ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P2.01
https://doi.org/10.26493/2590-9770.1403.1b1
(Also available at http://adam-journal.eu)
On 12-regular nut graphs*
Nino Bašić†
FAMNIT & IAM, University of Primorska, 6000 Koper, Slovenia, and
Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia
Martin Knor‡
Department of Mathematics, Faculty of Civil Engineering, Slovak University of
Technology in Bratislava, Radlinského 11, 810 05, Bratislava, Slovakia
Riste Škrekovski§
Faculty of Information Studies, 8000 Novo mesto, Slovenia, and
FMF, University of Ljubljana, 1000 Ljubljana, Slovenia, and
FAMNIT, University of Primorska, 6000 Koper, Slovenia
Received 8 February 2021, accepted 29 May 2021, published online 11 May 2022
Abstract
A nut graph is a simple graph whose adjacency matrix is singular with 1-dimensional
kernel such that the corresponding eigenvector has no zero entries. In 2020, Fowler et al.
characterised for each d ∈ {3, 4, . . . , 11} all values n such that there exists a d-regular nut
graph of order n. In the present paper, we resolve the first open case d = 12, i.e. we show
that there exists a 12-regular nut graph of order n if and only if n ≥ 16. We also present a
result by which there are infinitely many circulant nut graphs of degree d ≡ 0 (mod 4) and
no circulant nut graphs of degree d ≡ 2 (mod 4). The former result partially resolves a
question by Fowler et al. on existence of vertex-transitive nut graphs of order n and degree
d. We conclude the paper with problems, conjectures and ideas for further work.
Keywords: Nut graph, adjacency matrix, singular matrix, core graph, Fowler construction, regular
graph.
Math. Subj. Class.: 05C50, 15A18
*We would like to thank the two anonymous referees for their comments which helped to improve the presen-
tation of the paper.
†The work of the first author is supported in part by the Slovenian Research Agency (research program P1-
0294 and research projects J1-9187, J1-1691, N1-0140 and J1-2481).
‡The second author acknowledges partial support by Slovak research grants APVV-15-0220, APVV-17-0428,
VEGA 1/0206/20 and VEGA 1/0238/19.
§The research of the third author was partially supported by the Slovenian Research Agency (ARRS), research
program P1-0383 and research projects J1-1692 and J1-8130.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P2.01
1 Introduction
Let G be a simple graph with the vertex set V (G) = {0, 1, . . . , n−1}. Its adjacency matrix
A is a symmetric n×n matrix with entries ai,j , where 0 ≤ i, j ≤ n−1, such that ai,j = 1 if
{i, j} is an edge of G and ai,j = 0 otherwise. A graph G is a nut graph if A has eigenvalue
0 and no eigenvector has zero entries. As a consequence, the eigenspace corresponding
to the eigenvalue 0 is 1-dimensional. Observe that if the eigenspace corresponding to 0
has dimension greater than one, then there exists an eigenvector containing entry 0 that is
different from 0 = (0, 0, . . . , 0)T . For an introductory treatment of spectral graph theory,
which links graphs to linear algebra, see e.g. [3, 4, 7].
Nut graphs have been studied in [6, 9, 11, 12, 16, 17, 18, 19, 20, 22], see also the
webpage https://hog.grinvin.org/Nuts within the House of Graphs [2, 5]. Re-
cently, this concept was extended to signed graphs [1]. Nut graphs have chemical applica-
tions, see e.g. [9, 8, 21]. However, in the present paper we consider 12-regular graphs, so
our motivation is purely mathematical.
In [22], Gutman and Sciriha showed that the smallest non-trivial nut graph has order 7.
In [10], Fowler et al. determined all nut graphs on up to 10 vertices and all chemical nut
graphs on up to 16 vertices. The smallest order for which a regular nut graph exists is 8;
see also [9]. In [9], Fowler et al. presented the following question.
If there exists a d-regular graph of order n, then we say that the order n is admissible
regarding the degree d. Obviously, if d is even then every n ≥ d + 1 is admissible. If d is
odd then every even n ≥ d+ 1 is admissible.
Question 1.1. Is it true that for each degree d ≥ 3 there are only finitely many admissible
orders n such that there does not exist a d-regular nut graph of order n?
In the attempt to answer Question 1.1, the ‘Fowler Construction’ played an important
role; see also [11]. This construction implies the following theorem.
Theorem 1.2. Let G be a nut graph on n vertices and let u be a vertex of G of degree d.
Then there exists a nut graph of order n + 2d that is obtained from G by adding 2d new
vertices and rearranging the edges in a certain way. In the newly obtained nut graph the
degrees of the new vertices are d and the degrees of the original vertices are not changed.
Obviously, if G is a d-regular graph of order n, then the new graph is d-regular of order
n + 2d. Hence, to answer Question 1.1 positively for specific degree d, it suffices to find
d-regular graphs for 2d consecutive orders. In [11] (d = 3, 4) and [9] (5 ≤ d ≤ 11), the
authors found all pairs (d, n), such that d ≤ 11 and there exists a d-regular nut graph of
order n. In the present paper, we extend this result to d = 12. We prove the following
statement.
Theorem 1.3. There exists a 12-regular nut graph of order n if and only if n ≥ 16.
To prove the ‘positive part’ of Theorem 1.3, it suffices to find 12-regular nut graphs of
orders n ∈ {16, 17, . . . , 39}. We present these graphs in the following section. For odd
orders there is not much to say; we did a computer search and thus we provide a list of
graphs that we found. However, for even orders we can say more.
E-mail addresses: nino.basic@famnit.upr.si (Nino Bašić), knor@math.sk (Martin Knor),
skrekovski@gmail.com (Riste Škrekovski)
N. Bašić et al.: On 12-regular nut graphs 3
A graph G is called vertex-transitive if all vertices are equivalent under the action of the
automorphism group Aut(G). In other words, for each pair of vertices u, v ∈ V (G) there
exist an automorphism α ∈ Aut(G) such that α(u) = v. In [9], the following necessary
condition for a vertex-transitive nut graph was given.
Theorem 1.4. Let G be a vertex-transitive nut graph of degree d on n vertices. Then n and
d satisfy the following conditions. Either
(1) d ≡ 0 (mod 4), n ≡ 0 (mod 2) and n ≥ d+ 4, or
(2) d ≡ 0 (mod 2), n ≡ 0 (mod 4) and n ≥ d+ 6.
The existence of vertex-transitive nut graphs is interesting in its own right, see [9, Ques-
tion 9]. For our research it is important that, by Theorem 1.4, there may exist vertex-
transitive 12-regular graphs of even orders n ≥ 16. We found such graphs among circulant
graphs.
2 Results
We start with the ‘negative part’ of Theorem 1.3. There is only one 12-regular graph of
order 13, namely the complete graph K13, and it is not a nut graph. The unique 12-regular
graph of order 14 is obtained by removing a matching from K14, and again, this graph is
not a nut graph. Finally, there are 17 graphs of order 15 which are 12-regular. They are
obtained by removing a 2-factor from K15. Using the SageMath software [23], we analysed
all such graphs and concluded that none of them is a nut graph.
Now we turn our attention to the ‘positive part’ of Theorem 1.3. We start with more
general results for even orders. The following lemma is in fact implied in the text preceding
Proposition 1 in [11]. We present it here in a slightly more general setting together with its
short proof.
Lemma 2.1. Let G be a d-regular graph on n vertices such that its adjacency matrix A is
singular. Then for every eigenvector c = (c0, c1, . . . , cn−1)
T corresponding to eigenvalue
0, we have
n−1∑
i=0
ci = 0.
Proof. In every d-regular graph, the eigenvector 1 = (1, 1, . . . , 1) corresponds to the eigen-
value d. Since eigenspaces are mutually orthogonal, we have c · 1 = 0.
Let V = {0, 1, . . . , n−1} and let 1 ≤ a1 < a2 < · · · < at ≤ n2 . By C(n, {a1, a2, . . . ,
at}) we denote a graph on the vertex set V in which two vertices i, j ∈ V are adjacent
if and only if |i − j| = ak, where 1 ≤ k ≤ t. The graph C(n, {a1, a2, . . . , at}) is
called a circulant graph and it is regular. Its degree is 2t − 1 if at = n2 and 2t otherwise.
In fact, circulant graphs are vertex-transitive since ϕ : i → i + 1 is an automorphism of
C(n, {a1, a2, . . . , at}) (the addition is modulo n).
Circulant graphs are easy to describe and easy to handle. Therefore, it would be nice
if there were many nut graphs among them. We prove one positive and one negative result
about circulant graphs. We start with the following lemma.
Lemma 2.2. Let G = C(n, {a1, a2, . . . , at}) be a circulant nut graph, and let A be its ad-
jacency matrix. Then (1,−1, 1,−1, . . . )T is an eigenvector corresponding to eigenvalue 0.
4 Art Discrete Appl. Math. 5 (2022) #P2.01
Proof. We use the well-known fact that if b and c are eigenvectors corresponding to eigen-
value λ, then b+ c is also an eigenvector corresponding to eigenvalue λ.
Let b = (b0, b1, . . . , bn−1)T be an eigenvector corresponding to 0. Denote b0 = p and
b1 = q. Since ϕ : i → 2− i is an automorphism of G (the addition being modulo n), there
is an eigenvector c = (c0, c1, . . . , cn−1)T such that c2−i = −bi, 0 ≤ i ≤ n − 1. Then
c1 = −b1 = −q and c2 = −b0 = −p. Since b1 + c1 = 0 and b + c is an eigenvector,
we must have b + c = 0 because G is a nut graph. Hence, b2 + c2 = 0 and therefore
b2 = p. Now repeating the process we get b = (p, q, p, q, . . . ). Observe that n is even
by Theorem 1.4. Thus, by Lemma 2.1, we have q = −p and so (1,−1, 1,−1, . . . ) is an
eigenvector corresponding to eigenvalue 0.
Our negative result covers all circulant graphs of degree d ≡ 2 (mod 4).
Theorem 2.3. There is no circulant nut graph of degree d if d ≡ 2 (mod 4).
Proof. Let d ≡ 2 (mod 4). Denote t = d2 . Observe that t is an odd number. By way of
contradiction, assume that G = C(n, {a1, a2, . . . , at}) is a circulant nut graph. Then n
is even by Theorem 1.4. Let A = (a0,a1, . . . ,an−1)T be the adjacency matrix of G. By
Lemma 2.2, c = (1,−1, 1,−1, . . . )T is an eigenvector corresponding to eigenvalue 0, so
that Ac = 0, and in particular a0c = 0. However,
a0c = ca1 + ca2 + · · ·+ cat + cn−a1 + cn−a2 + · · ·+ cn−at .
Since cai = cn−ai for every i, 1 ≤ i ≤ t (observe that the difference between indices
ai and n − ai is even), we have a0c = 2(ca1 + ca2 + · · · + cat), which implies that
ca1 + ca2 + · · · + cat = 0. However, the sum of an odd number of odd numbers is odd, a
contradiction.
Now we prove the positive result. Notice that this result also partially resolves Ques-
tion 9 from [9] about the existence of vertex-transitive nut graphs of order n and degree d.
Theorem 2.4. Let d ≡ 0 (mod 4) and let n be even. Then C(n, {1, 2, . . . , d2}) is a nut
graph if and only if d2 + 1 is coprime to n and
d
4 is coprime to
n
2 .
Proof. Let t = d2 . Then t is even and the graph is G = C(n, {1, 2, . . . , t}).
Let A be the adjacency matrix of G. By Lemma 2.2, b = (1,−1, 1,−1, . . . )T is an
eigenvector of A corresponding to eigenvalue 0. Thus Ab = 0. Our aim is to show that if
t+1 is coprime to n and t2 is coprime to
n
2 , then Ac = 0 if and only if c is a multiple of b.
So let Ac = 0, where c = (c0, c1, . . . , cn−1)T . Let A = (a0,a1, . . . ,an−1)T . Then
atc = c0 + c1 + · · ·+ ct−1 + ct+1 + ct+2 + · · ·+ c2t = 0,
at+1c = c1 + c2 + · · ·+ ct + ct+2 + ct+3 + · · ·+ c2t+1 = 0.
Subtracting the two equations we get
atc− at+1c = c0 − ct + ct+1 − c2t+1 = 0,
and analogously
a2t+1c− a2t+2c = ct+1 − c2t+1 + c2t+2 − c3t+2 = 0.
N. Bašić et al.: On 12-regular nut graphs 5
This gives
c0 − ct = c2t+2 − c3t+2,
and analogously
c2t+2 − c3t+2 = c4t+4 − c5t+4,
c4t+4 − c5t+4 = c6t+6 − c7t+5, etc.
So if the odd number t+ 1 is coprime to the even number n, we get
c0 − ct = c2(t+1) − ct+2(t+1) = · · · = c2 − ct+2,
which gives
c2 − c0 = ct+2 − ct,
and analogously we get
ct+2 − ct = c2t+2 − c2t,
c2t+2 − c2t = c3t+2 − c3t, etc.
Here, t and n are both even. But if t2 is coprime to
n
2 then
c2 − c0 = ct+2 − ct = · · · = c4 − c2.
Hence,
c2 − c0 = c4 − c2 = c6 − c4 = · · ·
Now, if c2 > c0 then c0 < c2 < c4 < · · · < c0, a contradiction. Analogously, if c2 < c0
then c0 > c2 > c4 > · · · > c0, a contradiction. So c0 = c2 = · · · = cn−2 and analogously
c1 = c3 = · · · = cn−1. Hence if c0 = p, then c = (p,−p, p,−p, . . . ) by Lemma 2.1, and
the eigenspace corresponding to eigenvalue 0 is 1-dimensional.
Now suppose that t+1 is not coprime to n. Set b = 0. We will change some entries of
b. Since t+ 1 is odd, there is an even k such that (t+ 1)k ≡ 0 (mod n) and 1 ≤ k < n.
Set
b0 = 1, bt+1 = −1, b2(t+1) = 1, b3(t+1) = −1, . . . ,
where the indices are modulo n. We have changed k entries of b and since k is even, the
last changed entry has value −1. Thus some entries of b remained 0’s and nevertheless
Ab = 0, since if j-th entry of ai is 1, then either (j + (t + 1))-th or (j − (t + 1))-th
(modulo n) entry of ai is also 1 (while the other is 0). Hence, G is not a nut graph in this
case.
Finally, suppose that t2 is not coprime to
n
2 . Then there exists a number k such that
k | t2 , k |
n
2 and k > 1. Again, set b = 0. We will change some entries of b. Set
b0 = b2 = b4 = · · · = b2(k−2) = 1 and b2(k−1) = −(k − 1),
and repeat this pattern for all even indices of b. Since k | n2 , this pattern is repeated
exactly n2k times. And since every ai contains two disjoint sets of t consecutive 1’s, we
have Ab = 0. But half of the entries of b are 0’s and therefore G is not a nut graph.
6 Art Discrete Appl. Math. 5 (2022) #P2.01
Observe that the only requirement for n in Theorem 2.4 is that n is even and n > d.
However, if n = d + 2 then d2 + 1 is not coprime to n, and so n ≥ d + 4. Hence, by
Theorem 2.4, for d = 12 the following circulant graphs are nut graphs:
C(16, {1, 2, 3, 4, 5, 6}), C(20, {1, 2, 3, 4, 5, 6}), C(22, {1, 2, 3, 4, 5, 6}),
C(26, {1, 2, 3, 4, 5, 6}), C(32, {1, 2, 3, 4, 5, 6}), C(34, {1, 2, 3, 4, 5, 6}), and
C(38, {1, 2, 3, 4, 5, 6}).
Using the computer [23] we found the following graphs that are nut graphs:
C(18, {1, 2, 3, 4, 5, 8}), C(24, {1, 2, 3, 4, 5, 8}), C(28, {1, 2, 3, 4, 5, 10}),
C(30, {1, 2, 3, 4, 5, 8}), and C(36, {1, 2, 3, 4, 5, 8}).
3 Concluding remarks and further work
From the very beginning of our work on this paper, the nut circulant graphs were continu-
ously present, which fact motivates us explicitly to pose here the following problem.
Problem 3.1. Find which circulant graphs are nut graphs.
By the arguments in this paper, any circulant nut graph must satisfy the conditions of
Theorem 1.4(1), i.e. the order n is even, the degree d is divisible by 4, and n ≥ d+ 4. We
believe that for any such pairs n and d, there exists a circulant nut graph.
Conjecture 3.2. For every d, where d ≡ 0 (mod 4), and for every even n, n ≥ d + 4,
there exists a circulant nut graph C(n, {a1, a2, . . . , ad/2}) of degree d.
And, as a very particular case of the above conjecture, by restricting to 12-regular
graphs, we also propose.
Conjecture 3.3. For every even n, n ≥ 16, there exists a circulant nut graph C(n, {a1, a2,
. . . , a6}) of degree 12.
By Theorem 1.4, if n is odd then there is no vertex-transitive nut graph of order n and
degree 12. In this case all graphs were found by a computer search. If G is a regular graph
that contains edges u1v1 and u2v2 but does not contain edges u1v2, u2v1, then rewiring (i.e.
removing edges u1v1, u2v2 and adding edges u1v2, u2v1; it is also known as a Ryser switch
[15]) yields another regular graph. Our approach was to start with a ‘random’ 12-regular
graph of odd order and perform a sequence of rewirings. In this way all graphs in the
Appendix were obtained. For instance, the graph on 21 vertices, whose kernel eigenvector
contains only values 1 and −2, was obtained from C(21, {1, 2, 3, 4, 5, 6}) by removing the
edges (0, 16) and (2, 7) and adding the edges (0, 7) and (2, 16).
Note that kernel eigenvectors of all graphs in the Appendix on n = 3k vertices (for
k = 7, 9, 11, 13) contain only values 1 and −2. All those graphs have a special structure.
Let V = V1 ∪ V−2 be the partition of vertices with respect to the kernel eigenvector entry.
In each case, the graph induced by V−2 is isomorphic to a graph that can be obtained from
C(k, {1, 2}) by at most one rewiring, while the graph induced by V1 is isomorphic to a
graph that can be obtained from C(2k, {1, 2, 3, 4}) by at most one rewiring. Moreover,
let BiC(n, S) be the graph with the vertex set V = {v0, . . . , vn−1, u0, . . . , un−1} and the
edge set E = {viu(i+s) mod n : 0 ≤ i < n, s ∈ S}. This graph is a special kind of
N. Bašić et al.: On 12-regular nut graphs 7
bicirculant (see [13, 14] and references cited therein). The set V1 can be partitioned into
two subsets V1 = V ′1 ∪V
′′
1 , |V
′
1 | = |V
′′
1 |, such that the graph induced by edges from V−2 to
V ′1 is isomorphic to a graph that can be obtained from BiC(k, {0, 1, 2, 3}) by at most one
rewiring. Similarly, the graph induced by edges from V−2 to V ′′1 is also isomorphic to a
graph that can be obtained from BiC(k, {0, 1, 2, 3}) by at most one rewiring.
ORCID iDs
Nino Bašić https://orcid.org/0000-0002-6555-8668
Martin Knor https://orcid.org/0000-0003-3555-3994
Riste Škrekovski https://orcid.org/0000-0001-6851-3214
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N. Bašić et al.: On 12-regular nut graphs 9
Appendix A 12-regular nut graphs of odd orders
Here, we list one 12-regular nut graph of odd order n for each n ∈ {17, 19, . . . , 39}.
Each graph is given in the adjacency-lists (of neighbours of each vertex) representation,
formatted as a Python dictionary. We also give the corresponding kernel eigenvector c as a
list of integer entries.
Order n = 17.
{0: [1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 15, 16], 1: [0, 2, 3, 4, 6, 7, 8, 9, 10, 11, 15, 16], 2: [0, 1, 4, 5, 6, 7,
8, 9, 10, 11, 13, 15], 3: [0, 1, 4, 6, 7, 8, 9, 11, 12, 14, 15, 16], 4: [0, 1, 2, 3, 5, 6, 8, 9, 10, 11, 13, 16],
5: [0, 2, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15], 6: [1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 14, 15], 7: [1, 2, 3, 5, 6, 8,
10, 11, 12, 13, 14, 16], 8: [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 13, 14], 9: [0, 1, 2, 3, 4, 5, 6, 10, 12, 13, 14,
16], 10: [0, 1, 2, 4, 5, 7, 8, 9, 12, 14, 15, 16], 11: [0, 1, 2, 3, 4, 7, 8, 12, 13, 14, 15, 16], 12: [0, 3, 5, 6,
7, 9, 10, 11, 13, 14, 15, 16], 13: [2, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 16], 14: [3, 5, 6, 7, 8, 9, 10, 11, 12,
13, 15, 16], 15: [0, 1, 2, 3, 5, 6, 10, 11, 12, 13, 14, 16], 16: [0, 1, 3, 4, 7, 9, 10, 11, 12, 13, 14, 15]}
c = [3, −3, −2, 2, 1, 2, −1, −2, 3, −1, −1, 1, 1, −1, 1, −1, −2]
Order n = 19.
{0: [1, 2, 5, 7, 9, 10, 11, 12, 13, 14, 16, 18], 1: [0, 3, 5, 6, 7, 10, 12, 13, 14, 15, 17, 18], 2: [0, 4, 6, 7,
8, 9, 10, 11, 12, 16, 17, 18], 3: [1, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18], 4: [2, 5, 6, 7, 8, 11, 12, 13,
14, 15, 17, 18], 5: [0, 1, 4, 7, 8, 9, 11, 12, 13, 14, 15, 17], 6: [1, 2, 3, 4, 7, 8, 9, 10, 14, 15, 16, 17], 7:
[0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 15, 16], 8: [2, 3, 4, 5, 6, 7, 9, 11, 14, 15, 17, 18], 9: [0, 2, 5, 6, 7, 8, 10, 11,
12, 13, 16, 17], 10: [0, 1, 2, 3, 6, 9, 11, 12, 13, 14, 16, 18], 11: [0, 2, 3, 4, 5, 7, 8, 9, 10, 16, 17, 18],
12: [0, 1, 2, 3, 4, 5, 9, 10, 13, 14, 15, 16], 13: [0, 1, 3, 4, 5, 9, 10, 12, 14, 15, 16, 17], 14: [0, 1, 3, 4,
5, 6, 8, 10, 12, 13, 15, 18], 15: [1, 4, 5, 6, 7, 8, 12, 13, 14, 16, 17, 18], 16: [0, 2, 3, 6, 7, 9, 10, 11, 12,
13, 15, 18], 17: [1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 18], 18: [0, 1, 2, 3, 4, 8, 10, 11, 14, 15, 16, 17]}
c = [5, 10, 6, −10, −3, −1, 4, −1, −5, 1, 1, −5, −4, −3, −4, 2, −4, 7, 4]
Order n = 21.
{0: [1, 2, 3, 4, 5, 6, 7, 15, 17, 18, 19, 20], 1: [0, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20], 2: [0, 1, 3, 4, 5, 6,
8, 16, 17, 18, 19, 20], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 18, 19, 20], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 19, 20], 5:
[0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 20], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [0, 1, 3, 4, 5, 6, 8, 9, 10,
11, 12, 13], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15], 10: [4,
5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8, 9, 10,
11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19], 14: [8, 9, 10, 11, 12, 13,
15, 16, 17, 18, 19, 20], 15: [0, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20], 16: [1, 2, 10, 11, 12, 13, 14,
15, 17, 18, 19, 20], 17: [0, 1, 2, 11, 12, 13, 14, 15, 16, 18, 19, 20], 18: [0, 1, 2, 3, 12, 13, 14, 15, 16,
17, 19, 20], 19: [0, 1, 2, 3, 4, 13, 14, 15, 16, 17, 18, 20], 20: [0, 1, 2, 3, 4, 5, 14, 15, 16, 17, 18, 19]}
c = [1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1]
Order n = 23.
{0: [1, 2, 4, 6, 7, 8, 10, 11, 13, 19, 20, 21], 1: [0, 4, 5, 6, 7, 9, 11, 13, 16, 17, 20, 22], 2: [0, 3, 4, 6,
8, 11, 12, 13, 16, 19, 20, 21], 3: [2, 4, 5, 8, 9, 10, 12, 13, 14, 16, 17, 18], 4: [0, 1, 2, 3, 6, 7, 8, 14,
15, 16, 21, 22], 5: [1, 3, 7, 10, 11, 12, 14, 15, 17, 18, 19, 20], 6: [0, 1, 2, 4, 11, 12, 14, 17, 18, 19, 20,
22], 7: [0, 1, 4, 5, 10, 11, 12, 16, 18, 19, 21, 22], 8: [0, 2, 3, 4, 9, 10, 12, 13, 15, 16, 21, 22], 9: [1, 3,
10 Art Discrete Appl. Math. 5 (2022) #P2.01
8, 10, 11, 13, 14, 15, 18, 19, 21, 22], 10: [0, 3, 5, 7, 8, 9, 11, 13, 17, 18, 19, 21], 11: [0, 1, 2, 5, 6, 7,
9, 10, 13, 14, 15, 20], 12: [2, 3, 5, 6, 7, 8, 13, 14, 17, 19, 20, 22], 13: [0, 1, 2, 3, 8, 9, 10, 11, 12, 14,
15, 19], 14: [3, 4, 5, 6, 9, 11, 12, 13, 15, 16, 17, 20], 15: [4, 5, 8, 9, 11, 13, 14, 17, 18, 20, 21, 22],
16: [1, 2, 3, 4, 7, 8, 14, 17, 18, 20, 21, 22], 17: [1, 3, 5, 6, 10, 12, 14, 15, 16, 19, 20, 21], 18: [3, 5, 6,
7, 9, 10, 15, 16, 19, 20, 21, 22], 19: [0, 2, 5, 6, 7, 9, 10, 12, 13, 17, 18, 22], 20: [0, 1, 2, 5, 6, 11, 12,
14, 15, 16, 17, 18], 21: [0, 2, 4, 7, 8, 9, 10, 15, 16, 17, 18, 22], 22: [1, 4, 6, 7, 8, 9, 12, 15, 16, 18, 19,
21]}
c = [6, −24, −7, 13, 39, 1, 27, 4, −18, −4, 10, 3, −14, −14, 28, 1, −22, −2, 3, 6, −28,
2, −10]
Order n = 25.
