m(Q,W ) is a ∼-clique as in (a). If B /∈ Q, then |X| ≤ 2 by [19,
Proposition 4.4], a contradiction.
X ⊆ T0(B), B ∈ Fk+1,m(V,W )
Since |X| ≥ 3 we have B ∈ Q by [19, Proposition 4.4]. Any two points of M0 in
T(B) are ∼−-adjacent, so X = T(B) ∩ Fk,m(Q,W ) has form (b).
X ⊆ S0(H), H ∈ Fk−1,m(V,W )
Note that H ∈ Q as X is nonempty. This implies that any two points of M0 in S(H)
are ∼−-adjacent. Consequently, X = S(H) ∩ Fk,m(Q,W ) has form (c).
X ⊆ S0(H), H ∈ Fk−1,m−1(V,W )
As above H ∈ Q. The points of M0 in S0(H) are ∼-adjacent iff they are ∼+-
adjacent i.e. they are collinear in the surrounding polar Grassmann space where the
appropriate clique has form [H,Y ]k for some Y ∈ Qr (cf. [7, Section 3]). Hence
X = [H,Y ]k ∩ Fk,m(Q,W ) has form (d).
That way we obtain the desired list (a) – (d). ♢
Note that the λaf-cliques are essentially smaller than ∼-cliques.
Step 4. At least 3-element minimal intersections of the maximal ∼-cliques are lines of M.
Proof of Step 4. Let Kx be the family of cliques of the form (x) defined in Step 3. Let
X1, X2 be two distinct ∼-cliques and Z = X1 ∩X2. If X1, X2 ∈ K(a) ∪K(b), X1, X2 ∈
K(c), or X1 ∈ K(b) ∪ K(c) and X2 ∈ K(d), then Z contains at most a single point. If
X1 ∈ K(a) and X2 ∈ K(c), then Z is an affine line of M. If X1 ∈ K(b) and X2 ∈ K(c), then
Z is an α-line of M. If either X1 ∈ K(a) and X2 ∈ K(d) or X1, X2 ∈ K(d), then at least
3-element minimal Z is an ω-line of M. ♢
It is evident that every projective line of M can be presented as the intersection of
cliques enumerated in Step 3. So, applying Step 4 we get the line set of M recovered
which makes the proof of Theorem 6.12 complete.
Remark 6.13. The horizon of a star in M may have strange properties. Assume that
W ∈ Q and let H ∈ Qk−1, H ⊂ W⊥. Set m := dim(H ∩ W ). This means that
k− 1 +w−m < r. Then there is an Y0 ∈ Qr such that H ∪W ⊂ Y0. So, Y0 ∩W = W .
Write S0 = [H,Y0]k ∩Fk,m(Q,W ). Then S∞0 = [H,H +W ]k. Take any S = [H,Y ]k ∩
Fk,m(Q,W ) contained in S(H). Then S∞ = [H,H+(W ∩Y )]k ⊂ [H,H+W ]k = S∞0 .
So, in this case
S(H)∞ is the projective space [H,H +W ]k contained in M∞.
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 169
Nevertheless, S(H) contains affine subspaces of different dimensions.
Note that in this case k−m−1 = dim(T(B)∞) = dim(S(H)∞) = w−m−1 yields
w = k, so horizons of stars and tops may have equal dimensions only when M consists of
points in Qk that are at the fixed distance k −m from the fixed point W .
Remark 6.14. Theorem 6.12 for polar spine spaces and its counterpart [17, Proposi-
tion 4.12] for spine spaces both say that the respective geometry depends only on affine
lines together with parallelism, that is, projective lines can be recovered using affine line
structure. However, the idea of the proof presented in this paper is more general than that
in [17] because it does not rely on specific horizons and intersections of stars which are
completely different in M and in Ak,m(V,W ). As such it can be applied for spine spaces
and is expected to give less complex reasonings.
7 Classifications
Table 1: The classification of lines in a polar spine space Ak,m(Q,W ).
Class Representative line g = p(H,B) ∩ Fk,m(Q,W ) g∞
Ak,m(Q,W ) H ∈ Fk−1,m(Q,W ), B ∈ Fk+1,m+1(Q,W ) H + (B ∩W )
Lαk,m(Q,W ) H ∈ Fk−1,m(Q,W ), B ∈ Fk+1,m(Q,W ) –
Lωk,m(Q,W ) H ∈ Fk−1,m−1(Q,W ), B ∈ Fk+1,m+1(Q,W ) –
Each strong subspace X of a polar spine space is a slit space, that is a projective space P
with a subspace D removed. In the extremes D can be void, then X is basically a projective
space, or a hyperplane, then X is an affine space.
Table 2: The classification of stars and tops in a polar spine space Ak,m(Q,W ).