{0: [3, 4, 5, 7, 9, 10, 12, 13, 17, 19, 22, 23], 1: [2, 3, 5, 11, 12, 15, 16, 18, 19, 20, 21, 23], 2: [1, 3, 4,
5, 10, 13, 14, 17, 20, 21, 23, 24], 3: [0, 1, 2, 5, 8, 10, 14, 16, 20, 21, 23, 24], 4: [0, 2, 6, 8, 9, 10, 11,
13, 18, 21, 23, 24], 5: [0, 1, 2, 3, 10, 13, 14, 17, 18, 19, 20, 24], 6: [4, 8, 9, 10, 11, 12, 14, 17, 19, 20,
21, 22], 7: [0, 8, 9, 11, 12, 15, 16, 18, 19, 22, 23, 24], 8: [3, 4, 6, 7, 9, 10, 11, 13, 17, 18, 22, 23], 9:
[0, 4, 6, 7, 8, 10, 11, 12, 14, 15, 18, 21], 10: [0, 2, 3, 4, 5, 6, 8, 9, 15, 16, 17, 18], 11: [1, 4, 6, 7, 8, 9,
12, 13, 14, 17, 19, 20], 12: [0, 1, 6, 7, 9, 11, 13, 14, 15, 18, 21, 22], 13: [0, 2, 4, 5, 8, 11, 12, 16, 20,
21, 22, 23], 14: [2, 3, 5, 6, 9, 11, 12, 15, 16, 17, 19, 22], 15: [1, 7, 9, 10, 12, 14, 16, 17, 19, 20, 22,
24], 16: [1, 3, 7, 10, 13, 14, 15, 17, 18, 19, 20, 24], 17: [0, 2, 5, 6, 8, 10, 11, 14, 15, 16, 21, 23], 18:
[1, 4, 5, 7, 8, 9, 10, 12, 16, 21, 22, 24], 19: [0, 1, 5, 6, 7, 11, 14, 15, 16, 21, 22, 24], 20: [1, 2, 3, 5,
6, 11, 13, 15, 16, 22, 23, 24], 21: [1, 2, 3, 4, 6, 9, 12, 13, 17, 18, 19, 23], 22: [0, 6, 7, 8, 12, 13, 14,
15, 18, 19, 20, 24], 23: [0, 1, 2, 3, 4, 7, 8, 13, 17, 20, 21, 24], 24: [2, 3, 4, 5, 7, 15, 16, 18, 19, 20, 22,
23]}
c = [29, 20, −31, 7, 5, −13, 32, −19, −12, 1, 31, −12, −8, −6, −49, 17, 3, −17, −21,
20, 33, 7, 1, −2, −16]
Order n = 27.
{0: [2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 26], 1: [2, 3, 4, 5, 6, 7, 8, 22, 23, 24, 25, 26], 2: [0, 1, 3, 4, 5,
6, 7, 8, 23, 24, 25, 26], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 24, 25, 26], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 25, 26],
5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 26], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [0, 1, 2, 3, 4, 5, 6, 9,
10, 11, 12, 13], 8: [1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15],
10: [4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8,
9, 10, 11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19], 14: [8, 9, 10, 11, 12,
13, 15, 16, 17, 18, 19, 20], 15: [9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21], 16: [10, 11, 12, 13, 14,
15, 17, 18, 19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [12, 13, 14, 15, 16,
17, 19, 20, 21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17, 18,
19, 21, 22, 23, 24, 25, 26], 21: [0, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26], 22: [0, 1, 16, 17, 18, 19,
20, 21, 23, 24, 25, 26], 23: [0, 1, 2, 17, 18, 19, 20, 21, 22, 24, 25, 26], 24: [0, 1, 2, 3, 18, 19, 20, 21,
22, 23, 25, 26], 25: [0, 1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 26], 26: [0, 1, 2, 3, 4, 5, 20, 21, 22, 23, 24,
25]}
c = [1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1]
Order n = 29.
{0: [1, 3, 5, 6, 9, 10, 11, 13, 14, 19, 26, 28], 1: [0, 2, 4, 5, 11, 16, 17, 18, 19, 21, 26, 27], 2: [1, 3, 8,
9, 10, 11, 13, 24, 25, 26, 27, 28], 3: [0, 2, 12, 13, 17, 20, 21, 23, 24, 25, 26, 27], 4: [1, 5, 6, 9, 11, 15,
N. Bašić et al.: On 12-regular nut graphs 11
16, 17, 20, 22, 23, 28], 5: [0, 1, 4, 7, 12, 15, 16, 19, 20, 22, 24, 25], 6: [0, 4, 7, 8, 9, 11, 15, 17, 18,
19, 21, 22], 7: [5, 6, 8, 11, 12, 13, 15, 16, 18, 20, 22, 24], 8: [2, 6, 7, 10, 12, 15, 19, 20, 21, 24, 26,
27], 9: [0, 2, 4, 6, 12, 14, 15, 20, 22, 23, 24, 27], 10: [0, 2, 8, 13, 16, 17, 18, 20, 21, 23, 25, 26], 11:
[0, 1, 2, 4, 6, 7, 12, 16, 17, 19, 20, 23], 12: [3, 5, 7, 8, 9, 11, 14, 15, 18, 19, 21, 25], 13: [0, 2, 3, 7,
10, 14, 15, 21, 23, 25, 27, 28], 14: [0, 9, 12, 13, 15, 18, 22, 23, 24, 26, 27, 28], 15: [4, 5, 6, 7, 8, 9,
12, 13, 14, 18, 22, 27], 16: [1, 4, 5, 7, 10, 11, 18, 20, 21, 25, 27, 28], 17: [1, 3, 4, 6, 10, 11, 18, 19,
22, 24, 27, 28], 18: [1, 6, 7, 10, 12, 14, 15, 16, 17, 19, 23, 24], 19: [0, 1, 5, 6, 8, 11, 12, 17, 18, 23,
26, 27], 20: [3, 4, 5, 7, 8, 9, 10, 11, 16, 25, 26, 28], 21: [1, 3, 6, 8, 10, 12, 13, 16, 22, 23, 25, 26], 22:
[4, 5, 6, 7, 9, 14, 15, 17, 21, 24, 25, 27], 23: [3, 4, 9, 10, 11, 13, 14, 18, 19, 21, 24, 28], 24: [2, 3, 5,
7, 8, 9, 14, 17, 18, 22, 23, 28], 25: [2, 3, 5, 10, 12, 13, 16, 20, 21, 22, 26, 28], 26: [0, 1, 2, 3, 8, 10,
14, 19, 20, 21, 25, 28], 27: [1, 2, 3, 8, 9, 13, 14, 15, 16, 17, 19, 22], 28: [0, 2, 4, 13, 14, 16, 17, 20,
23, 24, 25, 26]}
c = [1, 1, 37, −13, −20, −42, 21, −5, −36, 25, 5, 30, 41, −25, 21, −6, 6, 17, 34, −34,
−14, −13, 7, −51, −16, 39, 5, −21, 6]
Order n = 31.
{0: [5, 10, 12, 13, 17, 18, 21, 22, 24, 26, 27, 29], 1: [3, 6, 7, 8, 10, 14, 17, 20, 23, 25, 27, 30], 2: [4,
7, 9, 10, 18, 21, 22, 23, 24, 25, 27, 28], 3: [1, 4, 5, 11, 13, 16, 17, 18, 19, 24, 25, 29], 4: [2, 3, 5, 11,
12, 13, 18, 21, 25, 26, 28, 29], 5: [0, 3, 4, 6, 7, 9, 11, 14, 17, 25, 27, 29], 6: [1, 5, 8, 9, 11, 13, 18, 20,
22, 26, 29, 30], 7: [1, 2, 5, 9, 10, 12, 20, 24, 25, 26, 27, 30], 8: [1, 6, 9, 14, 15, 17, 18, 20, 21, 22, 23,
30], 9: [2, 5, 6, 7, 8, 12, 14, 15, 19, 24, 27, 28], 10: [0, 1, 2, 7, 12, 13, 15, 18, 19, 21, 24, 28], 11: [3,
4, 5, 6, 12, 15, 17, 20, 22, 23, 29, 30], 12: [0, 4, 7, 9, 10, 11, 14, 16, 18, 21, 27, 30], 13: [0, 3, 4, 6,
10, 16, 20, 23, 24, 25, 26, 27], 14: [1, 5, 8, 9, 12, 15, 17, 18, 19, 20, 22, 23], 15: [8, 9, 10, 11, 14, 17,
19, 20, 21, 27, 28, 30], 16: [3, 12, 13, 18, 19, 21, 22, 23, 24, 26, 28, 29], 17: [0, 1, 3, 5, 8, 11, 14, 15,
20, 22, 23, 29], 18: [0, 2, 3, 4, 6, 8, 10, 12, 14, 16, 24, 25], 19: [3, 9, 10, 14, 15, 16, 20, 21, 22, 23,
26, 28], 20: [1, 6, 7, 8, 11, 13, 14, 15, 17, 19, 24, 25], 21: [0, 2, 4, 8, 10, 12, 15, 16, 19, 25, 27, 29],
22: [0, 2, 6, 8, 11, 14, 16, 17, 19, 23, 28, 30], 23: [1, 2, 8, 11, 13, 14, 16, 17, 19, 22, 26, 28], 24: [0,
2, 3, 7, 9, 10, 13, 16, 18, 20, 28, 30], 25: [1, 2, 3, 4, 5, 7, 13, 18, 20, 21, 26, 29], 26: [0, 4, 6, 7, 13,
16, 19, 23, 25, 27, 29, 30], 27: [0, 1, 2, 5, 7, 9, 12, 13, 15, 21, 26, 30], 28: [2, 4, 9, 10, 15, 16, 19, 22,
23, 24, 29, 30], 29: [0, 3, 4, 5, 6, 11, 16, 17, 21, 25, 26, 28], 30: [1, 6, 7, 8, 11, 12, 15, 22, 24, 26, 27,
28]}
c = [1, 91, −39, 14, 39, 33, 75, −48, −37, 2, 146, −14, −13, 23, 20, 6, −84, −32, 27, 38,
−93, −66, −43, 21, −79, −43, 18, −15, 59, 1, −8]
Order n = 33.
{0: [1, 2, 3, 4, 5, 6, 27, 28, 29, 30, 31, 32], 1: [0, 2, 3, 4, 5, 6, 7, 11, 28, 29, 31, 32], 2: [0, 1, 3, 4, 5,
6, 7, 8, 29, 30, 31, 32], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 30, 31, 32], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 31, 32], 5:
[0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 32], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [1, 2, 3, 4, 5, 6, 8, 9, 10,
11, 12, 13], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15], 10:
[4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 30], 11: [1, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8, 9, 10,
11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19], 14: [8, 9, 10, 11, 12, 13,
15, 16, 17, 18, 19, 20], 15: [9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21], 16: [10, 11, 12, 13, 14, 15,
17, 18, 19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [12, 13, 14, 15, 16, 17,
19, 20, 21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17, 18, 19,
21, 22, 23, 24, 25, 26], 21: [15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27], 22: [16, 17, 18, 19, 20, 21,
23, 24, 25, 26, 27, 28], 23: [17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29], 24: [18, 19, 20, 21, 22, 23,
12 Art Discrete Appl. Math. 5 (2022) #P2.01
25, 26, 27, 28, 29, 30], 25: [19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31], 26: [20, 21, 22, 23, 24, 25,
27, 28, 29, 30, 31, 32], 27: [0, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32], 28: [0, 1, 22, 23, 24, 25, 26,
27, 29, 30, 31, 32], 29: [0, 1, 2, 23, 24, 25, 26, 27, 28, 30, 31, 32], 30: [0, 2, 3, 10, 24, 25, 26, 27, 28,
29, 31, 32], 31: [0, 1, 2, 3, 4, 25, 26, 27, 28, 29, 30, 32], 32: [0, 1, 2, 3, 4, 5, 26, 27, 28, 29, 30, 31]}
c = [1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1,
1, −2, 1, 1, −2, 1]
Order n = 35.
{0: [1, 2, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34], 1: [0, 2, 3, 4, 5, 6, 7, 30, 31, 32, 33, 34], 2: [0, 1, 3, 4, 5,
6, 7, 8, 15, 31, 32, 33], 3: [0, 1, 2, 4, 5, 6, 8, 9, 15, 32, 33, 34], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 31, 33, 34],
5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 34], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [1, 2, 4, 5, 6, 8, 9, 10,
11, 12, 13, 21], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15],
10: [5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 25], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8,
9, 10, 11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 34], 14: [8, 9, 10, 11, 12,
13, 15, 16, 17, 18, 19, 20], 15: [2, 3, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20], 16: [10, 11, 12, 13, 14, 15,
17, 18, 19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [12, 13, 14, 15, 16, 17,
19, 20, 21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17, 18, 19,
21, 22, 23, 24, 25, 26], 21: [7, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27], 22: [16, 17, 18, 19, 20, 21,
23, 24, 25, 26, 27, 28], 23: [17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29], 24: [18, 19, 20, 21, 22, 23,
25, 26, 27, 28, 29, 30], 25: [10, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30], 26: [20, 21, 22, 23, 24, 25,
27, 28, 29, 30, 31, 32], 27: [21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33], 28: [22, 23, 24, 25, 26, 27,
29, 30, 31, 32, 33, 34], 29: [0, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34], 30: [0, 1, 24, 25, 26, 27, 28,
29, 31, 32, 33, 34], 31: [0, 1, 2, 4, 26, 27, 28, 29, 30, 32, 33, 34], 32: [0, 1, 2, 3, 26, 27, 28, 29, 30,
31, 33, 34], 33: [0, 1, 2, 3, 4, 27, 28, 29, 30, 31, 32, 34], 34: [0, 1, 3, 4, 5, 13, 28, 29, 30, 31, 32, 33]}
c = [1, −1, −1, −3, 3, 2, −1, −1, 1, 1, −2, 2, −2, −1, 3, −1, −1, 2, −2, −2, 5, −1, −1,
1, −2, −2, 6, −3, −1, 1, −1, 5, −1, −4, 1]
Order n = 37.
{0: [1, 2, 3, 4, 5, 6, 31, 32, 33, 34, 35, 36], 1: [0, 2, 3, 4, 5, 6, 7, 18, 22, 32, 33, 35], 2: [0, 1, 3, 4, 5,
6, 7, 8, 33, 34, 35, 36], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 34, 35, 36], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 35, 36],
5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 36], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [1, 2, 3, 4, 5, 6, 8, 9,
10, 11, 12, 13], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 15, 32],
10: [4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8,
9, 10, 11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 19, 36], 14: [8, 10, 11, 12,
13, 15, 16, 17, 18, 19, 20, 35], 15: [9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21], 16: [10, 11, 12, 13,
14, 15, 17, 18, 19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [1, 12, 14, 15,
16, 17, 19, 20, 21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17,
18, 19, 21, 22, 23, 24, 25, 26], 21: [15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27], 22: [1, 16, 17, 18,
19, 20, 21, 23, 24, 26, 27, 28], 23: [17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29], 24: [18, 19, 20, 21,
22, 23, 25, 26, 27, 28, 29, 30], 25: [19, 20, 21, 23, 24, 26, 27, 28, 29, 30, 31, 34], 26: [20, 21, 22, 23,
24, 25, 27, 28, 29, 30, 31, 32], 27: [21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33], 28: [22, 23, 24, 25,
26, 27, 29, 30, 31, 32, 33, 34], 29: [23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35], 30: [24, 25, 26, 27,
28, 29, 31, 32, 33, 34, 35, 36], 31: [0, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36], 32: [0, 1, 9, 26, 27,
28, 29, 30, 31, 33, 34, 36], 33: [0, 1, 2, 27, 28, 29, 30, 31, 32, 34, 35, 36], 34: [0, 2, 3, 25, 28, 29, 30,
31, 32, 33, 35, 36], 35: [0, 1, 2, 3, 4, 14, 29, 30, 31, 33, 34, 36], 36: [0, 2, 3, 4, 5, 13, 30, 31, 32, 33,
34, 35]}
N. Bašić et al.: On 12-regular nut graphs 13
c = [2, −3, −4, 5, 1, −1, −1, −4, 5, 2, −5, 1, 1, −1, 6, −5, −4, 7, −1, −5, 4, −5, 3, 6,
−5, −5, 8, −3, 1, 1, −4, 3, 4, −7, −1, 3, 1]
Order n = 39.
{0: [1, 2, 3, 4, 5, 6, 15, 33, 34, 36, 37, 38], 1: [0, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38], 2: [0, 1, 3, 4, 5,
6, 7, 8, 35, 36, 37, 38], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 36, 37, 38], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 37, 38],
5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 38], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [1, 2, 3, 4, 5, 6, 8, 9,
10, 11, 12, 13], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15],
10: [4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 17, 35], 12: [6, 7, 8,
9, 10, 11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19], 14: [8, 9, 10, 11, 12,
13, 15, 16, 17, 18, 19, 20], 15: [0, 9, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21], 16: [10, 11, 12, 13, 14,
15, 17, 18, 19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [12, 13, 14, 15, 16,
17, 19, 20, 21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17, 18,
19, 21, 22, 23, 24, 25, 26], 21: [15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27], 22: [16, 17, 18, 19, 20,
21, 23, 24, 25, 26, 27, 28], 23: [17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29], 24: [18, 19, 20, 21, 22,
23, 25, 26, 27, 28, 29, 30], 25: [19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31], 26: [20, 21, 22, 23, 24,
25, 27, 28, 29, 30, 31, 32], 27: [21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33], 28: [22, 23, 24, 25, 26,
27, 29, 30, 31, 32, 33, 34], 29: [23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35], 30: [24, 25, 26, 27, 28,
29, 31, 32, 33, 34, 35, 36], 31: [25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37], 32: [26, 27, 28, 29, 30,
31, 33, 34, 35, 36, 37, 38], 33: [0, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38], 34: [0, 1, 28, 29, 30, 31,
32, 33, 35, 36, 37, 38], 35: [1, 2, 11, 29, 30, 31, 32, 33, 34, 36, 37, 38], 36: [0, 1, 2, 3, 30, 31, 32, 33,
34, 35, 37, 38], 37: [0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 38], 38: [0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36,
37]}
c = [1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1,
1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1]
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P2.02
https://doi.org/10.26493/2590-9770.1297.f97
(Also available at http://adam-journal.eu)
Preferential attachment processes approaching
the Rado multigraph
Richard Elwes*
University of Leeds, Leeds, LS2 9JT, UK
Received 6 April 2019, accepted 2 July 2021, published online 20 May 2022
Abstract
We consider a preferential attachment process in which a multigraph is built one node
at a time. The number of edges added at stage t, emanating from the new node, is given by
some prescribed function f(t), generalising a model considered by Kleinberg and Klein-
berg in 2005 where f was presumed constant. We show that if f(t) is asymptotically
bounded above and below by linear functions in t, then with probability 1 the infinite limit
of the process will be isomorphic to the Rado multigraph. This structure is the natural
multigraph analogue of the Rado graph, which we introduce here.
Keywords: Preferential attachment, random graphs, multigraphs, Rado graph.
Math. Subj. Class.: 05C80
1 Introduction
In recent decades, there has been much interest in modelling and analysing the many net-
works which appear in the real world, in contexts such as the world wide web or online so-
cial networks. This work has drawn heavily on the mathematical study of random graphs, a
subject with its origins in the 1959 work of Erdős and Rényi, [15]. They principally studied
the graphs which emerge from the following process: begin with a collection of nodes, and
independently connect every pair with an edge, with some fixed probability p.
Erdős-Rényi random graph theory has two distinct facets. First, researchers have anal-
ysed the finite graphs which arise. Here, questions of interest include the emergence of a
giant component and the degree distribution of the nodes, and analyses are typically highly
sensitive to the value of p. In [3], Bollobás provides a comprehensive discussion of such
matters.
*The author thank this paper’s anonymous reviewers for their thoughtful and helpful remarks.
E-mail address: r.h.elwes@leeds.ac.uk (Richard Elwes)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P2.02
The second angle of approach is to consider this process on a countably infinite set of
nodes. In this case, a remarkable theorem of Erdős and Rényi guarantees that, irrespec-
tive of the value of p ∈ (0, 1), the resulting graph will with probability 1 be isomorphic
to the Rado graph. This famous graph is axiomatised by the following schema: given
any finite disjoint sets of nodes U and V , there exists a node v connected to each node
in V and none in U . This graph exhibits many properties which logicians and combina-
torists enjoy. To start with, it is universal in that it isomorphically embeds every finite and
countably infinite graph. It is also countably categorical, meaning that any two countable
models of the above axioms will be isomorphic. The graph is 1-transitive in that for any
any two nodes v1, v2 there is an automorphism α where α(v1) = v2. It is ultrahomoge-
neous: any isomorphism between finite induced subgraphs extends to an automorphism of
the whole structure. (Analogues of these facts are proved for a new structure, the Rado
multigraph, in Proposition 2.2 below.) The Rado graph continues to attract the attention of
today’s permutation group-theorists; it is known that its automorphism group is simple (in
the group-theoretic sense), and satisfies the strong small-index property. In [5], Cameron
provides a recent survey of such matters. Beyond this, the Rado graph satisfies several
subtler properties, notably rank-1 supersimplicity and 1-basedness, which make it a central
object of study for today’s model theorists. Wagner provides an authoritative account in
[17].
In more recent years, however, network science has grown beyond the Erdős-Rényi
approach, to embrace alternative methods for modelling real-world networks. The most
prominent of these is perhaps the preferential attachment (PA) mechanism introduced by
Barabási and Albert in [2]. Another notable class of models derive from the web-copying
mechanism introduced by Adler and Mitzenmacher in [1].
In PA models, a new node is introduced at each time step, and then connected to each
pre-existing node with a probability depending on the current degree of that node, accord-
ing to a rich-get-richer paradigm. PA processes can exhibit several properties observed in
real-world networks (but absent in Erdős-Rényi graphs), notably scale-freeness meaning
that the proportion of nodes of degree k is asymptotically proportional to k−γ for some
fixed γ and all k.
What can we say of the infinite limits of these processes? Results of Bonato and Janssen
[4] have made significant progress for web-copying models. Less work has been done in
the case of PA processes. The work of Oliveira and Spencer [14] studying the Growing Net-
work model of Krapivsky and Redner [11] and of Drinea, Enachescu, and Mitzenmacher
[7] is a notable exception. Of greatest relevance to the current paper, however, is the work
of Kleinberg and Kleinberg [10]. There the following PA process is considered: at each
time-step, a single node and a constant number C of edges are added. The new edges all
emanate from the new node, with their end-points independently chosen among the pre-
existing nodes, with probabilities proportional to their degrees. The resulting structures are
analysed as directed multigraphs: all edges are directed, two or more may share the same
start and end-points.
Kleinberg and Kleinberg prove that if C = 1 or C = 2, then there exists an infinite
structure, to which, with probability 1, the infinite limit of the process will be isomorphic.
However, the analogous result fails for C ≥ 3: given two instantiations of the process,
there is a positive probability that their infinite limits will fail to be isomorphic.
In this paper we extend the results and methods of [10], by considering a process which
adds f(t) many edges at stage t for some function f : N → N. Again the start-point of
R. Elwes: Preferential attachment processes approaching the Rado multigraph 3
every edge is the new node, and the end-points are chosen independently with probability
proportional to the nodes’ degrees. It follows easily from the results of [10] that when-
ever f is non-constant, or constant with value ≥ 3, there is a positive probability that the
infinite limiting structures of two instantiations will be non-isomorphic as directed multi-
graphs. However, by forgetting the directions of edges, and looking for isomorphisms as
multigraphs, we are able to recover a new categoricity result. In Theorem 3.2 we rigorously
establish a sufficient criterion for the resulting structure to be isomorphic to the Rado multi-
graph with probability 1. (This structure is the natural multigraph analogue of the Rado
graph, and is defined in Definition 2.1 below.) Our criterion is that f is asymptotically
bounded above and below by positive non-constant linear functions of t.
In [9], the author uses similar machinery to analyse a Preferential Attachment process
in which parallel edges are not permitted, and the new node t+1 is connected to each pre-
existing node u independently with probability du(t)t . Thus the number of new edges is not
prescribed, but is itself a random variable. It is shown in [9] that, so long as the initial graph
is neither edgeless nor complete, with probability 1 the infinite limit of the process will be
the Rado graph augmented with a finite number of either universal or isolated nodes.
We describe the structure of the remainder of the paper:
• In Section 2 we introduce the infinite Rado Multigraph.
• In Section 3 we introduce MPAf , our main variant of the preferential attachment
process, as well as a secondary variant GPAf . We describe suitable hypotheses on
the function f , and we prove some initial results. We state our main result, Theo-
rem 3.2, which asserts that under appropriate conditions MPAf approaches the Rado
multigraph.
• In Section 4 we develop the theory of martingales for the process MPAf , our main
probabilistic tool.
• In Section 5, we complete the proof of Theorem 3.2.
• In Section 6, we close with some discussion of possible further directions of study.
2 The Rado multigraph
We begin by defining the infinite structure which our finitary processes will be shown to
approach. So far as we are aware, this structure has not previously appeared in the literature.
However the reader familiar with the Rado graph will find little of surprise. (For clarity, we
work with the convention that 0 ∈ N.)
Definition 2.1. A finitary loopless multigraph is a structure (V,E) where V is a set of
vertices, and E is a finitary multiset of unordered pairs from V . That is to say every element
of E is of the form e = {vi, vj} (written vivj) where vi, vj ∈ V are distinct, E is itself
unordered, and each e has a multiplicity me ∈ N describing the number of occurrences of
e within E. (If e does not occur within E we consider it to have multiplicity 0.)
The Rado Multigraph is a finitary loopless multigraph where V is countably infinite
and which additionally satisfies the following axiom:
• For any n ≥ 1, any m1, . . . ,mn ∈ N, and any distinct u1, . . . , un ∈ V there exists
v ∈ V such that the multiplicity of vui is exactly mi.
4 Art Discrete Appl. Math. 5 (2022) #P2.02
We now show that the Rado Multigraph is unique up to isomorphism. We also take the
opportunity to observe that several other familiar properties of the Rado graph hold in our
context although we shall not use them directly:
Proposition 2.2. Let M and M′ be structures satisfying Definition 2.1 of the Rado Multi-
graph. Then the following hold:
1. ℵ0-categoricity: M ∼= M′.
2. 1-transitivity: Given vertices v1, v2 in M there exists some α ∈ Aut(M) where
α(v1) = v2.
3. Ultrahomogeneity: If A,B are finite substructures of M and γ : A ∼= B is a
multigraph-isomorphism, then there exists α ∈ Aut(M) where α ↾A= γ.
(Note: here we treat A and B as induced substructures: for any vertices u, v ∈ A
the multiplicity of uv within A equals that within M).
4. Universality: Any finite or countably infinite finitary loopless multigraph can be iso-
morphically embedded in M.
Proof. We concentrate on proving statement 1. (Statements 2-4 follow from minor al-
terations to our argument. We leave the reader to fill in the details.) We proceed by a
standard back-and-forth argument. First we list the elements of M as a1, a2, a3, . . . and
similarly b1, b2, b3, . . . for M′. Now we argue inductively. Suppose i is even, and suppose
(a′1, . . . , a
′
i)
∼= (b′1, . . . , b
′
i) have been chosen. Let a
′
i+1 = aj where j is minimum such
that aj ̸∈ {a′1, . . . , a
′
i}.
Let (m1, . . . ,mi) be the vector counting the edges between a′i+1 and (a
′
1, . . . , a
′
i).
Notice that each mj ∈ N by the assumption of finitariness. Then by hypothesis there exists
b′i+1 joined to (b
′
1, . . . , b
′
i) in a fashion described by (m1, . . . ,mi). Hence (a
′
1, . . . , a
′
i+1)
∼=
(b′1, . . . , b
′
i+1).
Odd steps are identical, exchanging the roles of M and M′. Thus we build an isomor-
phism M ∼= M′.
Our concern in the current work is on PA processes. However, we remark in passing
that the Rado multigraph also arises from the following process in the style of Erdős-Rényi.
We shall not use this result and leave the proof as an easy adaptation of the corresponding
classical result about the Rado graph.
Proposition 2.3. Let (pj)j≥1 be any sequence lying entirely in (0, 1). Let V be a countably
infinite set. Let M be multigraph arising from the following random process.
• At step 0, the structure has no edges.
• At step j ≥ 1, consider every pair of distinct v1, v2 ∈ V where vivj currently
has multiplicity j − 1, and connect v1, v2 with a new j
th edge with probability pj ,
independently of the behaviour of all other vertices.
Then with probability 1, M is isomorphic to the Rado multigraph.
R. Elwes: Preferential attachment processes approaching the Rado multigraph 5
3 Preferential attachment with prescribed edge growth
In this section we shall describe two variants of the preferential attachment process, es-
tablish some of their basic properties, and formally state our main result. The process of
principal interest will be MPAf which builds a directed multigraph. We will also mention
a natural variant GPAf , which builds a directed graph. Each proceeds by adding, at each
time step, a single node along with a prescribed number of directed edges emanating from
it. The number of these edges is determined by some fixed function f : N → N. (In fact the
directions of the edges will play no role in the theory: we shall analyse the resulting struc-
tures as undirected (multi)graphs. However in the interim it will be convenient to refer to
the ‘start-’ and ‘end-points’ of each edge, so we preserve directedness for the time being.)
We shall work over some initial directed (multi)graph G′ containing no isolated nodes (i.e.
nodes of degree 0). However our results will be independent of the choice of G′, so the
reader may choose to focus on the case where G′ is trivial.
Definition 3.1 (The process MPAf ). Let G′ = (V ′, E′) be a finite directed multigraph
containing no isolated nodes (so E′ is a multiset of ordered pairs from V ′). Suppose that
G′ contains |E′| = e′ edges and |V ′| = v′ nodes. We will assume V ′ = {1, . . . , v′}.
Suppose that the function f : N → N satisfies:
• f(0) = e′.
• f(t) = 0 whenever 1 ≤ t ≤ v′ − 1.
• f(t) ≥ 1 for all t ≥ v′.
At each time-step t ≥ 1, we shall construct a multigraph G(t) with vertex set V (t) and
edge multiset E(t).
First we impose G(1) = . . . = G(v′) = G′.
Whenever t ≥ v′, we will have V (t) := {1, . . . , t} and
E(t+ 1) = E(t) ∪ E(t+ 1)
where |E(t+ 1)| = f(t).
The start-point of each edge in E(t + 1) is the new node t + 1. The end-points are
chosen independently from V (t), with probabilities directly proportional to their degrees
in G(t).
Notice that, the degrees used to calculate the probabilities are taken from G(t), which
is to say the model does not notice any incremental updating of degrees between G(t)
and G(t + 1). One can imagine the endpoints of the f(t) many new edges being selected
simultaneously, and independently of each other.
Notice too that our assumption that f(t) ̸= 0 for t ≥ v′ (along with our assumption on
G′) serves to ensure that there are never any isolated nodes.
We may now state our main result. (Recall the asymptotic notation g1 = Θ(g2) for
functions g1, g2 as meaning that there exist c2 ≥ c1 > 0 so that for all large enough t we
have c1 · g2(t) ≤ g1(t) ≤ c2 · g2(t).)
Theorem 3.2. Suppose that G′ is a finite directed multigraph containing no isolated nodes,
that f satisfies the requirements from Definition 3.1, and also that f(t) = Θ(t). Then, with
probability 1, the infinite limit of MPAf (G
′) is isomorphic, as an undirected multigraph, to
the Rado multigraph.
6 Art Discrete Appl. Math. 5 (2022) #P2.02
Before we commence the proof of this theorem, we remark that we expect that a similar
result to apply to a graph variant of the process, which we briefly introduce:
Definition 3.3 (The process GPAf ). Let G′ = (V ′, E′) be a finite directed graph contain-
ing no isolated nodes. Suppose that G′ contains |E′| = e′ edges and |V ′| = v′ nodes. We
will assume V ′ = {1, . . . , v′}.
Suppose that the function f : N → N satisfies:
• f(0) = e′.
• f(t) = 0 whenever 1 ≤ t ≤ v′ − 1.
• 1 ≤ f(t) ≤ t for all t ≥ v′.
At each time-step t ≥ 1, we shall construct a graph G(t) with vertex set V (t) and edge
set E(t).
First we impose G(1) = . . . = G(v′) = G′.
Whenever t ≥ v′, we will have V (t) := {1, . . . , t} and
E(t+ 1) = E(t) ∪ E(t+ 1)
where |E(t+ 1)| = f(t).
The start-point of each edge in E(t + 1) is the new node t + 1. The end-points of
the edges are selected sequentially from V (t), without replacement, with the choice at
each step made from the remaining unselected elements of V (t) with probabilities directly
proportional to their degrees in G(t).
Conjecture 3.4. Suppose that G′ = (V ′, E′) be a finite directed graph containing no
isolated nodes, that f satisfies the conditions in Definition 3.3, and further that there are
constants 0 < c1 ≤ c2 < 1 where c1 · t ≤ f(t) ≤ c2 · t for all large enough t. Then, with
probability 1, the infinite limit of GPAf (G
′) is isomorphic as an undirected graph to the
Rado graph.
Our arguments will be independent of G′, and thus we shall largely suppress mention
of it. Let us now consider the distribution of edges at stage t+1. First notice that |E(t)| =
F (t) :=
∑t−1
i=0 f(i). Hence in MPAf , at stage t+ 1 given any pre-existing node u ≤ t, the
probability that any given edge in E(t + 1) has its end-point at u is exactly du(t)2F (t) , where
du(t) is the degree of u in G(t). Thus the expected number of edges in E(t + 1) with
endpoint at u is f(t)·du(t)2F (t) .
In GPAf this probability distribution is more complicated, and the expected number
of edges u receives at stage t + 1 depends in a more detailed way upon G(t). This is the
primary obstacle to extending the current work to a proof of Conjecture 3.4.
Our standing assumption will be that we are working in MPAf . We shall leave the case
of GPAf open, but make some remarks about it as we proceed.
Our assumption in Theorem 3.2 is that f(t) = Θ(t). However we shall be able to
develop much of the theory under the following weaker hypotheses:
Assumption 3.5.
∞∑
s=0
f(s)
F (s)
= ∞. (3.1)
R. Elwes: Preferential attachment processes approaching the Rado multigraph 7
∞∑
s=0
(
f(s)
F (s)
)2
< ∞. (3.2)
We briefly discuss this. Assumption 3.5 easily follows in full, for instance, if f(t) =
Θ(tα) for some α ≥ 0.
However part (2) fails in general for polynomially bounded functions, an example be-
ing:
f(t) =
{
t when t = 2n for n ∈ N
1 otherwise.
On the other hand, both parts do hold for some exponential functions, such as f(t) =
⌊ 14 t
−
3
4 e
1
4
t⌋.
In all cases, it will be useful to extend the domain of f to R≥0. We choose to do
this as a step function, via f(t) := f (⌊t⌋). (Of course there may be more natural ways
to achieve the same thing, however this choice will be convenient, as the fourth point in
the following Lemma makes clear.) We now gather together some observations about the
extended function f . These follow immediately from our previous conditions.
Corollary 3.6. The following hold:
• f(t) = e′ for 0 ≤ t < 1.
• f(t) = 0 whenever 1 ≤ t < v′.
• f(t) ≥ 1 for all t ≥ v′.
• f is Lebesgue-measurable with antiderivative
∫ t
0
f(s)ds =: F (t). (This notation is
consistent with the previous interpretation of F since the two functions coincide at
integer points.)
• F is monotonic increasing everywhere and strictly so for t ≥ v′.
Under our additional hypothesis we can say a little more:
Lemma 3.7. Suppose that Assumption 3.5(2) holds. Then for any β ≥ 1, there exists
Kβ > 0 so that for any t ≥ m ≥ 0:
∣∣∣∣∣
∫ t
m
f(s)
F (s)β
ds−
t∑
s=m
f(s)
F (s)β
∣∣∣∣∣ < Kβ .
Proof. Let M be such that whenever s ≥ M then f(s) < F (s). Such a value must exist
by Assumption 3.5(2).
It is enough to prove the result this for all m ≥ M , since one can then add
max
{∫ M
0
f(s)
F (s)β
ds,
M∑
s=0
f(s)
F (s)β
}
to Kβ to obtain the result for all m. Thus we shall assume m ≥ M .
8 Art Discrete Appl. Math. 5 (2022) #P2.02
Firstly, it is immediate by consideration of F ↾[s,s+1] that
t∑
s=m
f(s)
F (s+ 1)β
<
∫ t
m
f(s)
F (s)β
ds <
t∑
s=m
f(s)
F (s)β
.
Next we shall appeal to Newton’s generalised binomial theorem, that whenever a, b, β ∈
C with 0 < |b| < |a|, then (a + b)β =
∑
∞
j=0 C(β, j)a
β−jbj , where C(β, j) are the
generalised binomial coefficients.
When a = 1, the series has radius of convergence 1 in b. We shall also use the fact that
the series remains convergent for |b| = 1, so long as Re(β) > 0, which of course holds in
the context of this Lemma. (See [6] p.17, for example.) Now,
t∑
s=m
f(s)
F (s)β
−
t∑
s=m
f(s)
F (s+ 1)β
=
t∑
s=m
f(s)
F (s)β
−
f(s)
(F (s) + f(s))
β
<
t∑
s=m
f(s) (F (s) + f(s))
β
− f(s)F (s)β
F (s)2β
=
t∑
s=m
f(s)
(∑
∞
j=1 C(β, j)F (s)
β−jf(s)j
)
F (s)2β
<
t∑
s=m
∑
∞
j=1 C(β, j)F (s)
β−1f(s)2
F (s)2β
<
t∑
s=m
2βf(s)2
F (s)1+β
≤ 2β
t∑
s=m
f(s)2
F (s)2
< 2β ·K := Kβ
where K is the finite bound provided in Assumption 3.5(2).
The next two results hold in GPAf as well as MPAf :
Lemma 3.8. Suppose that Assumption 3.5(1) holds. Then for any node u, any stage t0,
and any state of the graph G(t0), the probability that v never receives another edge is 0.
Proof. Suppose that du(t0) = N ≥ 1. The probability that u never receives a further edge
is therefore given by (or in GPAf is bounded above by)
∞∏
t=t0
(
1−
N
2F (t)
)f(t)
.
We shall show that this is 0. It is clearly enough to do so in the case N = 1. Taking
logarithms, it is therefore enough to show that
∞∑
t=t0
f(t) ln
(
1 +
1
2F (t)− 1
)
R. Elwes: Preferential attachment processes approaching the Rado multigraph 9
diverges to ∞. Now as for small enough x, we know ln(1 + x) > 12x. Thus for large
enough t,
ln
(
1 +
1
2F (t)− 1
)
>
1
4F (t)
.
Thus the result follows from Assumption 3.5(1).
Corollary 3.9. Suppose that Assumption 3.5(1) holds. Then for any node u, given any
state of the graph G(t0), with probability 1 it will be true that d(t) → ∞ as t → ∞.
Proof. This follows automatically from Lemma 3.8 by the countable additivity of the prob-
ability measure.
4 Martingale theory
In this section, we apply some machinery from the theory of martingales to the process
MPAf , generalising the theory developed in [10]. We shall assume throughout that we
are working in MPAf , and begin with the following easy result, which does not transfer
immediately to GPAf .
Remark 4.1. Given any node u, define Uu(t + 1) := du(t + 1) − du(t) and µu(t) :=
E
(
Uu(t+ 1)
∣∣∣∣du(t)
)
. Then
µu(t)
du(t)
=
f(t)
2F (t)
.
In particular, if f(t) = Θ (tα) where α ≥ 0 then µu(t) = Θ
(
du(t)
t
)
.
The next two results are the key to our analysis, and generalise Proposition 3.1 of [10]
Proposition 4.2. Suppose that Assumption 3.5(2) holds. For any node u, define
A(t) = Au(t) :=
t−1∏
j=1
(
1 +
f(j)
2F (j)
)
and X(t) := Xu(t) =
du(t)
Au(t)
. Then
(i) X(t) is a martingale.
(ii) Thus, for any node u, with probability 1, there exists xu ≥ 0 such that
lim
t→∞
du(t)
A(t)
= xu.
(iii) A(t) = Θ
(
F (t)
1
2
)
10 Art Discrete Appl. Math. 5 (2022) #P2.02
Proof. Employing Remark 4.1, the first part is straightforward:
E(X(t+ 1)||X(t)) =
1
A(t+ 1)
E(d(t+ 1)||d(t))
=
1
A(t+ 1)
(d(t) + µ(t))
=
1
A(t+ 1)
d(t)
(
1 +
µ(t)
d(t)
)
=
1
A(t)
d(t) = X(t).
Part (ii) follows from (i) via Doob’s convergence theorem, which gives us that X(t) →
X for some random variable X .
Hence all that remains to understand the behaviour of A(t) for large t, to establish (iii).
By taking logarithms and employing the standard bounds x − 12x
2 < ln(1 + x) < x, we
see:
1
2
t−1∑
s=1
f(s)
F (s)
−
1
8
t−1∑
s=1
f(s)2
F (s)2
< lnA(t) <
1
2
t−1∑
s=1
f(s)
F (s)
.
Therefore by Assumption 3.5 and Lemma 3.7, it follows that
1
2
∫ t−1
s=1
f(s)
F (s)
ds−K < lnA(t) <
1
2
∫ t−1
s=1
f(s)
F (s)
ds+K ′
for some constants K and K ′ from which the result follows.
We need a little more information about the distribution of the xu provided by the
preceding result:
Proposition 4.3. Suppose that f satisfies Assumption 3.5 in full. Given any time t0, state
G0(t0), and node u, P (xu > 0) = 1.
Proof. Our proof closely follows that of Proposition 3.1 of [10].
We take u as fixed and shall suppress mention of it, writing X(n) for Xu(n), etc.,
throughout.
Given any n > m > 0 define X̃m(n) := (X(n)−X(m))
2. Then for fixed m, it is
an elementary fact that the sequence X̃m(n) forms a submartingale. We now proceed via a
sequence of claims.
Claim 1
E
(
X̃m(n)
∣∣∣∣X(m)
)
=
n−1∑
t=m
E
(
X(t+ 1)2
∣∣∣∣X(m)
)
−E
(
X(t)2
∣∣∣∣X(m)
)
.
Proof of Claim 1.
E
(
X̃m(n)
∣∣∣∣X(m)
)
= E
(
X(n)2 − 2X(n)X(m) +X(m)2
∣∣∣∣X(m)
)
= E
(
X(n)2
∣∣∣∣X(m)
)
− 2X(m)E
(
X(n)
∣∣∣∣X(m)
)
+X(m)2
= E
(
X(n)2
∣∣∣∣X(m)
)
−X(m)2.
Unpacking the sum in the statement of the claim gives the same result.
R. Elwes: Preferential attachment processes approaching the Rado multigraph 11
Claim 2 There exists K > 0 such that for all large enough m and all n > m
E(X̃m(n)||X(m)) < X(m) ·
K
F (m)
1
2
.
Proof of Claim 2
Proof of Claim 2. Recall U(t + 1) := d(t + 1) − d(t). Now U(t + 1) is binomially
distributed via b
(
f(t), d(t)2F (t)
)
meaning, as already observed, that E(U(t + 1)||d(t) =
d) = µ(t) = d·f(t)2F (t) and also Var(U(t + 1)||d(t) = d) =
d·f(t)
2F (t)
(
1− d2F (t)
)
. Thus,
writing f and F for f(t) and F (t) respectively,
E
(
U(t+ 1)2||d(t) = d
)
=
(
df
2F
)2
+
df
2F
(
2F − d
2F
)
<
fd
2F
+
f2d2
4F 2
.
At the same time,
E
(
d(t+ 1)2||d(t) = d
)
= E
(
(U(t+ 1) + d)
2
||d(t) = d
)
= E(U(t+ 1)2||d(t) = d) + 2dE(U(t+ 1)||d(t) = d) + d2
<
fd
2F
+
f2d2
4F 2
+ 2d ·
df
2F
+ d2
=
fd
2F
+
(
1 +
f
2F
)2
d2.
Recall the definition of the martingale X(t) := d(t)A(t) . Thus
E
(
X(t+ 1)2
∣∣∣
∣∣∣d(t) = d
)
=
(
1
A(t+ 1)2
)
·E
(
d(t+ 1)2
∣∣∣
∣∣∣ d(t) = d
)
<
1
A(t+ 1)2
(
fd
2F
+
(
1 +
f
2F
)2
d2
)
=
fA(t)
2F ·A(t+ 1)2
X(t) +
(
1 +
f
2F
)2(
A(t)
A(t+ 1)
)2
X(t)2
<
f
2F ·A(t)
X(t) +X(t)2.
Hence, by the law of total expectation,
E(X(t+ 1)2||X(m))−E(X(t)2||X(m)) <
f
2F ·A(t)
X(m).
12 Art Discrete Appl. Math. 5 (2022) #P2.02
Summing this up over successive terms (and appealing to Claim 1, Proposition 4.2(ii) and
Lemma 3.7) we get
E(X̃m(n)||X(m)) < X(m) ·
n−1∑
t=m
f
2FA(t)
< X(m) ·
n−1∑
t=m
f
2FA(t)
= O
(
X(m) ·
n−1∑
t=m
f
F
3
2
)
= O
(
X(m) ·
∫ n−1
t=m
f(t)
F (t)
3
2
dt
)
< X(m) ·
K
F (m)
1
2
for some K > 0.
Proof of Proposition 4.3, continued: We may now prove the proposition. We proceed by
defining a sequence of times: n0 = t0. Let ni+1 be the least n (if any exists) such that
X(n) < 12X (ni). Otherwise ni+1 = ∞.
The trick is to apply the Kolmogorov-Doob inequality (see for instance [10]) to X̃ni(n):
P(ni+1 < ∞||ni < ∞) = P
(
min
n≥ni
X(n) <
1
2
X(ni)
∣∣∣
∣∣∣X(ni)
)
≤ P
(
max
n≥ni
X̃ni(n) >
1
4
X(ni)
2
∣∣∣
∣∣∣X(ni)
)
= lim
N→∞
P
(
max
n:N≥n≥ni
X̃ni(n) >
1
4
X(ni)
2
∣∣∣
∣∣∣X(ni)
)
≤
4
X(ni)2
· lim
N→∞
E(X̃ni(N)||X(ni))
= O
(
4
X(ni)2
·
1
F (ni)
1
2
·X(ni)
)
= O
(
1
d(ni)
)
.
It follows from Corollary 3.9 that P(ni+1 < ∞||ni < ∞) → 0 as i → ∞, from which
the result follows.
We record one more result regarding the martingale X(t):
Corollary 4.4. Suppose that Assumption 3.5(2) holds. Then the martingale X(t) is bounded
in L2, that is to say supt E
(
X(t)2
)
< ∞.
Proof. By a standard result (see for example Theorem 12.1 of [18]), it is sufficient to show
that
∑
∞
j=0 E
(
|Xj+1 −Xj |
2
)
< ∞.
R. Elwes: Preferential attachment processes approaching the Rado multigraph 13
Notice that by Remark 4.1
|X(t+ 1)−X(t)| =
d(t+ 1)−
(
1 + µ(t)d(t)
)
d(t)
A(t+ 1)
=
U(t+ 1)− µ(t)
A(t+ 1)
Hence
E
(
|X(t+ 1)−X(t)|2
)
=
Var(U(t+ 1))
A(t+ 1)2
= O
(
d(t)f(t)
2F (t)
(
1−
d(t)
2F (t)
)
·
1
A(t+ 1)2
)
= O
(
d(t)
A(t+ 1)
·
f(t)
F (t)A(t+ 1)
)
= O
(
X(t) ·
f(t)
F (t)A(t+ 1)
)
= O
(
f(t)
F (t)A(t+ 1)
)
= O
(
f(t)
F (t)
3
2
)
.
Thus
t∑
j=0
E
(
|Xj+1 −Xj |
2
)
= O
(∫ t
j=0
f(j)
F (j)
3
2
dj
)
= O
(
K − F (t)−
1
2
)
= O(K).
5 Proof of main result
Definition 5.1. A witness request W is a set of pairs of the form W = {(ui,mi) |
1 ≤ i ≤ n} where (u1, . . . , un) is a sequence of nodes and (m1, . . . ,mn) an accom-
panying sequence of non-negative integers.
A witness for W is a node connected to each ui with multiplicity mi.
We write the event W [t] to mean that W is satisfied by some witness by time t.
Observe from the structure of the process that W [t] ⇒ W [t′] for all t′ ≥ t. The
following is the major step towards our goal:
Proposition 5.2. Suppose that f(t) = Θ(t), and that G(t0) is a state of the graph at
time t0. Let W be a witness request. Let ε > 0. Then there exist t1 > t0 such that
P
(
W [t1]
∣∣∣∣ G(t0)
)
> 1− ε.
Proof. We consider only stages from t0 + 1 onwards, and everything that occurs is condi-
tioned upon G(t0), which we shall therefore suppress.
Suppose W = {(ui,mi) | 1 ≤ i ≤ n}. We shall write m =
∑n
i=1 mi, and, abusing
notation, Ui = Uui(t+1), meaning the number of new edges which ui gains at the t+1st
14 Art Discrete Appl. Math. 5 (2022) #P2.02
stage, taking the dependency on t as given when the intended value is obvious. Similarly
we write di for dui(t). (We shall not consider dj for any j other than the ui, so this will
not cause confusion.)
We shall employ vector notation, writing U(t + 1) := U = (U1, . . . , Un) and m :=
(m1, . . . ,mn). Thus our focus is the event U = m. Let us first compute the probability of
this event in terms of the di. The relevant distribution is multinomial M(f, pi, . . . , pn, q)
where pi =
di
2F and q = 1−
∑
pi (again omitting the dependencies on t). Therefore
P
(
U = m
∣∣∣∣ d1, . . . dn
)
=
f !
m1! · . . .mn! · (f −m)!
· qf−m ·
∏
i
pi
mi
= Θ
((
1−
∑
i di
2F
)f−m
·
(
f
2F
)m
·
∏
i
d mii
)
noticing that f !(f−m)! ∼ f
m.
Now we employ our assumption that f(t) = Θ(t) from which it also follows that
1
2F = Θ
(
1
t2
)
and f2F = Θ
(
1
t
)
. Thus there exist constants c1, c2, C0, N > 0 depending
only on G0(t0) such that for all t ≥ N ,
P
(
U = m
∣∣∣∣ d1, . . . dn
)
≥ C0 ·
(
1−
∑
i di
c1t2
)c2t−m
· t−m ·
∏
i
d mii . (5.1)
Our aim is to bound this probability below, away from 0 over a long enough range of t.
We write Xi =
di
Ai
for the martingale supplied by Proposition 4.2, with xi := xui > 0 for
its limit supplied by Proposition 4.2 and Lemma 4.3. We will not attempt to condition on
the actual values xi, but only on the fact that these values are not extreme (NE).
First, choose κ1, κ2 > 0 such that
κ1t < A(t) < κ2t
for all large enough t. This is guaranteed to occur by Proposition 4.2(iii) since
F (t)
1
2 = Θ(t). We increase N if necessary to ensure that this holds. Notice that since
A(t) is entirely predictable in advance, the value of N remains dependent only on G0(t0).
Now, for any y2 > y1 > 0, define the following event:
NE(y1, y2) :
(
n∧
i=1
y1 < xi < y2
)
.
We shall apply this in the following case: given δ > 0 choose y2(δ) > y1(δ) > 0 so
that P(¬NE(y1, y2)) < δ. (We shall specify δ later, and will only need to consider one
such value. Thus we shall consider δ fixed for the purposes of what follows.)
By Corollary 4.4, Xi(t) → xi in expectation, and thus in probability. More precisely,
for any η > 0, we may increase N > 0 by some quantity depending only on η so that for
all t ≥ N and all i ≤ n
E
(
|Xi(t)− xi|
)
<
( η
n
)2
.
Thus, by Markov’s inequality
P
(
|Xi(t)− xi| >
η
n
)
<
η
n
.
R. Elwes: Preferential attachment processes approaching the Rado multigraph 15
Hence defining the event that all the Xi(t) are close (Cl) to their respective xi
Cl(t, η) :=
n∧
i=1
(
|Xi(t)− xi| <
η
n
)
we have for all t ≥ N
P (Cl(t, η)) > 1− η. (5.2)
Again, we shall pick a value of η later. Notice also that
P (Cl(t, η)) < P
(
Cl(t, η)
∣∣∣∣ NE(y1, y2)
)
+ δ.
So
P
(
Cl(t, η)
∣∣∣∣ NE(y1, y2)
)
> 1− η − δ. (5.3)
Next, we define a bound for di(t). Given δ, η > 0 as before, let b1(η) = b1(δ, η) :=
κ1 ·
(
y1 −
η
n
)
and b2(η) = b2(δ, η) := κ2 ·
(
y2 +
η
n
)
, insisting that η is small enough that
b1 > 0. Then we define the event
Bo(t, b1, b2) :=
n∧
i=1
(b1 · t < di(t) < b2 · t) .
Observe now that for t ≥ N
(NE(y1(δ), y2(δ)) & Cl(t, η)) ⇒ Bo(t, b1(η), b2(η)). (5.4)
Hence
P
(
Bo(t, b1, b2)
∣∣∣
∣∣∣NE(y1(δ), y2(δ))
)
≥ 1− η − δ.
Thus we obtain the unconditional bound:
P (Bo(t, b1, b2)) ≥ (1− η − δ)(1− δ). (5.5)
Now we use the bound obtained in (5.1) and see that whenever b1 ≤ b′1 < b
′
2 ≤ b2
P
(
U = m
∣∣∣
∣∣∣ Bo(t, b′1, b′2)
)
> C0 ·
(
1−
n · b2 · t
c1t2
)c2t−m
· t−m · (b1 · t)
m
= C0 · b
m
1 ·
(
1−
nb2
c1t
)c2t−m
= C0 · b
m
1 ·
(
1−
nb2c2
c1
·
1
c2t
)c2t
·
(
1−
nb2
c1t
)
−m
→ C0 · b
m
1 · e
−
nb2c2
c1 := C3 > 0.