Class Representative subspace
dim(P) D dim(D)
Sωk,m(Q,W )
[H, (H +W ) ∩ Y ]k : H ∈ Fk−1,m−1(Q,W ), Y ∈ Qr, H ⊂ Y
dim(W ∩ Y )−m ∅ -1
Sαk,m(Q,W )
[H,Y ]k ∩ Fk,m(Q,W ) : H ∈ Fk−1,m(Q,W ), Y ∈ Qr, H ⊂ Y
r − k [H, (H +W ) ∩ Y ]k dim(W ∩ Y )−m− 1
T αk,m(Q,W )
[B ∩W,B]k : B ∈ Fk+1,m(Q,W )
k −m ∅ -1
T ωk,m(Q,W )
[Θ, B]k ∩ Fk,m(Q,W ) : B ∈ Fk+1,m+1(Q,W )
k [B ∩W,B]k k −m− 1
170 Ars Math. Contemp. 20 (2021) 151–170
ORCID iDs
Krzysztof Petelczyc https://orcid.org/0000-0003-0500-9699
Krzysztof Prażmowski https://orcid.org/0000-0002-5352-5973
Mariusz Żynel https://orcid.org/0000-0001-9297-4774
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Jonathan E. Leech: Noncommutative Lattices:
Skew Lattices, Skew Boolean Algebras and Beyond
About the book: The extended study of non-commutative lattices was begun in 1949
by Ernst Pascual Jordan, a theoretical and mathematical physicist and co-worker of Max
Born and Werner Karl Heisenberg. Jordan introduced noncommutative lattices as algebraic
structures potentially suitable to encompass the logic of the quantum world. The modern
theory of noncommutative lattices began forty years later with Jonathan Leech’s 1989 paper
“Skew lattices in rings.” Recently, noncommutative generalizations of lattices and related
structures have seen an upsurge in interest, with new ideas and applications emerging,
from quasilattices to skew Heyting algebras. Much of this activity is derived in some way
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consists of seven chapters, mainly covering skew lattices, quasilattices and paralattices,
skew lattices of idempotents in rings and skew Boolean algebras. As such, it is the first
research monograph covering major results due to this renewed study of noncommutative
lattices. It will serve as a valuable graduate textbook on the subject, as well as a handy
reference to researchers of noncommutative algebras.
About the author: Jonathan Leech graduated from the University of Hawaii and earned
a PhD at the University of California, Los Angeles. He has taught mathematics at the
University of Tennessee, later at Missouri Western State University and finally at Westmont
College in Santa Barbara, California. He has been a Visiting Professor at Case Western
Reserve University, the Universidad de Granada in Spain and Universidade Mackenzie
vii
in Brazil, and a scholar in residence at both the University of Sidney and the University
of Tasmania in Australia. Throughout his academic career Professor Leech has studied
algebraic structures related to semigroups, with much of his emphasis being on the theory
of noncommutative lattices, and of skew lattices in particular. He laid the foundations
of the modern theory of noncommutative lattices in a number of (co)authored seminal
publications. His work has inspired many mathematicians around the world to pursue
research in this area.
J. E. Leech, Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and
Beyond, volume 4 of Famnit Lectures, Slovenian Discrete and Applied Mathematics
Society and University of Primorska Press, Koper, 2021, 284 pp., ISBN 978-961-
95273-0-6.
The paperback edition of the book was published on March 5, 2021 by SDAMS,
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viii
Petra Šparl Award 2022: Call for Nominations
The Petra Šparl Award was established in 2017 to recognise in each even-numbered year
the best paper published in the previous five years by a young woman mathematician in
one of the two journals Ars Mathematica Contemporanea (AMC) and The Art of Discrete
and Applied Mathematics (ADAM). It was named after Dr Petra Šparl, a talented woman
mathematician who died mid-career in 2016.
The award consists of a certificate with the recipient’s name, and invitations to give a
lecture at the Mathematics Colloquium at the University of Primorska, and lectures at the
University of Maribor and University of Ljubljana. The first award was made in 2018 to
Dr Monika Pilśniak (AGH University, Poland) for a paper on the distinguishing index of
graphs, and then two awards were made for 2020, to Dr Simona Bonvicini (Università di
Modena e Reggio Emilia, Italy) for her contributions to a paper giving solutions to some
Hamilton-Waterloo problems, and Dr Klavdija Kutnar (University of Primorska, Slovenia),
for her contributions to a paper on odd automorphisms in vertex-transitive graphs.
The Petra Šparl Award Committee is now calling for nominations for the next award.
Eligibility: Each nominee must be a woman author or co-author of a paper published in
either AMC or ADAM in the calendar years 2017 to 2021, who was at most 40 years old at
the time of the paper’s first submission.
Nomination Format: Each nomination should specify the following:
(a) the name, birth-date and affiliation of the candidate;
(b) the title and other bibliographic details of the paper for which the award is recom-
mended;
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most 500 words; and
(d) names and email addresses of one or two referees who could be consulted with regard
to the quality of the paper.
Procedure: Nominations should be submitted by email to any one of the three members
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Award Committee:
• Marston Conder, m.conder@auckland.ac.nz
• Asia Ivić Weiss, weiss@yorku.ca
• Aleksander Malnič, aleksander.malnic@guest.arnes.si
Marston Conder, Asia Ivić Weiss and Aleksander Malnič
Members of the 2022 Petra Šparl Award Committee
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