Hence, by letting C4 = C4(δ, η) := 12C3 and increasing N again if necessary (and
again by some predictable amount), we have for all t ≥ N
P
(
U = m
∣∣∣
∣∣∣ Bo(t, b′1, b′2)
)
> C4. (5.6)
Now for any ζ > 0, we may let M = M(ζ, δ, η) be large enough that (1− C4)
M
< ζ.
The goal therefore is to locate M places where Bo(t, b1(η), b2(η)) holds, and argue that
16 Art Discrete Appl. Math. 5 (2022) #P2.02
the probability that all of them fail to produce an instance of U = m is bounded above by
ζ.
Notice that bound (5.6) holds independently for all t ≥ N : the arguments are unaffected
by previous values of U so long as Bo(t, b′1, b
′
2) holds. However, the same is not true
for bound (5.5). By conditioning on whether or not U(t′) = m holds, we risk affecting
P (Bo(t, b1(η), b2(η))) for t > t′.
To navigate this obstacle, we shall locate a range [t2, t2 +M) within which the bound
Bo(t, b1(η), b2(η)) is guaranteed to hold, barring a certain extreme event ¬Sh defined be-
low, which will have a probability bounded above by θ for arbitrarily small θ.
We wish t2 to satisfy the tighter bound Bo(t2, b1(
η
2 ), b2(
η
2 )). Notice that appropriate
adaptations of (5.2), (5.4), and (5.5) above guarantee that for large enough t2,
P
(
¬Bo
(
t2, b1
(
η
2
)
, b2
(
η
2
)))
< η2 + 2δ. (5.7)
However, as already indicated, Bo
(
t2, b1
(
η
2
)
, b2
(
η
2
))
on its own is not quite enough to
guarantee Bo(t2 + j, b1(η), b2(η)) for j ≤ M . So let us describe the extra ingredient we
require. Notice that Ui(t) is a binomial distribution with a long right tail, since the number
of trials f(t) is of the order of t, and the probability of success per trial is di(t)2F (t) which is
of order 1t . We shall show that we may ignore the extremity of this tail, thus allowing us to
impose a tighter upper bound than f(t) on Ui(t) for all t ∈ [t2, t2 +M).
In Theorem 1.1 from [3], we find a useful bound for the right-tail of a binomial distri-
bution U ∼ b(f, p): if u > 1 and 1 ≤ S := ⌈ufp⌉ ≤ f − 1 then
P(U ≥ S) <
(
u
u− 1
)
·P(U = S).
Let us apply this in the case S = ⌈tα⌉ for some fixed α ∈
(
1
2 , 1
)
. (Its exact value does
not matter.) Then
u = u(t) =
tα
pf
=
tα2F (t)
d(t)f(t)
.
Assembling the bounds c1t ≤ f(t) ≤ c2t and c1t2 ≤ 2F (t) ≤ c2t2 and Bo(t, b′1, b
′
2)
where b1 ≤ b′1 < b
′
2 ≤ b2 and employing the standard bound for the binomial coefficiant(
f
S
)
≤
(
f · e
S
)S
, we find
P
(
Ui ≥ t
α
∣∣∣
∣∣∣ Bo(t, b′1, b′2)
)
<
(
c2
c1b1
tα
c1
c2b2
tα − 1
)
·
(
(ec2 + 1) · t
1−α
)tα
·
(
b2
c1t
)tα
·
(
1−
b1
c2t
)t−⌈tα⌉
<
(
B
tα
)tα
.
for some B > 0. Notice again that this holds independently of the specific values of b′1 and
b′2, so long as b1 ≤ b
′
1 < b
′
2 ≤ b2. Now we define a new event, that the tails are short (sh):
sh(t) :=
n∧
i=1
Ui(t+ 1) < t
α.
R. Elwes: Preferential attachment processes approaching the Rado multigraph 17
After increasing B to allow for the non-independence of the n different Ui we now see
that:
P
(
¬sh(t)
∣∣∣
∣∣∣ Bo(t, b1(η), b2(η))
)
< n ·
(
B
tα
)tα
. (5.8)
Putting these events together, define
Sh(t2) := ∀t ∈ [t2, t2 +M) sh(t).
To obtain a similar bound for P
(
¬Sh(t2)||Bo
(
t2, b1
(
η
2
)
, b2
(
η
2
)))
we first show that
for j ≤ M
(
Bo
(
t2, b1
(
η
2
)
, b2
(
η
2
))
&
j−1∧
i=0
sh(t2 + j − 1)
)
(5.9)
⇒ Bo(t2 + j, b1(η), b2(η)).
Suppose that Bo
(
t2, b1
(
η
2
)
, b2
(
η
2
))
holds. We address the lower bound first, for which
we do not require the hypothesis on sh. Instead, for all j ≤ M , clearly
d(t2 + j) ≥ d(t2) ≥ b1
(
η
2
)
· t2 = κ1 ·
(
y1 −
η
2n
)
· t2.
If additionally t2 ≥
2nMy1
η , then the final term above exceeds
κ1 ·
(
y1 −
η
n
)
· (t2 +M) ≥ b1(η) · (t2 + j).
Now we obtain the corresponding upper bound. By our assumption on sh,
d(t2 + j) ≤ d(t2) +M · (t2 +M)
α
≤ κ2 ·
(
y2 +
η
2n
)
· t2 +M · (t2 +M)
α
≤ κ2 ·
(
y2 +
η
n
)
· t2
if t2 ≥ max
{
M,
(
4Mn
κ2η
) 1
1−α
}
, which completes the proof of Implication (5.9).
Implication (5.9) allows us to take the M -fold sum of (5.8), finding
P
(
¬Sh(t2)
∣∣∣
∣∣∣ Bo
(
t2, b1
(
η
2
)
, b2
(
η
2
)))
<
t2+M∑
t=t2
n
(
B
tα
)tα
→ 0
as t2 → ∞. Thus for any θ > 0 for all large enough t2 we have
P
(
¬Sh(t2)
∣∣∣
∣∣∣ Bo
(
t2, b1
(
η
2
)
, b2
(
η
2
)))
< θ. (5.10)
Finally, we may complete the argument, setting δ = ε8 and θ = ζ =
ε
4 and η =
ε
2 and
t1 := t2 +M . For large enough t, we may update bound (5.6) to get
P
(
(¬U(t2 + 1) = m) & sh(t2)
∣∣∣
∣∣∣ Bo
(
t2, b1
(
η
2
)
, b2
(
η
2
)))
< 1− C4.
18 Art Discrete Appl. Math. 5 (2022) #P2.02
Similarly,
P
((
¬U(t2 + j + 1) = m
)
& sh(t2 + j)
∣∣∣
∣∣∣
j−1∧
i=0
sh(t2 + i) & Bo
(
t2, b1
(
η
2
)
, b2
(
η
2
)) )
< 1− C4.
As observed earlier, these bounds hold independently of the previous values of U, meaning
that
P
((
¬U(t2 + j + 1) = m
)
& sh(t2 + j)
∣∣∣
∣∣∣
j∧
i=0
¬U(t2 + i) &
j−1∧
i=0
sh(t2 + i) & Bo
(
t2, b1
(
η
2
)
, b2
(
η
2
)) )
< 1− C4.
Taking the product of these bounds, and denoting the failure of our desired result by
Fa(t2) := ∀t ∈ [t2, t2 +M) (U(t+ 1) ̸= m), we see that
P
(
Fa(t2) & Sh(t2)
∣∣∣
∣∣∣ Bo(t2, b1
(
η
2
)
, b2
(
η
2
) )
< ζ
and so by bounds (5.7) and (5.10)
P
(
Fa(t2)
)
< ζ + θ + η2 + 2δ = ε.
We may now complete the proof of our main result, which we first restate for the
reader’s convenience:
Theorem 5.3. Suppose that G′ is a finite directed multigraph containing no isolated nodes,
that f satisfies the requirements from Definition 3.1, and also that f(t) = Θ(t). Then, with
probability 1, the infinite limit of MPAf (G
′) is isomorphic, as an undirected multigraph, to
the Rado multigraph.
Proof. First notice that there are countably many witness requests. Thus we may organise
them into a list (Wj : j ≥ 1).
Let ε > 0. Again everything that occurs is conditioned upon G0(t0). We shall show
that the probability of all witness requests eventually being satisfied exceeds 1−ε. Suppose
inductively that we have found time tj so that so that P(
∧j
i=1 Wi[tj ]) > 1−
(
1− 12j
)
ε.
Let G = Gj be the set of all states G = G(tj) of the graph at time tj consistent with
G0(t0) and with
∧j
i=1 Wi[tj ]. Notice that G is a finite set, that P
(
G(tj)
∣∣∣∣ G0(t0)
)
> 0
for each G ∈ G, and by assumption that
∑
G∈G P
(
G(tj)
∣∣∣∣ G(t0)
)
> 1−
(
1− 12j
)
ε.
Consider now Wj+1 and let ε′ < 12j+1 ε. Now given each G
(k) ∈ G, by Proposition 5.2
there exist t(k) ≥ tj such that
P
(
Wj+1
[
t(k)
] ∣∣∣∣ G(k)(tj)
)
> 1− ε′.
R. Elwes: Preferential attachment processes approaching the Rado multigraph 19
Let tj+1 := max{t(k) | G(k) ∈ G}. Then
P
(
j+1∧
i=1
Wi[tj+1]
∣∣∣∣ G0(t0)
)
≥
∑
k
P
(
j+1∧
i=1
Wi[tj+1]
∣∣∣∣ G(k)(tj)
)
·P
(
G(k)(tj)
∣∣∣∣ G0(t0)
)
=
∑
k
P
(
Wj+1[tj+1]
∣∣∣∣ G(k)(tj)
)
·P
(
G(k)(tj)
∣∣∣∣ G0(t0)
)
>
∑
k
(1− ε′) ·P
(
G(k)(tj)
∣∣∣∣ G0(t0)
)
> (1− ε′)
(
1−
(
1−
1
2j
)
ε
)
> 1−
(
1−
1
2j+1
)
ε.
6 Future work
We close this paper with a short discussion of possible future directions of study. As noted
earlier, one goal is to translate the current work into the domain of graphs (rather than
multigraphs) by proving Conjecture 3.4, which we restate:
Conjecture 6.1. Suppose that G′ = (V ′, E′) be a finite directed graph containing no
isolated nodes, that f satisfies the conditions in Definition 3.3, and further that there are
constants 0 < c1 ≤ c2 < 1 where c1 · t ≤ f(t) ≤ c2 · t for all large enough t. Then, with
probability 1, the infinite limit of GPAf (G
′) is isomorphic as an undirected graph to the
Rado graph.
A second avenue to investigate is the limit of an MPAf process when f is strictly be-
tween the constant case (considered by Kleinberg and Kleinberg in [10]) and the linear
growth rate analysed here. A natural starting point would be the case f(t) = Θ
(√
t
)
.
One might hope somewhere within this regime to identify a connection to (a multigraph
analogue of) the theory of Shelah-Spencer sparse random graphs as elucidated in [16]. The
author recently established a connection between Shelah-Spencer graphs and the limits of
finitary random processes in [8], albeit in a context rather simpler than preferential attach-
ment.
Thirdly, a central role in the contemporary study of graph limits is played by the theory
of graphons, as developed, for instance, by Lovász in [13]. Thus it is natural to seek to
connect the current work to that body of knowledge. Although graphons as originally
conceived do not allow for multi-edges, in [12] a theory of convergence of sequences of
multigraphs is developed within the broader setting of decorated graphs and Banach space
valued graphons, so this is an initial point of contact to consider. (I am grateful to the
anonymous reviewer for bringing this to my attention.)
ORCID iDs
Richard Elwes https://orcid.org/0000-0002-6752-5501
20 Art Discrete Appl. Math. 5 (2022) #P2.02
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P2.03
https://doi.org/10.26493/2590-9770.1379.7a4
(Also available at http://adam-journal.eu)
Bracing frameworks
consisting of parallelograms*
Georg Grasegger†
Johann Radon Institute for Computational and Applied Mathematics (RICAM),
Austrian Academy of Sciences, Altenbergerstraße 69, 4040 Linz, Austria
Jan Legerský‡
Department of Applied Mathematics, Faculty of Information Technology,
Czech Technical University in Prague, Thákurova 9, 160 00 Prague, Czech Republic
Received 26 August 2020, accepted 7 July 2021, published online 11 May 2022
Abstract
A rectangle in the plane can be continuously deformed preserving its edge lengths,
but adding a diagonal brace prevents such a deformation. Bolker and Crapo characterized
combinatorially which choices of braces make a grid of squares infinitesimally rigid using
a bracing graph: a bipartite graph whose vertices are the columns and rows of the grid,
and a row and column are adjacent if and only if they meet at a braced square. Duarte
and Francis generalized the notion of the bracing graph to rhombic carpets, proved that the
connectivity of the bracing graph implies rigidity and stated the other implication without
proof. Nagy Kem gives the equivalence in the infinitesimal setting. We consider continuous
deformations of braced frameworks consisting of a graph from a more general class and its
placement in the plane such that every 4-cycle forms a parallelogram. We show that rigidity
of such a braced framework is equivalent to the non-existence of a special edge coloring,
which is in turn equivalent to the corresponding bracing graph being connected.
Keywords: Flexibility, rigidity, bracing, rhombic tiling.
Math. Subj. Class.: 52C25, 51K99, 70B99
*We thank Eliana Duarte for bringing us to this topic and sharing her ideas, and Matteo Gallet and an anony-
mous referee for their comments to the paper.
†Supported by the Austrian Science Fund (FWF): P31888 and by the Austrian Science Fund (FWF): W1214-
N15.
‡Supported by Ministry of Education, Youth and Sports of the Czech Republic, project no.
CZ.02.1.01/0.0/0.0/16_019/0000778 and by the Austrian Science Fund (FWF): P31061.
E-mail addresses: georg.grasegger@ricam.oeaw.ac.at (Georg Grasegger), jan.legersky@fit.cvut.cz (Jan
Legerský)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P2.03
1 Introduction
A planar framework is a graph together with a placement of its vertices in the plane. If
there is a non-trivial flex (a deformation of the placement preserving the distances between
adjacent vertices that is not induced by a rigid motion), then the framework is said to
be flexible, otherwise rigid. Bolker and Crapo [4] studied infinitesimal flexibility of a
framework corresponding to a grid of squares with some squares being braced by adding
diagonals, see Figure 1. They construct a bipartite graph by taking the columns and rows
of the grid to be the two parts of the vertex set; a column and row are connected if and only
if their common square is braced. They showed that a braced grid is infinitesimally rigid,
i.e., has no non-trivial first order flex, if and only if the bipartite graph is connected.
Figure 1: Grid frameworks can be deformed in a way that preserves the edge lengths. By
bracing it (i.e. by adding diagonal edges) we can reduce the number of degrees of freedom.
The left braced grid is rigid whereas the right one allows a flex.
Generalizations to rectangular grids with holes [14, 24] or placing longer diagonals [13]
than those for a single 4-cycle have been studied as well as bracing by cables [20]. Grids
with rectilinear boundary are discussed in [17]. Extensions to cubic grids have been stud-
ied [3, 21]. The papers [9, 28] describe the number of randomly added braces for which
the transition from flexible to rigid occurs. A related problem to the rigidity of a grid is the
rigidity of one- and multi-story building [19, 23], also with cables [5, 22, 25]. Simple forms
of bracing grids are also known to be suitable as a puzzle, for science communication and
for student’s exercises (see for instance [26]).
In this work, we focus on parallelograms instead of squares, and we allow a richer
combinatorial structure than grids. Flexibility of rhombic/parallelogramic tilings is studied
by physicists due to its relation with quasicrystals [29]. The bracing of rhombic carpets,
which are 1-skeleta of finite simply connected pieces of rhombic tilings, was investigated
by Wester [27]. Duarte and Francis [8] formalized the notions necessary to study the flexi-
bility of rhombic carpets: a natural step from columns and rows of a grid towards a rhombic
carpet is to take ribbons. These are sequences of rhombi such that every two consecutive
ones share an edge and all these edges are parallel. Following the idea of Bolker and Crapo,
Duarte and Francis construct a bracing graph whose vertices are the ribbons and two rib-
bons are adjacent if they have a common rhombus that is braced. They prove that if the
constructed graph is connected, then the braced rhombic carpet is rigid. Further, they state
the other implication without proof. We thank Eliana Duarte for pointing out this statement
to us and sharing some hints about a possible proof [7]. Nagy Kem [18] translates the
infinitesimal rigidity of a braced rhombic carpet to the rigidity of an auxiliary framework
which in turn corresponds to the connectivity of the bracing graph.
We formulate the problem of the flexibility of braced structures in terms of frameworks.
In particular, we define ribbons as equivalence classes on edges of the underlying graph us-
G. Grasegger and J. Legerský: Bracing frameworks consisting of parallelograms 3
ing its 4-cycles. We consider a special class of graphs, which we call ribbon-cutting graphs.
A connected graph is ribbon-cutting if every ribbon is an edge cut, i.e., removing the edges
of the ribbon makes the graph disconnected. Regarding the placement, we ask all 4-cycles
to form parallelograms, see Figure 2. Notice that the frameworks we consider — we call
them P-frameworks — form a proper superset of the frameworks corresponding to rhombic
carpets and rectangular grids (without holes). The question we address is analogous to the
one by Bolker, Crapo, Duarte and others, namely, characterization of choices of braces of
parallelograms yielding flexible/rigid P-frameworks. Contrary to Bolker, Crapo and Nagy
Kem, we consider continuous flexes, not infinitesimal ones. Furthermore, we use a recently
established method of special edge colorings to prove our results.
Figure 2: Carpet frameworks can be deformed in a way that preserves the edge lengths.
The notion of NAC-colorings was developed in our previous paper [11]. A NAC-
coloring is a surjective edge coloring of a graph by red and blue such that for every cycle of
the graph, either all edges have the same color or there are at least two edges of each color.
We proved that a graph has a flexible framework if and only if it has a NAC-coloring. It
appears that the techniques used to prove the theorem fit nicely to the context of bracing
P-frameworks if we restrict ourselves to certain NAC-colorings: a NAC-coloring is called
cartesian if there are no two vertices connected by a red path and blue path simultaneously.
The non-existence of a cartesian NAC-coloring serves as a bridge in the proof that a braced
P-framework is rigid if and only if the corresponding bracing graph (defined analogously
to [8]) is connected. Our results can be summarized as follows.
Theorem 1.1. For a braced P-framework (G, ρ), the following statements are equivalent:
(i) (G, ρ) is rigid,
(ii) G has no cartesian NAC-coloring, and
(iii) the bracing graph of G is connected.
In particular, the minimum number of braces making a framework rigid is one less than the
number of ribbons of its underlying graph.
Theorem 1.1 extends the result by Duarte and Francis [8], which is the implication
(iii) =⇒ (i) for rhombic carpets. These form a proper subclass of P-frameworks used
in this paper since all rhombic carpets are planar embeddings in graph theoretical sense,
contrary to P-frameworks.
We implemented the concepts introduced in the current paper by extending our SAGE-
MATH package FLEXRILOG [12]. We encourage the reader to experiment with the Jupyter
notebook available at https://jan.legersky.cz/bracingFrameworks.
4 Art Discrete Appl. Math. 5 (2022) #P2.03
The paper is organized as follows: Section 2 recalls the notions from rigidity theory
and NAC-colorings. We define ribbons, parallelogram placements and P-frameworks in
Section 3 and prove some basic results. We also formalize bracing and the notion of bracing
graph in our context. Section 4 provides the proofs yielding Theorem 1.1. We put additional
material to the appendix, for instance, we show that rhombic carpets are P-frameworks in
Appendix B.
2 Preliminaries
In this section we recall basic notions and definitions commonly used in rigidity theory.
The ideas are based on previous work using special edge colorings to find flexes of graphs.
We introduce these colorings here and describe what we mean by flexibility.
Definition 2.1. Let G = (VG, EG) be a connected graph. A map ρ : VG → R2 such that
ρ(u) ̸= ρ(v) for all edges uv ∈ EG is a placement. The pair (G, ρ) is called a framework.
Definition 2.2. Two frameworks (G, ρ) and (G, ρ′) are equivalent if
∥ρ(u)− ρ(v)∥ = ∥ρ′(u)− ρ′(v)∥
for all uv ∈ EG. Two placements ρ and ρ′ are congruent if there exists a Euclidean
isometry M of R2 such that Mρ′(v) = ρ(v) for all v ∈ VG.
Definition 2.3. A flex of the framework (G, ρ) is a continuous path t 7→ ρt, t ∈ [0, 1], in
the space of placements of G such that ρ0 = ρ and each (G, ρt) is equivalent to (G, ρ).
The flex is called trivial if ρt is congruent to ρ for all t ∈ [0, 1].
We define a framework to be (proper) flexible if there is a non-trivial flex in R2 (with
injective placements). Otherwise it is called rigid.
In a previous paper [11], we introduced a special edge coloring, which is called NAC-
coloring, in order to classify the graphs that have flexible frameworks.
Definition 2.4. Let G be a graph. A coloring of edges δ : EG → {blue, red} is called a
NAC-coloring, if it is surjective and for every cycle in G, either all edges have the same
color, or there are at least 2 edges in each color (see Figure 3). The NAC-coloring δ gives
subgraphs
Gδred = (VG, {e ∈ EG : δ(e) = red}) and G
δ
blue = (VG, {e ∈ EG : δ(e) = blue}) .
We remark that flexibility in [11] is defined in the following sense: an edge labeling
by positive real numbers (interpreted as lengths for edges) is called flexible if there are
infinitely many non-congruent placements inducing it. Notice that if a framework has a
flex, then the induced edge labeling (corresponding to the lengths) is flexible. On the other
hand, assuming we have a flexible edge labeling, then placements inducing the edge lengths
corresponding to the labeling form an algebraic variety containing an algebraic curve of
placements. Considering a nonsingular point in this curve, a local parametrization of the
curve around this point gives a flex. Therefore, we can state the result as follows.
Theorem 2.5 ([11]). A connected non-trivial graph allows a flexible framework if and only
if it has a NAC-coloring.
G. Grasegger and J. Legerský: Bracing frameworks consisting of parallelograms 5
Two non-adjacent vertices u and v overlap in the flex constructed in the proof of the
theorem in [11] if and only if there is a red path from u to v and a blue path from u to v. In
order to avoid overlapping vertices, we focus on a special type of NAC-colorings.
Definition 2.6. A NAC-coloring δ of a graph G is called cartesian if no two distinct ver-
tices are connected by a red and blue path simultaneously.
We chose the name cartesian due to its connection with cartesian products of graphs,
see Appendix A.
Figure 3: A coloring that is not NAC-coloring (left), cartesian NAC-coloring (middle) and
non-cartesian NAC-coloring (right).
Remark 2.7. A NAC-coloring δ of a graph G is cartesian if and only if for every connected
component R of Gδred and B of G
δ
blue, the intersection of the vertex sets of R and B contains
at most one vertex.
Notice that in a cartesian NAC-coloring, a 4-cycle subgraph is monochromatic, or the
opposite edges have the same color.
3 Ribbons and parallelogram placements
In this section we describe bracings of graphs (Section 3.2). We mainly consider a class
of graphs (Section 3.1) which essentially consists of four-cycles which we want to place
in the plane, forming parallelograms. Having these 4-cycles in mind we start by defining
an equivalence relation on the edges. The equivalence classes, called ribbons, generalize
the notion of rows and columns in a rectangular grid. Ribbons are a concept that is also
used in other places under various names (stripes, worms, de Bruijn lines) and for different
purpose (see for instance [2, 6, 10, 29]).
Definition 3.1. Let G be a graph. Consider the relation on the set of edges, where two
edges are in relation if they are opposite edges of a 4-cycle subgraph of G. An equivalence
class of the reflexive-transitive closure of the relation is called a ribbon. Figure 4 shows all
ribbons for some small graphs. A ribbon r is simple if the subgraph induced by r does not
contain any 4-cycle (see Figure 5a for an example of a non-simple ribbon).
In the case of rectangular grids, there is a natural way how to order the edges in a
ribbon, i.e., a row or column. In our context, there is no natural order of the edges in a
ribbon as Figure 5b indicates.
From now on, given a walk W , the notation (u, v) ∈ W means that the edge uv belongs
to W and u precedes v in W . Similarly, for a ribbon r, the notation (u, v) ∈ r ∩W means
(u, v) ∈ W and uv ∈ r. If (u, v) ∈ r ∩ W is used to iterate in a sum, the edges of W
6 Art Discrete Appl. Math. 5 (2022) #P2.03
Figure 4: The ribbons of the graphs are indicated by dashed lines. All edges intersecting
the line belong to the same ribbon.
(a) The only ribbon is not simple. (b) The yellow ribbon has three “ends”.
Figure 5: Special cases that might happen for ribbons.
must be considered as a multiset: the summand corresponding to (u, v) is included as many
times as uv occurs in W with u preceding v in W . Similarly for the cardinality of r ∩W :
|r ∩W | =
∑
(u,v)∈r∩W
1 .
Recall that a set of edges r is an edge cut of a connected graph G if the graph G \ r =
(VG, EG \ r) is disconnected.
Lemma 3.2. Let G be a connected graph with a simple ribbon r, which is an edge cut.
Then G \ r has exactly two connected components. If w,w′ ∈ VG and W is a walk from
w to w′, then |r ∩W | is odd if and only if r separates w and w′, i.e., w and w′ are in the
different connected components of G \ r. In particular, if W is a closed walk in G, then
|r ∩W | is even.
Proof. Let uv be an edge of r. For every edge u′v′ ∈ r, there exists a sequence of edges
u1v1, . . . , ukvk such that u = u1, v = v1, uk = u′, vk = v′ and (ui, vi, vi+1, ui+1) is a
4-cycle in G. Hence, there are walks (u1, . . . , uk) and (v1, . . . , vk) in G. An edge uiui+1
is in r if and only if vivi+1 is in r. But if uiui+1, vivi+1 ∈ r, then (ui, vi, vi+1, ui+1)
would be a 4-cycle in the subgraph induced by r, which is not possible since r is simple.
Hence, no edge of the two walks is in r. This shows that every vertex of an edge in r is
either connected to u, or v in G \ r, thus, G \ r has two connected components.
If W is a walk from w to w′, then |W ∩ r| is even if and only if w and w′ are in the
same connected component of G \ r.
We want to consider graphs that somehow consist of parallelograms. For interpreting
this idea we need to look at frameworks rather than graphs.
G. Grasegger and J. Legerský: Bracing frameworks consisting of parallelograms 7
Definition 3.3. Let G be a connected graph. A placement ρ : VG → R2 for G such that
ρ is injective and each 4-cycle in G forms a parallelogram1 in ρ is called a parallelogram
placement.
Remark 3.4. Let ρ be a parallelogram placement of a connected graph G. Edges of a
ribbon of G are parallel line segments of the same length in ρ.
Remark 3.5. By Remark 3.4, if there was a 4-cycle induced by a ribbon, then two opposite
vertices of the 4-cycle would coincide in a parallelogram placement, which contradicts
injectivity of the placement. Hence, if a graph allows a parallelogram placement, then all
its ribbons are simple.
The following properties of parallelogram placements are needed later on.
Lemma 3.6. Let G be a connected graph with a parallelogram placement ρ and ribbon r
which is an edge cut. If the vertex set of r is V1 ∪ V2, where all vertices of Vi belong to the
same connected component of G \ r, then ρ(V2) is a translation of ρ(V1). In particular, the
vector ρ(u2)− ρ(u1) is the same for all edges u1u2 ∈ r, ui ∈ Vi.
Proof. The ribbon r is simple by Remark 3.5. Lemma 3.2 gives the partition V1 ∪ V2
with |V1| = |V2|. The vector ρ(u2) − ρ(u1) is the same for all u1u2 ∈ r, ui ∈ Vi, by
Remark 3.4.
Lemma 3.7. Let G be a connected graph with a parallelogram placement ρ. Let r be a
ribbon of G which is an edge cut and W be a walk in G. If |r ∩W | is even, then
∑
(w1,w2)∈r∩W
(ρ(w2)− ρ(w1)) = 0 .
Proof. Let W = (u0, u1, . . . , um) be a walk. All ribbons are simple by Remark 3.5. Let
V1 ∪ V2 be as in Lemma 3.6. Let the edges of W that are in r be uj1uj1+1, . . . , ujkujk+1
with j1 < j2 < · · · < jk, k is even by assumption. We have that uj1 , uj2+1, uj3 , . . . ,
ujk+1 ∈ V1 and uj1+1, uj2 , uj3+1, . . . , ujk ∈ V2. By Lemma 3.6,
k∑
i=1
(ρ(uji+1)− ρ(uji)) = 0 .
3.1 Frameworks and graphs consisting of parallelograms
For proving the statements in this paper, we require frameworks with a parallelogram place-
ment where all ribbons of the underlying graph are edge cuts. This yields a so called
P-framework. We describe two different subclasses of P-frameworks: an illustrating ap-
proach is to start from a set of connected parallelograms with additional properties and
form a framework. This will be a carpet framework. The second one is a recursive con-
struction. Furthermore, we describe the relations between these classes. For streamlining
the paper, we put some results to Appendix B.
Definition 3.8. A graph G is called ribbon-cutting graph if it is connected and every ribbon
is an edge cut. If ρ is a parallelogram placement of G, we call the framework (G, ρ) a P-
framework.
1Here we consider only non-degenerate parallelograms, namely, not all vertices are collinear.
8 Art Discrete Appl. Math. 5 (2022) #P2.03
A rectangular lattice graph (grid graph) with its natural placement is a P-framework as
well as the frameworks in Figure 2 and the graphs in Figure 4 and 5b with the placements
given by their layouts.
There are ribbon-cutting graphs without any parallelogram placement. Figure 6 shows
such a graph, for which the non-existence of a parallelogram placement follows from failing
one of the necessary conditions given by Theorem 3.9. On the other hand, the graph in
Figure 7 is not ribbon-cutting but has a parallelogram placement.
Figure 6: The graph of the framework is ribbon-cutting, but it has no parallelogram place-
ment: if the red vertex and edges were placed forming a parallelogram, the two filled
vertices would coincide. Theorem 3.9 also shows this, since these two vertices are not sep-
arated by any ribbon.
Figure 7: A parallelogram placement of a graph with ribbons that are not edge cuts.
Theorem 3.9. If (G, ρ) is a P-framework, then there are no odd cycles in G, i.e., the graph
is bipartite, and every two vertices are separated by a ribbon.
Proof. Remark 3.5 guarantees that all ribbons of G are simple. Every ribbon intersects a
cycle in an even number of edges by Lemma 3.2, hence, the cycle is even.
Let u and v be two distinct vertices. Let W = (u = u0, u1, . . . , um = v) be a walk.
Let R be the set of ribbons which contain at least one edge of W . Since u and v are distinct
and ρ is injective, we have
0 ̸= ρ(um)− ρ(u0) =
m∑
i=1
(ρ(ui)− ρ(ui−1)) =
∑
r∈R
∑
(w1,w2)∈r∩W
(ρ(w2)− ρ(w1)) .
All ribbons r such that |r ∩ W | is even have a zero contribution by Lemma 3.7. Hence,
there must be a ribbon r′ such that |r′ ∩ W | is odd. The ribbon r′ separates u and v by
Lemma 3.2.
The following definition is a slight generalization of rhombic carpets used in [8] where
we allow parallelograms instead of rhombi.
Definition 3.10. Let S be a finite set of arbitrary parallelograms in R2 (including interiors)
such that:
G. Grasegger and J. Legerský: Bracing frameworks consisting of parallelograms 9
• if a point belongs to two parallelograms, then it is either a vertex of both, or an
interior point of an edge of both (in particular, if a point belongs to more than two
parallelograms, then it is a vertex),
• the boundary of the union
⋃
S is a simple polygon.
The framework obtained by taking the 1-skeleton of S together with the vertex positions is
called a carpet framework (see Figure 2 for an example).
Every carpet framework is a P-framework, see Corrolary B.6. In order to prove this
we introduce a class of graphs Grec, which contains the underlying graphs of carpet frame-
works. The definition is done recursively by adding vertices in a way that a parallelogram
placement can be extended. In order to streamline the paper, these discussions can be seen
in Appendix B.
3.2 Bracings
A general P-framework is flexible with many degrees of freedom. By adding edges to the
graph we can reduce this number. In particular we are interested in adding diagonal edges
of 4-cycles. This process is called the bracing of the graph or framework.
Definition 3.11. A braced ribbon-cutting graph is a graph G = (VG, Ec ∪ Ed) where Ec
and Ed are two non-empty disjoint sets such that the graph (VG, Ec) is a ribbon-cutting
graph and the edges in Ed correspond to diagonals of some 4-cycles of (VG, Ec). These
diagonals are also called braces. If r is a ribbon of (VG, Ec), then
r ∪ {u1u3 ∈ Ed : ∃ 4-cycle (u1, u2, u3, u4) of (VG, Ec) s.t. u1u2, u3u4 ∈ r}
is a ribbon of the braced ribbon-cutting graph G.
The framework (G, ρ) is called braced P-framework if G is a braced ribbon-cutting
graph and ρ is a parallelogram placement for (VG, Ec). Figure 8 shows an example.
Figure 8: An example of a braced P-framework (left) with the underlying ribbon-cutting
graph (right) and its bracing (middle).
Remark 3.12. A ribbon of a braced ribbon-cutting graph (V,Ec∪Ed) is an edge cut since
the corresponding ribbon of (V,Ec) is an edge cut.
We construct a new graph, which encodes the relations between the ribbons, i.e., we
ask whether they share 4-cycles. A subgraph of this graph indicates whether some of the
shared 4-cycles is braced.
10 Art Discrete Appl. Math. 5 (2022) #P2.03
Definition 3.13. Let G be a braced ribbon-cutting graph. The ribbon graph Γ of G is
the graph with the set of vertices being the set of ribbons of G and two ribbons r1, r2 are
adjacent if and only if there is a 4-cycle (u1, u2, u3, u4) in the underlying unbraced graph
of G such that u1u2, u3u4 ∈ r1 and u1u4, u2u3 ∈ r2. The subgraph (VΓ, Eb) of Γ, where
Eb = {r1r2 ∈ EΓ : r1 ∩ r2 is a non-empty subset of braces of G} ,
is called the bracing (sub)graph. See Figure 9 for an example of these definitions.
Ribbon-Cutting Graph
Braced
Ribbon-Cutting Graph Ribbon Graph Bracing Graph
Figure 9: Two ribbon-cutting graphs with an example of a bracing as well as the corre-
sponding ribbon graph and bracing graph. The vertices in the ribbon graph and the bracing
graph are colored in correspondence with the indicated ribbons.
We remark that the bracing subgraph according to the definition in [8] does not contain
the ribbons which have no brace. In our definition these ribbons are isolated vertices.
There are no loops in ribbon and bracing graphs if all ribbons of the underlying un-
braced ribbon-cutting graph are simple. An edge in a bracing graph does not determine
uniquely a braced 4-cycle (see the yellow and green ribbon in Figure 8).
Now we have all definitions to recall the main theorem of [8]. In the next section we
extend this theorem to P-frameworks and also prove the other direction.
Theorem 3.14 ([8]). Let (G, ρ) be a braced carpet framework. If the bracing graph of G
is connected, then (G, ρ) is rigid.
4 Flexibility of braced P-frameworks
In this section we determine when a bracing makes the framework rigid and in which cases
it remains flexible. We use cartesian NAC-colorings for that. The theory is therefore based
on [11]. Indeed, we show that a P-framework is flexible if and only if it has a cartesian
NAC-coloring. This finally leads to a proof of the main theorem.
Cartesian NAC-colorings of a subclass of ribbon-cutting graphs can be characterized
using ribbons.
G. Grasegger and J. Legerský: Bracing frameworks consisting of parallelograms 11
Lemma 4.1. Let G be a braced ribbon-cutting graph such that every two vertices are
separated by a ribbon. A NAC-coloring of G is cartesian if and only if each ribbon of G is
monochromatic.
Proof. Let δ be a NAC-coloring. If δ is cartesian, then all 4-cycles are either monochro-
matic or opposite edges have the same color. Since the edges of a braced 4-cycle have the
same color, ribbons are monochromatic. On the other hand, if the ribbons are monochro-
matic, then two vertices cannot be connected by a blue and red path simultaneously since
they are separated by a ribbon.
Theorem 4.2. If a braced P-framework (G, ρ) is flexible, then G has a cartesian NAC-
coloring.
Proof. A NAC-coloring for G can be constructed as in the proof of [11, Theorem 3.1]. The
zero set of the following system of equations for coordinates (xu, yu) for u ∈ VG describes
all placements of G inducing the same edge lengths as ρ:
(xu − xv)
2 + (yu − yv)
2 = ∥ρ(u)− ρ(v)∥2 for all uv ∈ EG . (4.1)
In order to remove rigid motions, we fix the position of an edge ūv̄ by setting
xū = 0 , yū = 0 , xv̄ = ∥ρ(ū)− ρ(v̄)∥ , yv̄ = 0 . (4.2)
We also impose that each 4-cycle (u1, u2, u3, u4) in G is a parallelogram:
xu2 − xu1 = xu3 − xu4 , xu4 − xu1 = xu3 − xu2 ,
yu2 − yu1 = yu3 − yu4 , yu4 − yu1 = yu3 − yu2 .
(4.3)
The existence of a flex of (G, ρ) implies that there are infinitely many placements in the zero
set of the system consisting of Equations 4.1, 4.2 and 4.3. Hence, there is an irreducible
algebraic curve C in the zero set. For every u, v ∈ VG such that uv ∈ EG, we define Wu,v
in the complex function field of C by
Wu,v = (xv − xu) + i(yv − yu) .
There exists a valuation ν of the function field of C yielding a NAC-coloring δ of G by
taking δ(uv) = red if ν(Wu,v) > 0 and δ(uv) = blue otherwise, see [11, Theorem 3.1]
for the details. Since Wu1,u2 = Wu4,u3 for the opposite edges u1u2 and u3u4 of a 4-cycle
(u1, u2, u3, u4), we have that δ(u1u2) = δ(u3u4). Therefore, ribbons are monochromatic
since a 4-cycle with a diagonal is monochromatic. The NAC-coloring δ is cartesian by
Lemma 4.1 since every two vertices are separated by a ribbon by Theorem 3.9 applied to
the underlying unbraced P-framework and Remark 3.12.
Lemma 4.3. Let (G, ρ) be a P-framework. Let u, v ∈ VG and W,W
′ be walks from u to v
in G. If G has a cartesian NAC-coloring δ and c ∈ {red, blue}, then
∑
(w1,w2)∈W
δ(w1w2)=c
(ρ(w2)− ρ(w1)) =
∑
(w1,w2)∈W
′
δ(w1w2)=c
(ρ(w2)− ρ(w1)) .
12 Art Discrete Appl. Math. 5 (2022) #P2.03
Proof. Let Ŵ be the walk obtained by concatenating W and the inverse of W ′. We con-
sider the sum ∑
(w1,w2)∈Ŵ
δ(w1w2)=c
(ρ(w2)− ρ(w1)) .
Since each ribbon r is monochromatic in a cartesian NAC-coloring and Ŵ is closed, the
number of edges in r ∩ Ŵ included in the sum is even by Lemma 3.2 (the ribbons are
simple by Remark 3.5). Hence, the sum is zero by Lemma 3.7.
Using the lemma we show the reverse direction of Theorem 4.2. The proof is con-
structive, i.e., it provides a flex. In order to do so, we adapt the “zigzag” grid construction
from [11]. A “zigzag” grid is determined by a column of points, whose copies are trans-
lated to other positions. Such a grid can flex so that the distances among all vertices in the
same column/row remain constant. The idea of the construction of a flexible framework for
a given graph with a NAC-coloring is to place the vertices of the graph to the “zigzag” grid
using the NAC-coloring so that every blue, resp. red, component is in one column, resp.
row of the grid, see Figure 10. Since there are no diagonals, the framework is flexible.
Figure 10: A flex of a “zigzag” grid and an example of a braced ribbon-cutting graph with
a NAC-coloring placed to the grid so that there are no diagonals.
In case of P-frameworks (G, ρ), the grid can be chosen so that the flex starts at ρ if the
graph allows a cartesian NAC-coloring, see Figure 11. We formalize these observations in
the following theorem.
Figure 11: Any parallelogram placement of a braced ribbon cutting graph with a cartesian
NAC-coloring can be obtained via some “zigzag” grid.
Theorem 4.4. If a braced P-framework has a cartesian NAC-coloring, then it is flexible.
Proof. Let (G′, ρ) be a braced P-framework and δ′ be a cartesian NAC-coloring of G′. We
can assume that ρ(ū) = (0, 0) for a fixed vertex ū ∈ VG′ . Let G be the graph G′ with
braces removed and δ be the NAC-coloring of G obtained by restricting δ′.
Let R1, . . . , Rm, resp. B1, . . . , Bn, be the vertex sets of the connected components
of Gδred, resp. G
δ
blue. We define a map ρred : {R1, . . . , Rm} → R
2 as follows: for Ri, let W
G. Grasegger and J. Legerský: Bracing frameworks consisting of parallelograms 13
be any walk in G from ū to a vertex of G in Ri and
ρred(Ri) =
∑
(w1,w2)∈W
δ(w1w2)=blue
(ρ(w2)− ρ(w1)) .
Lemma 4.3 guarantees that it is well-defined, namely, the sum is independent of the choice
of W and the vertex in Ri. We define ρblue : {B1, . . . , Bn} → R2 analogously by swapping
red and blue.
For t ∈ [0, 2π] and v ∈ VG = VG′ , where v ∈ Ri ∩Bj , let
ρt(v) =
(
cos(t) sin(t)
− sin(t) cos(t)
)
· ρred(Ri) + ρblue(Bj) .
If W is a walk in G from ū to v, then
ρ0(v) = ρred(Ri) + ρblue(Bj)
=
∑
(w1,w2)∈W
δ(w1w2)=blue
(ρ(w2)− ρ(w1)) +
∑
(w1,w2)∈W
δ(w1w2)=red
(ρ(w2)− ρ(w1))
=
∑
(w1,w2)∈W
(ρ(w2)− ρ(w1)) = ρ(v)− ρ(ū) = ρ(v) .
We follow the argument from [11] that the lengths of the edges in EG′ are constant along ρt.
Notice that the vertex sets of Gδred, resp. G
δ
blue, and G
′δ′
red, resp. G
′δ′
blue, are the same since each
brace has the same color as the 4-cycle it braces. Let uv be an edge in EG′ with u ∈ Ri∩Bj
and v ∈ Rk ∩Bℓ. If uv is red, then i = k and hence
∥ρt(v)− ρt(u)∥ = ∥ρblue(Bℓ)− ρblue(Bj)∥ .
On the other hand, if uv is blue, then j = ℓ and
∥ρt(v)− ρt(u)∥ =
∥∥∥∥
(
cos(t) sin(t)
− sin(t) cos(t)
)
· (ρred(Rk)− ρred(Ri))
∥∥∥∥
= ∥ρred(Rk)− ρred(Ri)∥ .
Since ρ0 = ρ and ρ is injective, none of the edges has length zero. Therefore, ρt is a flex
of (G′, ρ).
Finally, we connect the results of flexibility and NAC-colorings with the connectivity
of the bracing graph.
Theorem 4.5. Let G be a braced ribbon-cutting graph such that every two vertices are
separated by a ribbon. The bracing graph of G is connected if and only if G does not have
a cartesian NAC-coloring.
Proof. Let B be the bracing graph of G. In a cartesian NAC-coloring of G, ribbons are
monochromatic by Lemma 4.1. Hence, if two ribbons are adjacent in B, then the union of
their edges is monochromatic. Therefore, if B is connected, all edges of G must have the
same color, namely, no cartesian NAC-coloring exists.
14 Art Discrete Appl. Math. 5 (2022) #P2.03
For the opposite implication, assume B is not connected. We color the edges of the
ribbons of one connected component by red and the rest by blue. To show that this sur-
jective edge coloring is a NAC-coloring, consider a cycle C. Let uv be an edge of C and
r be the ribbon containing uv. Since r separates G, r contains another edge u′v′ of C.
Since ribbons are monochromatic, either all edges of C have the same color or there are
two edges of each color. The obtained NAC-coloring is cartesian by Lemma 4.1.
This forms the last part of the proof of Theorem 1.1.
Proof of Theorem 1.1. Let (G, ρ) be a braced P-framework. Every two vertices are sepa-
rated by a ribbon by Theorem 3.9 and Remark 3.12. Hence, (G, ρ) is rigid if and only if G
has no cartesian NAC-coloring (Theorems 4.2 and 4.4) if and only if the bracing graph of
G is connected (Theorem 4.5).
Each edge of the bracing graph corresponds to at least one brace. The minimum number
of braces making the framework rigid follows from the fact that the number of edges of a
spanning tree of the bracing graph is one less than the number of vertices, i.e., ribbons. The
result is also illustrated in Figure 12.
r3
r4 r5 r6
r7
r8
r2
r1
r1
r2
r3
r4
r5
r6
r7
r8
r3
r4 r5 r6
r7
r8
r2
r1
r1
r2
r3
r4
r5
r6
r7
r8
Figure 12: Two bracings of a P-framework where the first one is rigid as visible by the
connectivity of the bracing graph. The second bracing yields a flexible framework since the
bracing graph is not connected. We show three instances of the flex that is possible with the
bracing and the unique resulting cartesian NAC-coloring thereof (shaded parallelograms
preserves their shapes).
G. Grasegger and J. Legerský: Bracing frameworks consisting of parallelograms 15
A consequence of Theorem 1.1 is that rigidity of a braced P-framework is a combina-
torial property, not a geometric one.
Corollary 4.6. If G is a braced ribbon-cutting graph admitting a parallelogram placement,
then either
(i) (G, ρ) is rigid for all parallelogram placements ρ of G, or
(ii) (G, ρ) is flexible for all parallelogram placements ρ of G.
Conclusion
We have applied the theory of NAC-colorings to P-frameworks generalizing previous re-
sults in the area of bracing grids. In fact, we have shown that a P-framework is rigid if
and only if it has no cartesian NAC-coloring if and only if the bracing graph is connected.
Notice that a consequence of this statement is that a braced rectangular grid/rhombic carpet
is rigid if and only if it is infinitesimally rigid. This is not the case for grids with holes as
there are instances which are rigid but not infinitesimally rigid (an example can be obtained
by bracing all squares besides those with the indicated ribbons in Figure 7).
Similarly as in rectangular grids, there are plenty of interesting questions for further
generalizations such as graphs with holes, different types of diagonals or higher dimen-
sions. For P-frameworks these questions are subject to further research.
ORCID iDs
Georg Grasegger https://orcid.org/0000-0001-7421-8115
Jan Legerský https://orcid.org/0000-0002-8122-668X
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G. Grasegger and J. Legerský: Bracing frameworks consisting of parallelograms 17
Appendix
A Cartesian products of graphs and NAC-colorings
In this section, we show a connection of cartesian NAC-colorings with cartesian products
of graphs. Recall that the cartesian product of graphs G and H is given by
G□H = (VG × VH , {(u, u
′)(v, v′) : (u = v ∧ u′v′ ∈ EH) ∨ (u
′ = v′ ∧ uv ∈ EG)}) .
By coloring edges coming from G by red and the rest blue, the following holds.
Theorem A.1 ([1, 15]). The cartesian product of any two nontrivial graphs G and H has
a cartesian NAC-coloring.
We remark that the statement of Theorem A.1 has been pointed out by [15] indepen-
dently of [1]. In [1] a cartesian NAC-coloring is called good since applying the grid con-
struction described in [11] yields a proper flexible framework, whereas for a non-cartesian
NAC-coloring there are overlapping vertices. Our naming is motivated by the fact that the
converse statement can be proved using ideas from [16] about embeddings of graphs into
cartesian products.
Theorem A.2. If a graph G has a cartesian NAC-coloring, then there are graphs Q1, Q2
with at least two vertices each and an injective graph morphism h : G → Q1 □ Q2 such
that each vertex in VQ1 ∪ VQ2 occurs as a coordinate of a vertex in h(G). In particular, G
can be viewed as a subgraph of Q1 □Q2.
Proof. Let δ be a cartesian NAC-coloring of G. Let R1, . . . , Rm, resp. B1, . . . , Bn, be the
vertex sets of the connected components of Gδred, resp. G
δ
blue. Since δ is surjective and no
blue edge can connect vertices of the same red component [11, Lemma 2.4], m ≥ 2 and
n ≥ 2. Let πred : VG → {R1, . . . , Rm} and πblue : VG → {B1, . . . , Bm} map a vertex to
the vertex set of its red, resp. blue, component, namely, πred(v) = Ri and πblue(v) = Bj if
v ∈ Ri ∩Bj . We define the following quotient graphs
Q1 = ({R1, . . . , Rm}, {πred(u)πred(v) : uv ∈ EG and δ(uv) = blue}) ,
Q2 = ({B1, . . . , Bn}, {πblue(u)πblue(v) : uv ∈ EG and δ(uv) = red}) .
Let Q be the cartesian product of Q1 and Q2, and h : VG → VQ be the graph morphism
given by
h(v) = (πred(v), πblue(v)) .
We check that it is indeed a morphism: if uv is an edge of G, w.l.o.g. red, then πred(u) =
πred(v) and πblue(u) ̸= πblue(v) from the properties of NAC-colorings. Thus, h(u)h(v) =
(πred(u), πblue(u))(πred(u), πblue(v)) which is an edge of Q. The morphism h is injective
by Remark 2.7 since δ is cartesian. Each vertex in VQ1 ∪ VQ2 occurs as a coordinate of a
vertex in h(G), since πred and πblue are surjective.
B Carpet frameworks are P-frameworks
As indicated in Section 3.1, we prove here that a carpet framework is P-framework. For
this, we define the following class of graphs.
18 Art Discrete Appl. Math. 5 (2022) #P2.03
Definition B.1. We define the class of graphs Grec recursively. The 4-cycle graph is in Grec.
There are two types of construction (see also Figure 13):
ADD4-CYCLE: If G ∈ Grec with uv ∈ EG, then the graph (VG ∪ {w1, w2}, EG ∪
{uw1, w1w2, w2v}) is in Grec, where w1, w2 /∈ VG.
CLOSE4-CYCLE: If G ∈ Grec with uv, vw ∈ EG and the vertex v is separated from any
vertex in VG \{u, v, w} by a ribbon which does not contain uv or vw, then the graph
(VG ∪ {w
′}, EG ∪ {uw
′, w′w}), where w′ /∈ VG, is in Grec.
Note that the separation assumption is needed for avoiding situations as described in
Figure 6.
u v u v
w1 w2
ADD4-CYCLE
u
v
w
u
v
w
w′
CLOSE4-CYCLE
Figure 13: Two recursive Grec constructions.
Figure 5b gives an example of a graph in Grec that is not the underlying graph of a carpet
framework. It is easy to use the construction to show that the class has the ribbon-cutting
property.
Proposition B.2. Every graph in Grec is ribbon-cutting.
Proof. By structural induction: the 4-cycle graph is ribbon-cutting. ADD4-CYCLE pre-
serves the property since {uw1, w2v} is a new ribbon and w1w2 belongs to the ribbon
of uv. CLOSE4-CYCLE does so as well: the edges uw′ and w′w belong to the ribbons of
vw and uv respectively. If any ribbon of the extended graph were not an edge cut, than it
would not be an edge cut in the original graph. Notice that the separation assumption is not
needed for this.
Recall that for a P-framework (G, ρ), any ribbon r is simple by Remark 3.5 and G \ r
has two connected components by Lemma 3.2. This allows us to translate the vertices of
one of the components by a constant vector.
Remark B.3. Let (G, ρ) be a P-framework and r be a ribbon of G. Let V1 and V2 be the
vertex sets of the two connected components of G \ r. For every vector t ∈ R2 \ {ρ(u1)−
ρ(u2) : u1 ∈ V1, u2 ∈ V2}, the placement ρ′ of G given by ρ′(v) = ρ(v) + t if v ∈ V2 and
ρ′(v) = ρ(v) otherwise is a parallelogram placement.
G. Grasegger and J. Legerský: Bracing frameworks consisting of parallelograms 19
We are going to show the relation between P-frameworks, carpet frameworks and the
graphs in Grec. Namely, the underlying graphs of carpet frameworks are in Grec, which is
in turn a subset of the underlying graphs of P-frameworks. For this we need an equivalent
condition to the separation assumption in CLOSE4-CYCLE.
Lemma B.4. For a P-framework (G, ρ) and uv, vw ∈ EG, the following are equivalent:
(i) The vertex v is separated from any vertex in VG \ {u, v, w} by a ribbon which does
not contain uv or vw.
(ii) There exists a parallelogram placement ρ′ of the graph G′ = (VG ∪ {w′}, EG ∪
{uw′, w′w}), where w′ /∈ VG.
Proof. (i) =⇒ (ii) If we want to extend ρ to a parallelogram placement of G′, the position
ρ(w′) of the new vertex w′ is uniquely determined by the requirement that
(ρ(u), ρ(v), ρ(w), ρ(w′)) is a parallelogram. We can assume that ρ(u), ρ(v), ρ(w) are not
collinear, hence, ρ(v) ̸= ρ(w′). If it is not so, we replace ρ by a parallelogram placement
obtained by Remark B.3 for the ribbon of uv and a non-zero translation.
If ρ : VG′ → R2 is injective, we are done. Otherwise, ρ(w′) = ρ(u′) for a unique
vertex u′ ∈ VG \ {u, v, w}. By assumption, there is a ribbon r separating v from u such
that uv, vw /∈ r. Thus, u, v, w are in the same connected component of G \ r, whereas u′
is in the other one. Using Remark B.3, there is a parallelogram placement ρ′ of G such that
ρ(w′) ̸= ρ′(u′). Moreover, the translation vector can be chosen so that the whole image
ρ′(VG) avoids ρ(w′). Therefore, ρ′ uniquely extends to a parallelogram placement of G′
by setting ρ′(w′) = ρ(w′).
¬(i) =⇒ ¬(ii) Assume that u′ ∈ VG \ {u, v, w} is a vertex such that it is separated
from v only by the ribbon of uv or vw. Let W = (v = u0, u1, . . . , um = u′) be a walk
from v to u′. Let R be the set of ribbons which contains at least one edge of W . All ribbons
are simple by Remark 3.5. By the assumption and Lemma 3.2, |r ∩ W | is even for every
ribbon r avoiding uv and vw. For any parallelogram placement ρ of G, we have
ρ(u′)− ρ(v) =
m∑
i=1
(ρ(ui)− ρ(ui−1)) =
∑
r∈R
∑
(w1,w2)∈r∩W
(ρ(w2)− ρ(w1))
3.7
=
∑
r∈R
uv∈r∨vw∈r
∑
(w1,w2)∈r∩W
(ρ(w2)− ρ(w1))
3.6
= α(ρ(w)− ρ(v)) + β(ρ(u)− ρ(v)) ,
where α, β ∈ {0, 1}. Actually, α = β = 1, otherwise ρ(u′) = ρ(w) or ρ(u′) = ρ(u),
which violates injectivity. Hence, ρ(u′) = ρ(w) + ρ(u)− ρ(v). Assume for contradiction
that there is a parallelogram placement ρ′ of G′. Since ρ′|VG is a parallelogram placement
of G, we have by the previous ρ′(u′) = ρ′(w) + ρ′(u)− ρ′(v). But this is a contradiction
since ρ′(w′) = ρ′(w) + ρ′(u)− ρ′(v) as well and w′ ̸= u′.
Corollary B.5. There exists a P-framework (G, ρ) for every G ∈ Grec.
Proof. We proceed by structural induction. The 4-cycle can be placed as a parallelogram.
For a graph G′ constructed using ADD4-CYCLE from G, a parallelogram placement of G
can be extended to a parallelogram placement of G′ by placing the two new vertices to
20 Art Discrete Appl. Math. 5 (2022) #P2.03
Figure 14: P-frameworks whose underlying graphs are not in Grec.
form a parallelogram so that the placement is injective. If G′ is constructed from G by
CLOSE4-CYCLE, then there exists a parallelogram placement of G′ by Lemma B.4.
Note that there are P-frameworks whose underlying graphs are not in Grec, see Fig-
ure 14.
Corollary B.6. If (G, ρ) is a carpet framework, then G ∈ Grec. In particular, (G, ρ) is a
P-framework.
Proof. By the definition of carpet framework, ρ is a parallelogram placement. Once we
show that G ∈ Grec, the fact that (G, ρ) is a P-framework follows from Proposition B.2.
We proceed by induction on the number of parallelograms yielding a carpet framework.
Let S be the set of parallelograms in R2 giving a carpet framework (G′, ρ′) according to
Definition 3.10. If |S| = 1, then (G′, ρ′) is the 4-cycle with a parallelogram placement,
hence, G′ ∈ Grec. Suppose that |S| ≥ 2. The boundary of
⋃
S is a simple polygon M with
k edges. We divide the parallelograms having an edge in the polygon M into the following
categories (see Figure 15):
• K1 — parallelograms with one edge in M ,
• K2 — parallelograms with two incident edges in M such that the vertex that is not
in these two edges is not in M ,
• K ′2 — parallelograms with two incident edges in M that are not in K2,
• K ′′2 — parallelograms with two opposite edges in M ,
• K3 — parallelograms with three edges in M .
Clearly, k = |K1| + 2|K2| + 2|K ′2| + 2|K
′′
2 | + 3|K3|. The sum of the interior an-
gles of the simple polygon M equals (k − 2)π. Considering contributions to the sum for
parallelograms in the categories above (see Figure 15), we have
|K1|π + |K2|π + 2|K
′
2|π + 2|K
′′
2 |π + 2|K3|π ≤ (k − 2)π
⇐⇒ 2 ≤ |K2|+ |K3| .
For a parallelogram s in K2 ∪K3, S \ s satisfies the assumptions of Definition 3.10. Thus,
we have a carpet framework (G, ρ) and G is in Grec by induction assumption. If s ∈ K3,
then G can be extended to G′ by ADD4-CYCLE. If s ∈ K2, then G can be extended to
G′ by CLOSE4-CYCLE, since the separation assumption is satisfied by Lemma B.4 and the
placement ρ′.
G. Grasegger and J. Legerský: Bracing frameworks consisting of parallelograms 21
K0
K2
K2
K2
K2
K
′
2
K
′′
2
K1
K1
K1
K1
K1
K3
Figure 15: An example of dividing parallelograms into categories according to their inter-
section with the boundary. The angles whose contribution to the sum of the interior angles
is considered are indicated. Note that the parallelogram labeled K0 belongs to S but is not
part of the boundary.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P2.04
https://doi.org/10.26493/2590-9770.1401.a6a
(Also available at http://adam-journal.eu)
Constructing integer-magic graphs via the
combinatorial nullstellensatz*
Richard M. Low†
San Jose State University, San Jose, CA 95192, USA
Dan Roberts‡
Illinois Wesleyan University, Bloomington, IL 61701, USA
Received 31 December 2020, accepted 7 July 2021, published online 11 May 2022
Abstract
Let A be a nontrivial additive abelian group and A∗ = A \ {0}. A graph is A-magic
if there exists an edge labeling f using elements of A∗ which induces a constant vertex
labeling of the graph. Such a labeling f is called an A-magic labeling and the constant
value of the induced vertex labeling is called an A-magic value. In this paper, we use the
Combinatorial Nullstellensatz to construct nontrivial classes of Zp-magic graphs, prime
p ≥ 3. For these graphs, some lower bounds on the number of distinct Zp-magic labelings
are also established.
Keywords: integer-magic graph, integer-magic labeling, Combinatorial Nullstellensatz
Math. Subj. Class.: 05C78
1 Introduction
Let G = (V,E) be a graph, where G might be disconnected and/or a multigraph. For any
nontrivial additive abelian group A, let A∗ = A\{0}. A mapping f : E(G) → A∗ is called
an edge labeling of G. Any such edge labeling induces a vertex labeling f+ : V (G) → A,
where the label at a vertex is the sum of the edge labels incident to that vertex. Here, a
loop label is counted only once. An edge labeling f whose induced mapping f+ on V (G)
*Part of this work was conducted while the second author was visiting the first author during a sabbatical.
The authors are grateful to the anonymous referee, whose valuable comments and suggestions improved the final
manuscript.
†Corresponding author. Although Stephen G. Hartke was unable to join us on this project, the first author
wishes to thank him for his contribution in formulating the original Hartke polynomial.
‡The second author was supported by a grant from Illinois Wesleyan University.
E-mail addresses: richard.low@sjsu.edu (Richard M. Low), drobert1@iwu.edu (Dan Roberts)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P2.04
is a constant is called an A-magic labeling of G. In this case, the constant is called the
A-magic value of f and G is called an A-magic graph. If G has a Zk-magic labeling (for
some k ≥ 2), then G is an integer-magic graph. The integer-magic spectrum of a graph G
is the set IM(G) = {k ≥ 2 : G is Zk-magic}. Generally speaking, it is quite difficult to
determine the integer-magic spectrum of a graph. Note that the integer-magic spectrum of
a graph is not to be confused with the set of achievable magic values.
The concept of an A-magic graph was first introduced in [12]. Since then, A-magic
graph labelings have been studied in [15, 20, 22, 37, 39, 41] and Zk-magic graphs were
investigated in [11, 13, 14, 16, 17, 18, 19, 21, 24, 25, 30, 31, 32, 33, 42, 38, 40]. Z-
magic graphs were considered by Stanley [43, 44], where he pointed out that the theory of
magic labelings could be studied in the general context of linear homogeneous Diophantine
equations. They were also considered in [2, 34].
Labelings form a large and important area of study in graph theory. First formally in-
troduced by Rosa [29] in the 1960s, graph labelings have captivated the interest of many
mathematicians in the ensuing decades. In addition to the intrinsic beauty of the sub-
ject matter, graph labelings have applications (discussed in papers by Bloom and Golomb
[4, 3]) in graph factorization problems, X-ray crystallography, radar pulse code design, and
addressing systems in communication networks. The interested reader is directed to Gal-
lian’s [6] dynamic survey, which contains 2900+ references to research papers and books
on the topic of graph labelings.
2 Preliminaries
All graph-theoretic terms (which are not explicitly defined) are standard ones and can be
found in [7]. Throughout this paper, we consider general graphs which might be discon-
nected and/or a multigraph. We first note a few important facts which are known about
Zk-magic labelings. Lemmas 2.2 and 2.4 are found in [20], whereas Lemma 2.1 is a slight
generalization of a lemma found in [20].
Lemma 2.1. For a graph G, let i(v) denote the number of edges, multiedges and loops
incident to v ∈ V (G). Then, G is Z2-magic ⇐⇒ i(v) are of the same parity, for all
v ∈ V (G).
Lemma 2.2. If G is Zk-magic and k|n, then G is Zn-magic.
Remark 2.3. The converse of Lemma 2.2 is not true, in general. For example, it was
shown in [13] that IM(K4 −{uv}) = {4, 6, 8, . . . }. In particular, K4 −{uv} is Z6-magic.
However, K4 − {uv} is not Z3-magic.
Lemma 2.4. Let p be prime. If G is Zp-magic for some magic value t ̸= 0, then G is
Zp-magic with magic value t
′ for any nonzero t′ ∈ Zp.
Proof. Let b = t′t−1. Multiply all of the edge labels by b. Since Zp is a field, this gives
edge labels which are non-zero. Hence, we have the desired Zp-magic labeling.
Lemmas 2.1 and 2.2 allow us to focus on primes p ≥ 3. Because of Lemma 2.4, it
suffices to look at Zp-magic labelings with magic values equal to 0 and 1.
R. M. Low and D. Roberts: Constructing integer-magic graphs 3
3 The Combinatorial Nullstellensatz
In [1], Alon proved the following result and successfully applied it to problems in additive
number theory and graph theory.
Theorem 3.1 (Combinatorial Nullstellensatz). Let f = f(x1, . . . , xm) be a polynomial of
degree d over a field F. Suppose that the coefficient of the monomial xt11 · · ·x
tm
m in f is
nonzero and t1 + · · · + tm = d. If S1, . . . , Sm are subsets of F with |Si| ≥ ti + 1, then
there exists an x′ = (x′1, x
′
2, . . . , x
′
m) ∈ S1 × · · · × Sm for which f(x
′) ̸= 0.
Example 3.2. Let f(x1, x2, x3, x4) = x
4
1x2x3− 2x
5
1+x
2
1x
2
2x
2
3+x
2
4 ∈ Z3[x1, x2, x3, x4].
We will apply Theorem 3.1 on the term x21x
2
2x
2
3 in f . Note that deg(f) = 6 = deg(x
2
1x
2
2x
2
3).
We choose S1 = {0, 1, 2}, S2 = {0, 1, 2}, S3 = {0, 1, 2} and S4 = {2}. Then, Theo-
rem 3.1 implies that there exist si ∈ Si, where 1 ≤ i ≤ 4, such that f(s1, s2, s3, s4) ̸= 0.
Note that the Combinatorial Nullstellensatz cannot be applied to any of the other monomial
terms in f .
After its discovery, the Combinatorial Nullstellensatz would soon become a powerful
tool in extremal combinatorics [10]. With regards to graph labeling and coloring problems,
it has been used to prove theorems on anti-magic labelings, neighbor sum distinguishing
total colorings, and list colorings [27, 36, 8]. For a recent research monograph on the
Combinatorial Nullstellensatz and graph coloring problems, the reader is directed to [45].
4 The Hartke polynomials
Let G = (V,E), where |V (G)| = n, |E(G)| = m, and the edges, multiedges and loops
of G are identified with variables x1, x2, . . . , xm. As mentioned previously, we will focus
on Zp-magic labelings (prime p ≥ 3) and magic values equal to 0 and 1. For fixed prime
p ≥ 3 and t ∈ {0, 1}, define the polynomials ft in Zp[x1, . . . , xm] as
ft(x) = ft(x1, . . . , xm) =
∏
v∈V (G)

1−

t−
∑
v∈xj
xj


p−1

 , (4.1)
where the addition and multiplication are taken modulo p. The given factorization of (4.1)
and its factors are called the canonical factorization and canonical factors of ft, respec-
tively. The ft are called Hartke polynomials and were introduced in [23]. Note that each
Hartke polynomial describes a unique graph G up to isomorphism.
In this section, we recall the basic properties of ft (see [23]) and give additional analysis
of these polynomials. This is used in conjunction with Theorem 3.1 in subsequent sections
of this paper, where we construct Zp-magic graphs.
Remark 4.1. Note that deg(ft(x)) = (p− 1) · |V (G)|. This follows from:
1. There are |V (G)| canonical factors of ft(x).
2. Each of the canonical factors is of degree p− 1.
3. Theorem in [9]: Let R be a commutative ring with unity and g, h ∈ R[x1, x2, . . . , xm].
If R has no zero divisors, then deg(gh) = deg(g) + deg(h).
4 Art Discrete Appl. Math. 5 (2022) #P2.04
Observations 4.2. Let x′ be an m-tuple in Z∗p×Z
∗
p×· · ·×Z
∗
p. Then, we note the following:
1. ft(x) is defined for all connected multigraphs G.
2. The range of ft(x) is {0, 1}. This follows from the fact that each canonical factor of
ft takes on a value of 0 or 1, due to Fermat’s Little Theorem [9]: If p is prime, then
ap = a for all a ∈ Zp.
3. f0(x
′) = 1 ⇒ x′ is a Zp-magic labeling of G with magic value 0.
4. f1(x
′) = 1 ⇒ x′ is a Zp-magic labeling of G with magic value 1.
5. f0(x
′) = 0 and f1(x
′) = 0 ⇒ x′ is not a Zp-magic labeling of G with magic value
0 or 1.
6. f0(x
′) = 1 ⇒ f1(x
′) = 0. If f0(x
′) = 1, then x′ is a Zp-magic labeling of G with
magic value 0. Thus, x′ is not a Zp-magic labeling of G with magic value 1.
7. f1(x
′) = 1 ⇒ f0(x
′) = 0. This is the contrapositive of Observation 6.
Two techniques are often used to establish results in graph labeling problems. Either
a construction of a desired labeling is obtained through ingenuity, or one shows the non-
existence of the labeling (via proof by contradiction). In practice, these methods can be
time-consuming and difficult to use.
In [23], the Combinatorial Nullstellensatz and Hartke polynomials were used to prove
that certain graphs were Zp-magic (prime p ≥ 3), without having to construct an actual
Zp-magic labeling. As far as the authors know, it was the first time that a nonconstructive
method was used to analyze integer-magic graph labelings. The focus of this paper is to
use the Combinatorial Nullstellensatz to construct Zp-magic graphs, for prime p ≥ 3. In
particular, we construct Hartke Zp-magic graphs.
Definition 4.3. Let p ≥ 3 be prime. A graph G is called Hartke Zp-magic if Theorem 3.1
can be used on a Hartke polynomial ft of G to prove that G is Zp-magic. In this case, a
nonvanishing monomial term M of degree (p − 1) · |V (G)| in the expansion of ft (where
Theorem 3.1 is applied in such a manner) is called a Hartke term.
Example 4.4. Let p = 3 and G7 be the graph illustrated in Figure 1. Note that G7 is the
graph F4 in [28]. Using Mathematica 12, we see that deg(f0(x)) = 16 and that f0(x) con-
tains the monomial term 14336x5x6 · · ·x20 ≡ 2x5x6 · · ·x20 (mod 3). Let Si = {1, 2}
for i = 5, 6, . . . , 20, and Si = {1} for i = 1, 2, 3 and 4. So by Theorem 3.1, we have that
f0(x
′) ̸= 0, for some x′ ∈ S1 × S2 × · · · × S20. Thus, f0(x
′) = 1 and we conclude that
G7 is a Hartke Z3-magic graph with magic value 0.
Proposition 4.5. Let p ≥ 3 be prime and G be a graph with Hartke polynomial ft. Then,
G is Hartke Zp-magic with magic value t⇐⇒ ft has a nonvanishing monomial term M of
degree (p− 1) · |V (G)|, where all the exponents ti satisfy 0 ≤ ti ≤ p− 2.
Proof. (=⇒). Suppose that G is a Hartke Zp-magic graph with Hartke polynomial ft.
Then, there exists a Hartke term M of degree (p − 1) · |V (G)| in ft. Since G is a Hartke
Zp-magic graph, there exist nonempty subsets S1, S2, . . . , Sm of Z
∗
p = {1, 2, . . . , p − 1}
corresponding to the m variables in ft, which satisfy the hypothesis of Theorem 3.1 (when
applied to M ). In particular, all of the exponents ti of M satisfy 0 ≤ ti ≤ p− 2.
R. M. Low and D. Roberts: Constructing integer-magic graphs 5
v
4
v
5
v
6
v
7
v
1
v
2
v
3
v
8
x
1
x
2
x
3
x
4
x
5
x
6
x
7
x
8
x
9
x
10
x
11
x
12
x
13
x
14
x
15
x
16x17
x
18
x
19
x
20
Figure 1: G7 has a Z3-magic labeling with magic value 0.
(⇐=). Suppose that ft has a nonvanishing monomial term M of degree (p−1)·|V (G)|,
where all of the exponents ti satisfy 0 ≤ ti ≤ p− 2. For each exponent ti (associated with
variable xi) appearing in M , choose a nonempty subset Si of Z
∗
p = {1, 2, . . . , p−1} where
|Si| ≥ ti + 1. Thus by Theorem 3.1, G has a Zp-magic labeling. In particular, G is Hartke
Zp-magic with magic value t.
Proposition 4.6. Let p ≥ 3 be prime. If G is a Hartke Zp-magic graph, then
|E(G)| ≥ p−1
p−2 · |V (G)|.
Proof. We prove the contrapositive. If |E(G)| < p−1
p−2 · |V (G)|, then a straightforward
counting argument shows that every nonvanishing monomial of degree (p − 1) · |V (G)|
in ft has an exponent ti ≥ p − 1. Thus by Proposition 4.5, G is not a Hartke Zp-magic
graph.
Remark 4.7. The converse of Proposition 4.6 is not true, in general. For example, let
p = 3 and consider the graph G comprised of P3 with K6 attached at an end-vertex. Then,
G has 8 vertices and 17 edges. Thus, the inequality |E(G)| ≥ p−1
p−2 · |V (G)| is satisfied.
However, G is not Z3-magic since P3 is not Z3-magic; hence, G is not Hartke Z3-magic.
Theorem 4.8. Suppose p ≥ 3 is prime and G is a graph. Let M0 and M1 denote the sets
of nonvanishing monomial terms of degree (p−1)·|V (G)|, of f0(x) and f1(x), respectively.
Then, M0 = M1.
Proof. Let M ∈ M0. For each vertex v ∈ V (G), let bv denote the sum inside the corre-
sponding canonical factor in Equation (4.1). Observe that every term in the expansion of
(0− bv)
p−1 = bp−1v is of degree p− 1. Thus in the expansion of f0(x), M arises from the
product of terms Πv∈V (G)b
p−1
v . More specifically, M is equal to a product consisting of
one term from each bp−1v .
Now, let us examine f1(x) carefully. First, we note that every term in the expansion
of (1 − bv)
p−1 is of the form 1kbp−1−kv , where 0 ≤ k ≤ p − 1. In the expansion of
the f1(x), every nonvanishing monomial term of degree (p− 1) · |V (G)| will arise from a
product of terms, one from each of the bp−1v . Monomial terms of f1(x) which do not arise
6 Art Discrete Appl. Math. 5 (2022) #P2.04
in this manner have degree at most (p−1) · (|V (G)|−1)+(p−2) = (p−1) · |V (G)|−1.
In particular, M ∈ M1.
This argument is reversible. Thus, the claim is established.
Corollary 4.9. Let p ≥ 3 be prime. Then, G is a Hartke Zp-magic graph with magic value
0 ⇐⇒ G is a Hartke Zp-magic graph with magic value 1.
Proof. This follows immediately from Theorems 3.1 and 4.8.
Example 4.10. This example illustrates the proof of Corollary 4.9. Let p = 3. Consider
the graph G7 (from Example 4.4) illustrated in Figure 1. In that example, the monomial
term 14336x5x6 · · ·x20 ≡ 2x5x6 · · ·x20 (mod 3) of degree 8·2 = 16 was found in f0(x).
This was then used to show that G7 is a Hartke Z3-magic graph with magic value 0. By
Theorem 4.8, f1(x) must also contain this particular Hartke term. This is easily verified
by using Mathematica 12. Hence, we conclude that G7 is a Hartke Z3-magic graph with
magic value 1.
5 Constructing Zp-magic graphs
Definition 5.1. Let p ≥ 2 be prime, t ∈ {0, 1} and G have a Zp-magic labeling with magic
value t. Then, G is called an edge-stable Zp-magic graph if the addition of any number of
simple edges, multiedges and/or loops to G results in a Zp-magic graph with magic value t.
Example 5.2. The 1-vertex loop graph is an edge-stable Zp-magic graph. In [13], it was
shown that IM(K4 − {e}) = {4, 6, 8, . . . }. Thus, C4 is Zp-magic but not edge-stable, for
all primes p.
Theorem 5.3. Let p ≥ 3 be prime. Adding simple edges, multiedges and/or loops to
a Hartke Zp-magic graph results in a new Hartke Zp-magic graph. In particular, every
Hartke Zp-magic graph is edge-stable.
Proof. Suppose that G is a Hartke Zp-magic graph with Hartke polynomial ft. Let G
∗
(with Hartke polynomial f∗t ) be obtained by adding simple edges, multiedges and/or loops
to G. First, note that deg(ft) = deg(f
∗
t ). Since G is Hartke Zp-magic, there exists a Hartke
term M in ft. By Proposition 4.5, all of the exponents ti of M satisfy 0 ≤ ti ≤ p − 2.
Furthermore, M also appears in the expansion of f∗t . By Proposition 4.5, G
∗ is a Hartke
Zp-magic graph with magic value t.
Example 5.4. Let p = 5 and G5 be the first graph illustrated in Figure 2. Then, f1(x) ∈
Z5[x1, x2, . . . , x8], where
f1(x) = [1− (1− (x1 + x3))
4] · [1− (1− (x1 + x2 + x6 + x7))
4] ·
[1− (1− (x2 + x8))
4] · [1− (1− (x3 + x4))
4] ·
[1− (1− (x4 + x5 + x7 + x8))
4] · [1− (1− (x5 + x6))
4].
Using Mathematica 12, we see that deg(f1(x)) = 24 and that f1(x) contains the mono-
mial term 1069056x31x
3
2 · · ·x
3
8 ≡ x
3
1x
3
2 · · ·x
3
8 (mod 5). Let Si = {1, 2, 3, 4}, for i =
1, 2, . . . , 8. By Theorem 3.1, we have that f1(x
′) ̸= 0, for some x′ ∈ S1 × S2 × · · · × S8.
Thus, f1(x
′) = 1 and we conclude that G5 is a Hartke Z5-magic graph with magic value 1.
R. M. Low and D. Roberts: Constructing integer-magic graphs 7
v
1
v
3
v
2
v
5
v
6
v
4
x
1
x
2
x
3
x
7
x
5
x
6
x
4
x
8
v
1
v
3
v
2
v
5
v
6
v
4
3
3
2
33
3
3 3
Figure 2: A Z5-magic labeling of G5 with magic value 1.
With some considerable effort (by hand), one can obtain a Z5-magic labeling of G5 with
magic value 1, as illustrated in the second graph of Figure 2.
Suppose we add a loop or an additional edge to G5. Then, there exist Z5-magic label-
ings with magic value 1, for these new graphs. Figures 3, 4 and 5 illustrate Theorem 5.3.
Remark 5.5. Theorem 5.3 says that any graph which contains a Hartke Zp-magic graph
as a spanning subgraph is also a Hartke Zp-magic graph. Note that the converse of The-
orem 5.3 is not necessarily true. The 1-vertex loop graph in Example 5.2 illustrates an
edge-stable Zp-magic graph which is not Hartke Zp-magic.
We now establish some lower bounds for the number of Zp-magic labelings with magic
value t for a given graph. Symmetry is ignored when counting these labelings. For exam-
ple, if one labeling can be obtained by “rotating” another labeling, then these two labelings
are counted separately.
Theorem 5.6. Let p ≥ 3 be prime and G be a Hartke Zp-magic graph. Suppose that G
∗
is obtained by adding z simple edges, multiedges and/or loops to G. Then, the number of
different (ignoring symmetry) Zp-magic labelings of G
∗ (with magic value t) is greater
than or equal to (p− 1)z .
Proof. Let G,G∗, ft, f
∗
t and M be defined as in the proof of Theorem 5.3. There, we saw
that M is also a term in the expansion of f∗t . The variables in f
∗
t corresponding to the
8 Art Discrete Appl. Math. 5 (2022) #P2.04
v
1
v
3
v
2
v
5
v
6
v
4
3
2
4
42
4
2 4
1
Figure 3: A Z5-magic labeling (magic value 1) of G5 with a loop (labeled 1) at v1.
v
1
v
3
v
2
v
5
v
6
v
4
2
2
2
33
4
4 2
2
Figure 4: A Z5-magic labeling (magic value 1) of G5 with a loop (labeled 2) at v1.
additional z simple edges, multiedges, and/or loops do not appear in M . Hence we can
apply Theorem 3.1 to M in f∗t , where each of the z new variables can take on any of the
p − 1 non-zero elements from Zp. Thus for each t ∈ {0, 1}, there are at least (p − 1)
z
different Zp-magic labelings of G
∗ with magic value t.
Example 5.7. In Example 4.4, we saw that graph G7 (Figure 1) is a Hartke Z3-magic graph
with magic value 0. By Theorem 4.8 and Corollary 4.9, G7 is also a Hartke Z3-magic graph
with magic value 1. Let G∗7 denote the graph obtained by adding loop x21 and multiple
edge x22 to G7, as illustrated in Figure 6. Since the monomial term 14336x5x6 · · ·x20 ≡
2x5x6 · · ·x20 (mod 3) is a Hartke term in the ft of G
∗
7, there exist Z3-magic labelings of
G∗7 (with magic value t), with x21 and x22 having labels 1 or 2. Thus, there are at least
(3 − 1)2 = 4 different Z3-magic labelings of G
∗
7 with magic value t. For this particular
example, even more can be said. The Hartke term 2x5x6 · · ·x20 (mod 3) does not involve
x1, x2, x3 and x4. Each of these particular edges can be labeled with 1 or 2. Hence for
each t ∈ {0, 1}, there are at least (3 − 1)6 = 64 different Z3-magic labelings of G
∗
7 with
magic value t.
Definition 5.8. Let p ≥ 3 be prime, G be a Hartke Zp-magic graph, and M be a Hartke
term of ft. Then, the excess set of M (denoted by EM ) is the set of variables that have
R. M. Low and D. Roberts: Constructing integer-magic graphs 9
v
1
v
3
v
2
v
5
v
6
v
4
2
3
3
23
3
3 3
1
Figure 5: A Z5-magic labeling (magic value 1) of G5 with edge v1v6 (labeled 1).
exponent zero in M .
Theorem 5.9. Let p ≥ 3 be prime, G be a Hartke Zp-magic graph, and M be a Hartke
term of ft. Then for each t ∈ {0, 1}, G has at least (p−1)
|EM | different Zp-magic labelings
with magic value t. Furthermore if G − E is connected, where E is any subset of edges
corresponding to variables in EM , then G − E has a Zp-magic labeling with magic value
t.
Proof. Suppose that p ≥ 3 is prime, G is a Hartke Zp-magic graph, and M is a Hartke
term of ft. The variables in EM do not appear in M . Thus, we can apply Theorem 3.1 to
M in ft, where each variable in EM can take on any of the p − 1 non-zero elements from
Zp. Thus for each t ∈ {0, 1}, there are at least (p − 1)
|EM | different Zp-magic labelings
of G with magic value t. Finally, M will still be a Hartke term of the Hartke polynomials
of G − E, where E is any subset of edges (corresponding to variables in EM ). Hence,
Theorem 3.1 can be applied to the Hartke polynomials of G − E and we conclude that
G− E has a Zp-magic labeling with magic value t.
Example 5.10. Consider the graph G7 in Example 4.4. Then G7 − E, where E is any
subset of edges (corresponding to x1, x2, x3, x4), has a Z3-magic labeling with magic
value t. This is because its associated Hartke polynomial contains the same Hartke term
14336x5x6 · · ·x20 ≡ 2x5x6 · · ·x20 (mod 3).
Corollary 5.11. Let p ≥ 3 be prime and G be a Hartke Zp-magic graph. Suppose that
|E(G)| ≥ (p − 1) · |V (G)|. Then, G has at least (p − 1)[|E(G)|−(p−1)·|V (G)|] different
Zp-magic labelings with magic value t, for each t ∈ {0, 1}.
Proof. Suppose that p ≥ 3 is prime, G is a Hartke Zp-magic graph, and |E(G)| ≥ (p −
1) · |V (G)|. Let M be a Hartke term of ft. Note that when |E(G)| =
p−1
p−2 · |V (G)|,
the corollary follows from Theorem 5.9. When |E(G)| > p−1
p−2 · |V (G)|, M includes at
most (p − 1) · |V (G)| distinct variables. Since |E(G)| ≥ (p − 1) · |V (G)|, we have that
|EM | ≥ |E(G)| − (p− 1) · |V (G)|. By Theorem 5.9, the result follows.
Theorem 5.12. Let p ≥ 3 be prime. Then, the disjoint union of Hartke Zp-magic graphs
is a Hartke Zp-magic graph.
10 Art Discrete Appl. Math. 5 (2022) #P2.04
v
4
v
5
v
6
v
7
v
1
v
2
v
3
v
8
x
1
x
2
x
3
x
4
x
5
x
6
x
7
x
8
x
9
x
10
x
11
x
12
x
13
x
14
x
15
x
16x17
x
18
x
19
x
20
x
21
x
22
Figure 6: For each t ∈ {0, 1}, G∗7 has at least (3 − 1)
6 = 64 different Z3-magic labelings
with magic value t.
Proof. It suffices to show the claim is true for the disjoint union of two Hartke Zp-magic
graphs H1 and H2, with magic value t ∈ {0, 1}. Note that the degrees of the Hartke
polynomials of H1, H2, and the disjoint union of H1 and H2 are (p − 1) · |V (H1)|, (p −
1) · |V (H2)|, and (p− 1) · (|V (H1)|+ |V (H2)|), respectively. Let M1 and M2 be Hartke
terms in the Hartke polynomials of H1 and H2, respectively. Note that M1 ·M2 does not
vanish, since the coefficients come from a field. Thus, the degree of M1 ·M2 is (p − 1) ·
(|V (H1)| + |V (H2)|) and M1 ·M2 appears in the expansion of the Hartke polynomial of
the disjoint union of H1 and H2. We also see that M1 ·M2 is a Hartke term, since M1 and
M2 individually are Hartke terms. Therefore, the disjoint union of Hartke Zp-magic graphs
is a Hartke Zp-magic graph.
Definition 5.13. A weak join of graphs H1, H2, . . . , Hr is defined to be a connected graph
with vertex set
⋃r
i=1 V (Hi) and edge set Z ∪ (
⋃r
i=1 E(Hi)), where Z is a set of simple
and/or multiedges of the form uv with u ∈ V (Hi) and v ∈ V (Hj), where i ̸= j.
Example 5.14. Figure 7 illustrates a weak join of C6, W6 (of order six) and P2.
Theorem 5.15. Let p ≥ 3 be prime. Then, a weak join of Hartke Zp-magic graphs is a
Hartke Zp-magic graph.
Proof. Let H1, H2, . . . , Hr be Hartke Zp-magic graphs. By Theorem 5.12, the disjoint
union
⋃r
i=1 Hi is a Hartke Zp-magic graph. Since a weak join of H1, H2, . . . , Hr is formed
by adding simple edges and/or multiedges between the Hi in
⋃r
i=1 Hi, the claim is estab-
lished by Theorem 5.3.
Example 5.16. Let p = 3 and G3 be the top half of the graph in Figure 8. Then,
R. M. Low and D. Roberts: Constructing integer-magic graphs 11
Figure 7: A weak join of C6, the wheel graph W6 and P2.
f0(x) ∈ Z3[x1, x2, . . . , x15], where
f0(x) = [1− (0− (x1 + x7 + x8 + x14 + x15))
2] · [1− (0− (x1 + x2 + x11 + x12))
2] ·
[1− (0− (x2 + x3 + x8 + x9))
2] · [1− (0− (x3 + x4 + x12 + x13))
2] ·
[1− (0− (x4 + x5 + x9 + x10 + x15))
2] · [1− (0− (x5 + x6 + x13 + x14))
2] ·
[1− (0− (x6 + x7 + x10 + x11))
2].
Using Mathematica 12, we see that deg(f0(x)) = 14 and that f0(x) contains the monomial
term −6400x1x2 · · ·x11x12x14x15 ≡ 2x1x2 · · ·x11x12x14x15 (mod 3). Let Si = {1, 2},
for i = 1, 2, . . . , 12, 14, 15 and S13 = {1}. So by Theorem 3.1, we have that f0(x
′) ̸= 0,
for some x′ ∈ S1 × S2 × · · · × S15. Thus, f0(x
′) = 1 and G3 is a Hartke Z3-magic graph
with magic value 0.
Now, let G4 (graph G1121 from [28]) be the bottom half of the graph in Figure 8. Then,
f0(y) ∈ Z3[y1, y2, . . . , y14], where
f0(y) = [1− (0− (y1 + y5 + y6 + y9 + y10 + y14))
2] · [1− (0− (y1 + y2 + y11))
2] ·
[1− (0− (y2 + y3 + y7 + y8 + y10))
2] · [1− (0− (y3 + y4 + y13 + y14))
2] ·
[1− (0− (y4 + y5))
2] · [1− (0− (y6 + y7 + y11 + y12))
2] ·
[1− (0− (y8 + y9 + y12 + y13))
2].
Using Mathematica 12, we see that deg(f0(y)) = 14 and that f0(y) contains the mono-
mial term −4096y1y2 · · · y13y14 ≡ 2y1y2 · · · y13y14 (mod 3). Let Si = {1, 2}, for i =
1, 2, . . . , 14. So by Theorem 3.1, we have that f0(y
′) ̸= 0, for some y′ ∈ S1×S2×· · ·×S14.
Thus, f0(y
′) = 1 and G4 is a Hartke Z3-magic graph with magic value 0.
The graph G in Figure 8 is a weak join of G3 and G4, where z1, z2 and z3 are the addi-
tional simple and multiedges used to create the weak join. Let f0(r) ∈ Z3[r1, r2, . . . , r32],
where
ri =



xi if 1 ≤ i ≤ 15;
yi−15 if 16 ≤ i ≤ 29;
zi−29 if 30 ≤ i ≤ 32,
be a Hartke polynomial of G. Using Mathematica 12, we see that deg(f0(r)) = 28 and
12 Art Discrete Appl. Math. 5 (2022) #P2.04
that f0(r) contains the monomial term
(−6400r1r2 · · · r11r12r14r15) · (−4096r16r17 · · · r28r29)
≡ (2r1r2 · · · r11r12r14r15) · (2r16r17 · · · r28r29) (mod 3)
≡ r1r2 · · · r11r12r14r15r16r17 · · · r28r29 (mod 3).
Let Si = {1, 2}, for i = 1, 2, . . . , 12, 14, 15, 16, 17 . . . 28, 29 and S13 = S30 = S31 =
S32 = {1}. So by Theorem 3.1, we have that f0(r
′) ̸= 0, for some r′ ∈ S1×S2×· · ·×S32.
Thus, f0(r
′) = 1 and G is a Hartke Z3-magic graph with magic value 0.
v
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6
v
7
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1
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2
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3
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5
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6
x
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1
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2
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3
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5
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6
y
1
y
2 y
3
y
4y
6
y
5
y
7
w
7y
8
y
9y10
y
11
y
12
y
13
y
14
z
1 z2 z
3
Figure 8: A weak join of Hartke Z3-magic graphs G3 and G4 is a Hartke Z3-magic graph.
In [22], the Zk-magic property was analyzed for various classical graph products.
There, it was shown that if G and H are connected Zk-magic graphs, then the Cartesian
and lexicographic products of G and H are Zk-magic, for k ∈ {2, 3, 4, . . . }. However, if
instead we strengthen the restriction on G and weaken the restriction on H , then we obtain
additional results. To this end, recall the following definitions [5].
Definition 5.17. Let G and H be connected graphs. Then, the Cartesian product G□H is
a graph which has vertex set V (G□H) = {(g, h) : g ∈ V (G) and h ∈ V (H)} and edge
R. M. Low and D. Roberts: Constructing integer-magic graphs 13
set E(G□H), where two vertices (g, h) and (g′, h′) are adjacent if (g = g′ and h adj h′)
or (h = h′ and g adj g′).
Definition 5.18. Let G and H be connected graphs. Then, the lexicographic product G◦H
is a graph which has vertex set V (G ◦H) = {(g, h) : g ∈ V (G) and h ∈ V (H)} and edge
set E(G ◦H), where two vertices (g, h) and (g′, h′) are adjacent if (g = g′ and h adj h′)
or (g adj g′).
Definition 5.19. Let G and H be connected graphs. Then, the strong product G ⊠H is a
graph which has vertex set V (G⊠H) = {(g, h) : g ∈ V (G) and h ∈ V (H)} and edge set
E(G ⊠H), where two vertices (g, h) and (g′, h′) are adjacent if (g = g′ and h adj h′) or
(h = h′ and g adj g′) or (h adj h′ and g adj g′).
Of these three graph products, only the lexicographic product is not commutative.
Example 5.20. Figure 9 illustrates P2□P3, P2 ◦ P3, P3 ◦ P2 and P2 ⊠ P3.
Corollary 5.21. Let p ≥ 3 be prime. Suppose that G is a Hartke Zp-magic graph and H
is a graph. Then, the Cartesian product G□H is a Hartke Zp-magic graph.
Proof. Let T be a spanning tree of H . G□T is obtained by replacing each vertex of T
with a copy of G and replacing each edge of T with edges connecting the corresponding
vertices of copies of G. Since G is a Hartke Zp-magic graph, G□T is a weak join of Hartke
Zp-magic graphs. By Theorem 5.15, G□T is a Hartke Zp-magic graph. Finally, for each
of the edges in H which are not in T , add edges connecting the corresponding vertices of
copies of G in G□T to obtain G□H . Since G□T is a Hartke Zp-magic graph, we see that
G□H is a Hartke Zp-magic graph, by Theorem 5.3.
Corollary 5.22. Let p ≥ 3 be prime. Suppose that G is a Hartke Zp-magic graph and H
is a graph. Then, G ◦H , H ◦G and G⊠H are Hartke Zp-magic graphs.
Proof. First, note that V (G□H) = V (G ◦ H) = V (H ◦ G) = V (G ⊠ H). We also
see that the edge sets of G ◦H , H ◦ G, and G ⊠H contain the edge set of G□H . Since
G□H is Hartke Zp-magic by Corollary 5.21, these other products are Hartke Zp-magic, by
Theorem 5.3.
6 Further directions and some open questions
Throughout this paper, we used the Combinatorial Nullstellensatz in the construction of
Hartke Zp-magic graphs, for prime p ≥ 3. Graphs of this type were found to have an
edge-stability property. This was used to further construct non-trivial Hartke Zp-magic
graphs.
It is natural for the reader to wonder if connected simple Hartke Zp-magic graphs exist,
for all orders n ≥ 6 and prime p ≥ 3. The authors of this paper believe that this is true.
Conjecture 6.1. Let p ≥ 3 be prime. Then, there exists a connected simple Hartke Zp-
magic graph G, for all |V (G)| ≥ 6.
14 Art Discrete Appl. Math. 5 (2022) #P2.04
Cartesian product of P2 and P3
Lexicographic product of P2 and P3
Lexicographic product of P3 and P2
Strong product of P2 and P3
Figure 9: P2□P3, P2 ◦ P3, P3 ◦ P2 and P2 ⊠ P3.
The Combinatorial Nullstellensatz can be generalized in different ways. Theorem 3.1
is true over integral domains. The Generalized Combinatorial Nullstellensatz [35] sharpens
Theorem 3.1; instead of analyzing a monomial with degree = deg(f), it suffices to consider
a monomial that does not divide any other monomial term in f . In [26], Michalek remarks
that the Combinatorial Nullstellensatz is true over any commutative ring R with unity, as
long as a − b is not a zero divisor in R, for any a, b ∈ Si (i = 1, 2, . . . ,m). Can any
of these generalizations of the Combinatorial Nullstellensatz help us in analyzing the Zp-
magic graph labeling problem (prime p ≥ 3)?
Here are some other questions one might consider.
1. Are there natural classes of Hartke Zp-magic graphs?
2. Let p ≥ 3 be prime and G be a connected simple graph satisfying |E(G)| ≥ p−1
p−2 ·
|V (G)|. What is the probability that G is a Hartke Zp-magic graph?
R. M. Low and D. Roberts: Constructing integer-magic graphs 15
3. Are there other types of graph labeling problems where the Combinatorial Nullstel-
lensatz or its various generalizations can be used?
ORCID iDs
Richard M. Low https://orcid.org/0000-0003-2320-8296
Dan Roberts https://orcid.org/0000-0001-8456-9533
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P2.05
https://doi.org/10.26493/2590-9770.1382.2ad
(Also available at http://adam-journal.eu)
On generalised Petersen graphs of girth 7 that
have cop number 4*
Harmony Morris
Winston Churchill High School, 1605 15 Ave N, Lethbridge, AB, T1H 1W4, Canada
Joy Morris†
Department of Mathematics and Computer Science, University of Lethbridge
Lethbridge, AB, T1K 3M4, Canada
Received 31 December 2020, accepted 7 July 2021, published online 11 May 2022
Abstract
We show that if n = 7k/i with i ∈ {1, 2, 3} then the cop number of the generalised
Petersen graph GP (n, k) is 4, with some small previously-known exceptions. It was pre-
viously proved by Ball et al. (2015) that the cop number of any generalised Petersen graph
is at most 4. The results in this paper explain all of the known generalised Petersen graphs
that actually have cop number 4 but were not previously explained by Morris et al. in a
recent preprint, and places them in the context of infinite families. (More precisely, the
preprint by Morris et al. explains all known generalised Petersen graphs with cop number
4 and girth 8, while this paper explains those that have girth 7.)
Keywords: Cops and robbers, generalised Petersen graphs, girth, cop number.
Math. Subj. Class.: 05C57, 91A46
1 Introduction
Cops and robbers is a game that can be played on any graph. There are two players:
one playing the cops, with one or more pieces and the other playing the robber, with a
single piece. Both players have perfect information: they can see the graph and the current
locations of all pieces. The two players take turns, with the cop player going first. On
*The authors thank Dave Morris and the anonymous referees for their careful reading of their paper and their
helpful suggestions.
†The second author was supported by the Natural Science and Engineering Research Council of Canada (grant
RGPIN-2017-04905).
E-mail addresses: harmony.morris23@lethsd.ab.ca (Harmony Morris), joy.morris@uleth.ca (Joy Morris)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P2.05
their first turns, each player chooses a vertex on which to place each of their pieces. On all
subsequent turns, they may move any number of their pieces to any neighbouring vertex.
The object of the game for the cops is to capture the robber by landing on the same vertex
as them with any of the cop pieces. The object for the robber is to avoid being captured
forever. The cop number of a graph is the minimum number of cops required in order for
the cop player to have a winning strategy.
This game first appears in [5] and [6], and has been studied extensively on many fami-
lies of graphs (see [3] as an excellent reference).
We are interested in studying this game only on the well-known family of generalised
Petersen graphs. The generalised Petersen graph GP (n, k) is a graph on 2n vertices whose
vertex set is the union of A = {a0, . . . , an−1} and B = {b0, . . . , bn−1}. The edges have
one of three possible forms:
• {ai, ai+1};
• {ai, bi}; and
• {bi, bi+k},
where subscripts are calculated modulo n. This family generalises the Petersen graph,
which is GP (5, 2). We require n ≥ 5, and 0 < k < n/2. This ensures that the graphs are
cubic, and avoids some isomorphic copies of graphs.
The girth of a generalised Petersen graph is at most 8, and is well-understood from
the parameters n and k (see [2, Theorem 5]). Up to isomorphisms, a generalised Petersen
graph has girth 7 if and only if its girth is not 6 or less, and its parameters satisfy one of the
following conditions:
• n = 7k/i where i ∈ {1, 2, 3};
• k = 4;
• n = 2k + 3; or
• n = 3k ± 2.
Generalised Petersen graphs of girth 6 or less are characterised (up to isomorphisms)
as those having k ≤ 3, n = 2k + 2, or n = jk where j ∈ {5/2, 3, 4, 5, 6}. By carefully
checking these with the girth at most 7 conditions for compatibility, we obtain the following
precise characterisation. Up to isomorphisms, a generalised Petersen graphs has girth 7 if
and only if k ≥ 4 and its parameters satisfy one of the following conditions:
• k = 4 and n /∈ {10, 12, 16, 20, 24};
• n = 7k/i where i ∈ {1, 2, 3}, and if i = 3 then k ̸= 6;
• n = 2k + 3 and k ̸= 6; or
• n = 3k ± 2 and (n, k) ̸= (10, 4).
In [1], Ball et al. showed that the cop number of every generalised Petersen graph is
at most 4. They also provided a list of all generalised Petersen graphs with n ≤ 40 that
attain this bound. (Minor corrections to this list were included in [4], which also proved
that every generalised Petersen graph of girth 8 that appears on this list falls into an infinite
class of generalised Petersen graphs of girth 8 that have cop number 4.) Almost all graphs
on this list have girth 8, but there are three graphs of girth 7 on the list:
H. Morris and J. Morris: On generalised Petersen graphs of girth 7, . . . 3
ai
ai−1
bi
ai+1
ai−2 bi−1 bi−k
b
i+k bi+1
ai+2
ai−3 bi−2 bi−k−1
b
i+k−1
b
i−2k
a
i−k
a
i+k bi+2k
b
i−k+1
b
i+k+1 bi+2
ai+3
a
i
−
4
b
i
−
3
b
i
−
k
−
2
b
i
+
k
−
2
b
i
−
2
k
−
1
a
i
−
k
−
1
a
i
+
k
−
1
b
i
+
2
k
−
1
b
i
−
3
k
a
i
−
2
k
a
i
−
k
−
1
a
i
−
k
+
1
a
i
+
k
−
1
a
i
+
k
+
1
a
i
+
2
k
b
i
+
3
k
b
i
−
2
k
+
1
a
i
−
k
+
1
a
i
+
k
+
1
b
i
+
2
k
+
1
b
i
−
k
+
2
b
i
+
k
+
2
b
i
+
3
a
i
+
4
Figure 1: The vertices within distance 4 of ai, in our families of graphs.
• GP (28, 8);
• GP (35, 10); and
• GP (35, 15).
Notably, all of these graphs have parameters of the form n = 7k/2 or n = 7k/3. In this
paper, we show that with the exception of the generalised Petersen graphs with n < 42 that
do not appear above, every generalised Petersen graph whose parameters have the form
n = 7k/i where i ∈ {1, 2, 3} has cop number 4. Thus, in light of the results of [4], we
show that all known generalised Petersen graphs with cop number 4 are included in infinite
families that have this property.
For graphs in the family we are studying (generalised Petersen graphs with n = 7k/i,
i ∈ {1, 2, 3} and n ≥ 42 or (n, k) ∈ {(28, 8), (35, 10), (35, 15)}), the neighbourhood (up
to distance 4) of an arbitrary vertex ai ∈ A is shown in Figure 1, while that for an arbitrary
vertex bi ∈ B is shown in Figure 2. Note that at distance 4, some vertices may be the
same as others; a pair of nodes that are depicted by the same non-standard shape (e.g. a
diamond) represent the same vertex. In our results, we will assume that either n ≥ 42, or
(n, k) ∈ {(28, 8), (35, 10), (35, 15)}. Careful calculations (left to the reader) verify that
this assumption avoids any additional vertices at distance 4 being the same as each other,
so that the graphs are as depicted. This will be of critical importance in our proofs.
2 Main result
We begin by defining what it means for a robber to be trapped.
Definition 2.1. The cops have trapped the robber if there is a cop on or adjacent to each vi
(i ∈ {1, 2, 3}), where v1, v2, and v3 are the neighbours of the vertex the robber is on (no
matter whose turn it is), or if at least one cop is adjacent to the robber on the cops’ turn.
Observe that if the cops have trapped the robber, then the robber will be caught on the
cops’ next move if the robber moves, or within the next two cop moves if the robber passes,
and this is the only configuration for which this is true.
4 Art Discrete Appl. Math. 5 (2022) #P2.05
bi
b
i−k
ai
b
i+k
b
i−2k
a
i−k ai−1 ai+1
a
i+k bi+2k
b
i−3k
a
i−2k
a
i−k−1
a
i−k+1 bi−1
ai−2 ai+2 bi+1
a
i+k−1
a
i+k+1
a
i+2k
b
i+3k
a
i
−
3
k
a
i
−
2
k
−
1
a
i
−
2
k
+
1
b
i
−
k
−
1
a
i
−
k
−
2
b
i
−
k
+
1
a
i
−
k
+
2
b
i
−
k
−
1
b
i
+
k
−
1
b
i
−
2
a
i
−
3
a
i
+
3
b
i
+
2
b
i
−
k
+
1
b
i
+
k
+
1
a
i
+
k
−
2
b
i
+
k
−
1
a
i
+
k
+
2
b
i
+
k
+
1
a
i
+
2
k
−
1
a
i
+
2
k
+
1
a
i
+
3
k
Figure 2: The vertices within distance 4 of bi, in our families of graphs. The dotted lines
represent an edge between bi+3k and bi−3k.
We intend to demonstrate that three cops do not have a winning strategy on the graphs
we are studying. Together with the upper bound of [1], this is sufficient to show that the
cop number for these graphs is 4. In order to prove that three cops do not have a winning
strategy, we will first show that if three cops have not already trapped the robber, the robber
always has a legal move whereby the cops cannot trap the robber on the cops’ next move.
After setting up some notation, we will define three cases that together encompass all possi-
ble configurations the cops can ever be in relative to the robber’s position, and will explain
that these cases are exhaustive. Our first lemma will show that one of these three cases can
only arise in our graphs if the robber is already trapped. Our second lemma shows that if
either of the other cases applies, the robber has a legal move whereby the cops cannot trap
the robber on the cops’ next turn. These lemmas set up an inductive argument showing that
the robber can always avoid becoming trapped unless they begin the game trapped. In the
proof of our main theorem, we show that the robber can always find a starting position that
is not trapped, thus completing the proof.
We now introduce the labelling system used throughout the rest of this paper and all
the proofs within. We assume throughout that the game is being played with 3 cops, C1,
C2, and C3. We will use r to denote the vertex the robber starts on. Label the three vertices
adjacent to r with v1, v2, and v3. When we mention a “branch” from vi (i ∈ {1, 2, 3}), we
are referring to vi and all of the other 7 vertices along any paths from the robber’s vertex
to the vertices at distance 4 from the robber, using only a fixed one of vi’s neighbours
(excluding r from our count). All of this is illustrated in Figure 3.
We now outline the three cases included in [4] as the only possible sets of configurations
for the robber to be in, relative to the positions of the three cops. They are as follows:
• Case 1. For at least one vertex vi (i ∈ {1, 2, 3}), vi has no cop on one of its branches,
and no cop within distance 2 of the robber on the other branch.
• Case 2. For every cop Ci (i ∈ {1, 2, 3}), Ci is not within distance 2 of the robber.
• Case 3. For at least one vertex vi (i ∈ {1, 2, 3}), vi has at least one cop on its
branches who is at distance 2 or less from the robber. If any vertices vj (j ∈ {1, 2, 3})
H. Morris and J. Morris: On generalised Petersen graphs of girth 7, . . . 5
r
v1
v2
v3
Figure 3: This figure shows our basic notation. In the illustration, r is a vertex in A. The
two branches from v1 are circled.
are not covered by that clause, then for each such vj , vj has at least one cop on each
of its branches.
Now that we have stated our three cases, we will explain why these cases cover all of
the possibilities. Suppose we have a set-up that does not conform to Case 2. This means
that there is at least one cop who is at distance 2 or less from the robber. This cop is on
vi (i ∈ {1, 2, 3}), or one of its adjacent vertices (other than r). If the situation also does
not satisfy the assumptions of Case 3, then there must be at least one vertex (say vj , where
j ∈ {1, 2, 3} and j ̸= i) that has no cop on one of its branches, and the closest cop through
vj is at distance 3 or more from the robber. This means that Case 1 applies to vertex vj .
We will now prove our first lemma, which will be used in our main lemma.
Lemma 2.2. Suppose that n = 7k/i where i ∈ {1, 2, 3}, and either n ≥ 42 or (n, k) ∈
{(28, 8), (35, 10), (35, 15)}. If we play cops and robbers with 3 cops on GP (n, k), then
the only way for the configuration of the robber with respect to the cops to fall into Case 3
is if the robber is trapped.
Proof. To prove this lemma, we will use a case analysis on the possible configurations
within Case 3, and show why each is impossible unless the robber is trapped. Our proof
holds regardless of whether the robber is on a vertex in A or a vertex in B. We assume that
both the cop and robber players use their moves optimally.
We now move to our case analysis.
Case A. For exactly one vertex vi (i ∈ {1, 2, 3}), vi satisfies the conditions of Case 3
by having at least one cop on its branches who is at distance 2 or less from the robber.
In order for our situation to fall under Case A, vi satisfies the conditions of Case 3
already, but there are two other vertices, vj and vk (where {i, j, k} = {1, 2, 3}), that still
need to satisfy these conditions. This means that each of vj and vk requires both of its
branches to be covered by cops. There is no way for two cops to cover all four branches,
unless those two cops are directly on vj and vk, which does not fall under Case A. This
means that it is impossible for a configuration of cops and robber on our graphs to fall
under Case A. ■
6 Art Discrete Appl. Math. 5 (2022) #P2.05
Case B. For exactly two vertices vi and vj (i, j ∈ {1, 2, 3}), vi and vj satisfy the
conditions of Case 3 by each having at least one cop on its branches who is at distance
2 or less from the robber.
Assuming our scenario meets the parameters of Case B, there is only one vertex vk
(where {i, j, k} = {1, 2, 3}) which does not yet satisfy the conditions of Case 3. This final
vertex is required to have at least one cop on each of its branches. Since two of the three
cops have already been placed, we only have one cop left. No vk has a vertex that is a part
of both of its branches, other than vk itself, which cannot have a cop on it because of the
conditions of Case B. ■
Case C. For every vertex vi (i ∈ {1, 2, 3}), vi satisfies the conditions of Case 3 by
having at least one cop on its branches who is at distance 2 or less from the robber.
In order for the parameters of Case C to be met, there must be a cop on each of v1,
v2, and v3, or the adjacent vertices (excluding r, of course). In this case, the robber is
trapped. ■
We now prove our main lemma, which will be used inductively in the proof of our
theorem.
Lemma 2.3. Suppose that n = 7k/i where i ∈ {1, 2, 3}, and either n ≥ 42 or (n, k) ∈
{(28, 8), (35, 10), (35, 15)}. If we play cops and robbers on GP (n, k) with three cops,
then unless the robber starts their turn trapped, they always have a legal move so that they
cannot be trapped by the cops on the cops’ turn.
Proof. To prove this lemma, we will use a case analysis of the current positions of the cops
compared to that of the robber (on the robber’s turn). We will use the techniques from [4]
to prove this lemma. We assume that the three cops have not yet trapped the robber. We
also assume that both the cop and robber players use their moves optimally. We will go on
to prove that unless the robber starts their turn trapped, they always have a legal move so
that the robber cannot be trapped at the end of the cops’ turn.
In Lemma 2.2, we proved that Case 3 cannot apply to our graphs. This means that
we need only prove this lemma under the assumption that the scenario falls into Case 1 or
Case 2. These proofs are below.
Case 1. For at least one vertex vi (i ∈ {1, 2, 3}), vi has no cop on one of its branches,
and no cop within distance 2 of the robber on the other branch.
Let us assume that the vertex satisfying the requirements of this case is v1 (the other
cases are similar). Equally, let us assume that the left branch is the one with no cop. This
situation is shown in Figure 4. Since there is no cop within distance 2 of the robber on
vertex v1, the robber can go to v1 safely. No cop can catch the robber on this move, since
no cop can be within distance 1 of v1 at the start of this turn. Since there are no cops on
the left branch from v1, there are no cops within distance 4 of the robber in that direction
before the robber’s move, so even after the cops’ turn there cannot be a cop within distance
2 of the robber in that direction. Therefore, the robber is not trapped at the end of the cops’
turn. ■
Case 2. For every cop Ci (i ∈ {1, 2, 3}), Ci is not within distance 2 of the robber.
Figure 5 illustrates this case. In this scenario, the robber can move to any of the three
vertices adjacent to their position (v1, v2, or v3). Let’s have the robber choose v1. Since
no cops were within distance 2 of the robber at the beginning, once the robber has moved,
H. Morris and J. Morris: On generalised Petersen graphs of girth 7, . . . 7
r
v1
v2
v3
Figure 4: The vertices that we assume do not have cops on them in Case 1 are circled. The
illustration shows this when r ∈ A.
r
v1
v2
v3
Figure 5: The vertices that cannot have cops on them in Case 2 are circled. The illustration
shows this when r ∈ A.
they cannot be within distance 1 of the robber, so cannot catch the robber. Furthermore,
since no cop was on vertex v1, v2, v3, or any of their neighbours, after the cops’ move no
cop is on r or any of its neighbours (and r is a neighbour of the robber’s new position).
This means that the robber is not trapped. ■
We have now proven that the robber cannot be trapped at the end of the cops’ turn if
they did not start their turn trapped.
This allows us to prove our main result.
Theorem 2.4. Suppose that n = 7k/i where i ∈ {1, 2, 3}, and n ≥ 42 or (n, k) ∈
{(28, 8), (35, 10), (35, 15)}. Then the cop number of the graph GP (n, k) is 4.
Proof. We begin by showing that there is always a vertex the robber can choose for their
first move, so that they do not begin the game trapped.
Let w be an arbitrary vertex with neighbours v1, v2, and v3. Recall that for the robber
to be trapped, there must be cops on each of its neighbours or their adjacent vertices, or
8 Art Discrete Appl. Math. 5 (2022) #P2.05
w
v1
v2
v3
C1 C2 C3
Figure 6: The circles in this figure show the ideal positions for the cops at the beginning of
the game, when w ∈ A.
there must be at least one cop directly adjacent to the robber. In order for the cops to be
in a position where they could have the robber trapped, a cop should be placed on each of
v1, v2, and v3 or their adjacent vertices. Without loss of generality, let us assume that Ci
(i ∈ {1, 2, 3}) is on vi or an adjacent vertex.
Suppose momentarily that all three cops choose to go on vertices adjacent to w. Under
this scenario, should the robber go on any vertex adjacent to a cop, they will be trapped.
This gives us 10 vertices where the robber cannot go without being trapped (including those
with cops already on them).
Now, suppose that all three cops choose to go on vertices at distance 2 from w. In this
set-up, in addition to the robber being trapped if they are adjacent to a cop, they will also
be trapped if they go on w, since there are cops adjacent to v1, v2, and v3. Therefore, in
this case, there are 13 different vertices where the robber cannot go without being trapped.
This means that our second set of positions, as shown in Figure 6, are the ideal choices
for the cops when there are only three of them, and if they go in the ideal positions, they
can still only ensure that the robber will be trapped if placed on one of 13 different vertices.
Since n ≥ 28, our graph has at least 56 vertices, and the robber still has many choices that
do not leave them trapped, meaning that the robber will not start trapped.
Combining this with the results of Lemma 2.3, we conclude that the robber never has
to become trapped, so c(G) > 3.
By [1], if G is a generalised Petersen graph, then c(G) ≤ 4, so c(G) = 4.
ORCID iDs
Harmony Morris https://orcid.org/0000-0002-6348-137X
Joy Morris https://orcid.org/0000-0003-2416-669X
References
[1] T. Ball, R. W. Bell, J. Guzman, M. Hanson-Colvin and N. Schonsheck, On the cop number of
generalized Petersen graphs, Discrete Math. 340 (2017), 1381–1388, doi:10.1016/j.disc.2016.
10.004.
H. Morris and J. Morris: On generalised Petersen graphs of girth 7, . . . 9
[2] M. Boben, T. Pisanski and A. Žitnik, I-graphs and the corresponding configurations, J. Combin.
Des. 13 (2005), 406–424, doi:10.1002/jcd.20054.
[3] A. Bonato and R. J. Nowakowski, The game of cops and robbers on graphs, volume 61 of
Student Mathematical Library, American Mathematical Society, Providence, RI, 2011, doi:10.
1090/stml/061.
[4] J. Morris, T. Runte and A. Skelton, Most Generalized Petersen graphs of girth 8 have cop number
4, 2020, arXiv:2009.00693 [math.CO], https://arxiv.org/abs/2009.00693.
[5] R. Nowakowski and P. Winkler, Vertex-to-vertex pursuit in a graph, Discrete Math. 43 (1983),
235–239, doi:10.1016/0012-365X(83)90160-7.
[6] A. Quilliot, Problèmes de jeux, de point fixe, de connectivité et de représentation sur des graphes,
des ensembles ordonnés et des hypergraphes, Ph.D. thesis, Université de Paris VI, 1983.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P2.06
https://doi.org/10.26493/2590-9770.1434.bf4
(Also available at http://adam-journal.eu)
Topology of clique complexes of line graphs*
Shuchita Goyal
Chennai Mathematical Institute, Chennai, India
Samir Shukla
Indian Institute of Technology Mandi, India
Anurag Singh†
Indian Institute of Technology Bhilai, India
Received 2 October 2020, accepted 27 July 2021, published online 11 May 2022
Abstract
The clique complex of a graph G is a simplicial complex whose simplices are all the
cliques of G, and the line graph L(G) of G is a graph whose vertices are the edges of G
and the edges of L(G) are incident edges of G. In this article, we determine the homotopy
type of the clique complexes of line graphs for several classes of graphs including triangle-
free graphs, chordal graphs, complete multipartite graphs, wheel-free graphs, and 4-regular
circulant graphs. We also give a closed form formula for the homotopy type of these
complexes in several cases.
Keywords: H-free complex, clique complex, line graph, chordal graphs, wheel-free graphs, circulant
graphs.
Math. Subj. Class.: 05C69, 55P15
1 Introduction
For a fixed graph H , a subgraph K of a graph G is called H-free if K does not contain
any subgraph isomorphic to H . Finding H-free subgraphs of a graph G is a well studied
*We would like to thank Russ Woodroofe for his insightful comments and suggestions. We are also thankful
to the anonymous referee for pointing out the connection between the nerve of a cover of ∆L and the 2-skeleton
of the clique complex, which helped us in shortening many proofs of Section 3 significantly. The first and third
authors are partially supported by a grant from the Infosys Foundation.
†Corresponding author.
E-mail addresses: shuchitagoyal23@gmail.com (Shuchita Goyal), samir@iitmandi.ac.in (Samir Shukla),
anurags@iitbhilai.ac.in (Anurag Singh)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P2.06
notion in extremal graph theory. It is easy to observe that H-freeness is a hereditary prop-
erty, thereby giving a simplicial complex for any fixed graph G, denoted F(G,H). These
complexes have already been studied for different graphs H , for example see [2, 7, 11, 14,
15, 22] when H is a complete graph, [9, 19, 21] when H is a star graph, [8, 13] when H is
r disjoint copies of complete graphs on 2 vertices, denoted rK2. Replacing H by a class of
trees on fixed number of vertices has also gained attention in the last decade, for instance
see [6, 18, 20].
In this article, our focus will be on the complex F(G, 2K2). In [13], Linusson et. al
have studied F(G, rK2) for complete graphs and complete bipartite graphs. They showed
that F(G, rK2) is homotopy equivalent to a wedge of (3r − 4)-dimensional spheres when
G is a complete graph, and (2r − 3)-dimensional spheres if G is a complete bipartite
graph. Further, Holmsen and Lee [8] studied these complexes for general graph G and
they showed that F(G, rK2) is (3r − 3)-Leray1, and it is (2r − 2)-Leray if G is bipartite.
It is worth noting that F(G, 2K2) is the clique complex of the line graph of G, denoted
∆L(G). Clique complexes of graphs have a very rich literature in topological combina-
torics, for instance see [1, 10]. Recently, Nikseresht [17] studied some algebraic properties
like Cohen-Macaulay, sequentially Cohen-Macaulay, Gorenstein, etc., of clique complexes
of line graphs. In this article, we determine the exact homotopy type of ∆L( ) for vari-
ous classes of graphs including triangle-free graphs, chordal graphs, complete multipartite
graphs, wheel-free graphs and 4-regular circulant graphs. Moreover, we show that ∆L( )
in each of these cases is homotopy equivalent to a wedge of equidimensional spheres, ex-
cept for the 4-regular circulant graphs for which it is homotopy equivalent to a wedge of
circles and 2-spheres.
This article is organized as follows. In Section 2, we recall various definitions and
results which are needed. In the next section, we analyze ∆L(G) as the 2-skeleton of
the clique complex of G and then use it to compute the homotopy type of ∆L(G) for
various classes of graphs including triangle free graphs and chordal graphs. In Section 3.1,
by realizing ∆L as a functor from the category of graphs to the category of simplicial
complexes, we compute the homotopy type of ∆L(G) when G is a complete multipartite
graph. Section 4 is devoted towards the study of ∆L of wheel-free graphs and 4-regular
circulant graphs. In the last section, we discuss a few questions that arise naturally from
the work done in this article.
2 Preliminaries
A (simple) graph is an ordered pair G = (V (G), E(G)), where V (G) is called the set of
vertices and E(G) ⊆
(
V (G)
2
)
, the set of (unordered) edges of G. The vertices v1, v2 ∈
V (G) are said to be adjacent, if (v1, v2) ∈ E(G) and this is also denoted by v1 ∼ v2. For
v ∈ V (G), the set of its neighbours, NG(v), in G is {x ∈ V (G) : x ∼ v}. The degree
of a vertex v ∈ V (G) is the cardinality of the set of its neighbours. A vertex v of degree
1 is called a leaf vertex and the vertex adjacent to v is called the parent of v. An edge
adjacent to a leaf vertex is called a leaf/hanging edge. A graph H with V (H) ⊆ V (G)
and E(H) ⊆ E(G) is called a subgraph of the graph G. For a nonempty subset U ⊆
V (G), the induced subgraph G[U ], is the subgraph of G with vertices V (G[U ]) = U and
E(G[U ]) = {(a, b) ∈ E(G) : a, b ∈ U}. Similarly, for a nonempty subset F ⊆ E(G),
1A simplicial complex K is called d-Leray (over a field F) if H̃i(L,F) = 0 for all i ≥ d and for every induced
subcomplex L ⊆ K.
S. Goyal, S. Shukla and A. Singh: Topology of clique complexes of line graphs 3
the induced subgraph G[F ], is the subgraph of G with vertices V (G[F ]) = {v ∈ V (G) :
v ∈ e for some e ∈ F} and E(G[F ]) = F . In this article, the graph G[V (G) \ A] will be
denoted by G−A for A ⊊ V (G).
For n ≥ 1, the complete graph on n vertices is a graph where any two distinct vertices
are adjacent, and is denoted by Kn. For n ≥ 3, the cycle graph Cn is the graph with
V (Cn) = {1, . . . , n} and E(Cn) = {(i, i + 1) : 1 ≤ i ≤ n − 1} ∪ {(1, n)}. For r ≥ 1,
the path graph of length r is a graph with vertex set V (Pr) = {0, . . . , r} and edge set
E(Pr) = {(i, i + 1) : 0 ≤ i ≤ r − 1}, and is denoted by Pr. The line graph, L(G),
of a graph G is the graph with V (L(G)) = E(G) and E(L(G)) = {((v1, v2), (v′1, v
′
2)) :
|{v1, v2} ∩ {v
′
1, v
′
2}| = 1}.
An (abstract) simplicial complex K is a collection of finite sets such that if τ ∈ K and
σ ⊂ τ , then σ ∈ K. The elements of K are called the simplices (or faces) of K. If σ ∈ K
and |σ| = k + 1, then σ is said to be k-dimensional. The maximal simplices of K are also
called facets of K. The set of 0-dimensional simplices of K is denoted by V (K), and its
elements are called vertices of K. A subcomplex of a simplicial complex K is a simplicial
complex whose simplices are contained in K. For k ≥ 0, the k-skeleton of a simplicial
complex K is a subcomplex consisting of all the simplices of dimension ≤ k and it is
denoted by K(k). The cone of a simplicial complex K with apex w, denoted Cw(K), is a
simplicial complex whose facets are σ∪{w} for each facet σ of K. In this article, we always
assume empty set as a simplex of any simplicial complex and we consider any simplicial
complex as a topological space, namely its geometric realization. For the definition of
geometric realization, we refer to Kozlov’s book [12].
The clique complex, ∆(G), of a graph G is the simplicial complex whose simplices are
subsets σ ⊆ V (G) such that G[σ] is a complete graph.
The clique complex of line graph of a graph G, ∆L(G), has also been studied by
Linusson, Shareshian, and Welker in [13] and Holmsen and Lee in [8]. They denoted these
complexes by NM2(G) and proved the following results.
Theorem 2.1 (Theorem 1.1, [13]). Let n be a positive integers. Then NM2(Kn) is homo-
topy equivalent to a wedge of spheres of dimension 2.
Theorem 2.2 (Theorem 1.1, [8]). For a graph G, the complex NM2(G) is 3-Leray.
It is worth mentioning here that the Lerayness of the clique complex of the complement
of a claw-free graph has also been studied in [3] and [16].
2.1 (Homotopy) Pushout
Let X,Y, Z be topological spaces, and p : X ! Y and q : X ! Z be continuous maps.
The pushout of the diagram Y
p
 − X
q
−! Z is the space
(
Y
⊔
Z
)
/ ∼,
where ∼ denotes the equivalence relation p(x) ∼ q(x), for x ∈ X .
The homotopy pushout of Y
p
 − X
q
−! Z is the space
(
Y ⊔ (X× [0, 1])⊔Z
)
/ ∼, where
∼ denotes the equivalence relation (x, 0) ∼ p(x), and (x, 1) ∼ q(x), for x ∈ X . It can be
shown that homotopy pushouts of any two homotopy equivalent diagrams are homotopy
equivalent.
4 Art Discrete Appl. Math. 5 (2022) #P2.06
Remark 2.3. If spaces are simplicial complexes and maps are subcomplex inclusions, then
their homotopy pushout and pushout spaces are equivalent up to homotopy. For elaborate
discussion on this, we refer interested reader to [5, Chapter 7].
Remark 2.4. Consider a diagram Y
p
 − X
q
−! Z and let W be its pushout object. If p, q are
null-homotopic, then W ≃ Y
∨
Z
∨
Σ(X), where ≃ and
∨
respectively denotes homo-
topy equivalence and the wedge of topological spaces; and Σ(X) denotes the suspension
of X .
2.2 Simplicial Collapse
Let K be a simplicial complex and τ, σ ∈ K be such that σ ⊊ τ and τ is the only maximal
simplex in K that contains σ. A simplicial collapse of K is the simplicial complex L
obtained from K by removing all those simplices γ of K such that σ ⊆ γ ⊆ τ . Here σ is
called a free face of τ and (σ, τ) is called a collapsible pair. We denote this collapse by
K↘ L. A complex is called collapsible if it collapses onto a point by applying a sequence
of simplicial collapses. It is a simple observation that, if K ↘ L then K ≃ L (in fact, K
deformation retracts onto L). Throughout this article, we write K ↘ ⟨A1, A2, . . . , Ar⟩ to
mean that K collapses onto a subcomplex whose faces are A1, . . . , Ar and all its subsets.
We now give a result about collapsing which will be used throughout this article. To
the best of our knowledge, this result is not present in writing anywhere in literature.
Lemma 2.5. Let K be a simplicial complex and σ be a facet of K. Let A,B,C ⊂ σ be
such that A,B ̸= ∅ and σ = A ⊔ B ⊔ C. For each a ∈ A and b ∈ B, let {a, b} be a free
face of σ in K.
1. If C ̸= ∅, then σ ↘ ⟨σ \A, σ \B⟩.
2. If C = ∅, |A| = 1 and |B| ≥ 2, then for each a ∈ A and b ∈ B, σ ↘ ⟨B, {a, b}⟩.
3. If C = ∅ and |A|, |B| ≥ 2, then for each a ∈ A and b ∈ B, σ ↘ ⟨A,B, {a, b}⟩.
Proof. Let A = {a1, . . . , ap} and B = {b1, . . . , bq}. Without loss of generality, we can
assume that ap = a and bq = b.
1. We first do collapsing using the vertex a1 from set A and then use similar arguments
for the rest. Since {a1, b1} is a free face of σ, σ ↘ ⟨σ \ {b1}, σ \ {a1}⟩. Now
({a1, b2}, σ \ {b1}) is a collapsible pair and therefore σ \ {b1}↘ ⟨σ \ {b1, b2}, σ \
{a1, b1}⟩. Hence σ ↘ ⟨σ \ {b1, b2}, σ \ {a1}⟩. Inductively, assume that for some
1 ≤ j < q, σ ↘ ⟨σ \ {b1, . . . , bj}, σ \ {a1}⟩. Now ({a1, bj+1}, σ \ {b1, . . . , bj})
is a collapsible pair and therefore σ ↘ ⟨σ \ {b1, . . . , bj , bj+1}, σ \ {a1}⟩. Using
induction, we get that σ ↘ ⟨σ \ {b1, . . . , bq} = σ \B, σ \ {a1}⟩.
If p = 1, then this completes the proof, otherwise we proceed as follows:
By doing similar collapsing using vertices a2, a3, . . . , ap−1 from set A, we get that
σ ↘ ⟨σ\B, σ\{a1, . . . , ap−1}⟩. Now ({ap, b1}, σ\{a1, . . . , ap−1}) is a collapsible
pair and therefore σ \ {a1, . . . , ap−1} ↘ ⟨σ \ {a1, . . . , ap−1, ap} = σ \ A, σ \
{a1, . . . , ap−1, b1}⟩. Thus σ ↘ ⟨σ \ B, σ \ A, σ \ {a1, . . . , ap−1, b1}⟩. We now
show that σ \ {a1, . . . , ap−1, b1}↘ ⟨σ \A, σ \B⟩.
S. Goyal, S. Shukla and A. Singh: Topology of clique complexes of line graphs 5
Since C ̸= ∅, it is easy to observe that by using collapsible pairs in the following
order:
({ap, b2}, σ \ {a1, . . . , ap−1, b1}), . . . , ({ap, bq}, σ \ {a1, . . . , ap−1, b1, . . . , bq−1}),
and applying the collapses, we get that σ\{a1, . . . , ap−1, b1}↘ ⟨σ\{a1, . . . , ap−1,
ap, b1, . . . , bq−1}, σ\{a1, . . . , ap−1, b1, . . . , bq−1, bq}⟩. Since σ\{a1, . . . , ap−1, ap,
b1, . . . , bq−1} ⊆ σ \A and σ \{a1, . . . , ap−1, b1, . . . , bq−1, bq} ⊆ σ \B, we get that
σ ↘ ⟨σ \A, σ \B⟩.
2. Here σ = {a, b1, . . . , bq}. Using similar arguments as in the first paragraph of previ-
ous case and using the collapsible pairs in the following order:
({a, b1}, σ), ({a, b2}, {a, b2, . . . , bq}), . . . , ({a, bq−1}, {a, bq−1, bq}),
and doing the collapses, we get the desired result.
3. Using similar arguments as in the proof of case 1, we get that σ ↘ ⟨σ \ B, σ \
{a1, . . . , ap−1}⟩. Observe that σ \ {a1, . . . , ap−1} = {ap, b1, . . . , bq}. We now use
case 2 to collapse σ \ {a1, . . . , ap−1} and get the result.
Induction along with Lemma 2.5 gives the following result.
Corollary 2.6. Let K be a simplicial complex and let σ = A1 ⊔A2 . . .⊔Ak ⊔C be a facet
of K, where Aj = {a
j
1, . . . , a
j
lj
} for 1 ≤ j ≤ k. Let {ais, a
j
t} be a free face of σ for each
i ̸= j, s ∈ [li] and t ∈ [lj ].
(1) If C = ∅, then σ ↘ ⟨A1, . . . , Ak, {a1l1 , a
k
lk
}, {a2l2 , a
k
lk
}, . . . , {ak−1lk−1 , a
k
lk
}⟩.
(2) If C ̸= ∅, then σ ↘ ⟨A1 ⊔ C, . . . , Ak ⊔ C⟩.
3 Structural properties of ∆L
In this section, we first analyze ∆L(G) as the 2-skeleton of the clique complex of G and
then we use it to compute the homotopy type of ∆L(G) for various classes of graphs
including triangle free graphs and chordal graphs. Later we realize ∆L as a functor from
the category of graphs to the category of simplicial complexes and compute the homotopy
type of ∆L(G), when G is a complete multipartite graph, using the functoriality of ∆L.
The nerve of a family of sets (Ai)i∈I is the simplicial complex N = N({Ai}) defined
on the vertex set I so that a finite subset σ ⊆ I is in N precisely when
⋂
i∈σ Ai ̸= ∅.
Theorem 3.1 ([4, Theorem 10.6(i)]). Let K be a simplicial complex and (Ki)i∈I be a
family of subcomplexes such that K =
⋃
i∈I Ki. Suppose every nonempty finite interSec-
tion Ki1 ∩ . . . ∩ Kit for ij ∈ I, t ∈ N is contractible, then K and N({Ki}) are homotopy
equivalent.
Remark 3.2. Since any non-empty interSection of simplices is again a simplex, Theo-
rem 3.1 implies that the nerve of the facets of a simplicial complex K is homotopy equiva-
lent to K.
6 Art Discrete Appl. Math. 5 (2022) #P2.06
Lemma 3.3. For any graph G, the complex ∆L(G) is homotopy equivalent to the 2-
skeleton of the clique complex of G, i.e.,
∆L(G) ≃ (∆(G))(2).
Proof. Let F be the set of facets of ∆L(G). It is easy to observe that any F ∈ F is either
a collection of three edges of G forming a triangle in G or of the form σa = {(a, v) : v ∈
NG(a)} for some a ∈ V (G). From Remark 3.2, N(F) ≃ ∆L(G). It is therefore enough
to show that N(F) ≃ (∆(G))(2).
Let K be a simplicial complex obtained from (∆(G))(2) by adding a barycenter to every
2-simplex of (∆(G))(2). Clearly K ≃ (∆(G))(2) and K ∼= N(F).
Let v(G) and e(G) denote the number of vertices and edges in a graph G, respectively.
For a given graph G, define u(G) = e(G) − e(T ), where T is a spanning tree of G. The
following is an immediate consequence of Lemma 3.3.
Corollary 3.4. Let G be a triangle free graph. Then ∆L(G) is homotopy equivalent to G
(considered as a 1-dimensional simplicial complex). In particular
∆L(G) ≃
∨
u(G)
S1,
A graph G is called chordal if it has no induced cycle of length greater than 3, i.e., each
cycle of length more than 3 has a chord.
Theorem 3.5. Let G be a connected chordal graph. Then ∆L(G) is homotopy equivalent
to a wedge of 2-dimensional spheres.
Proof. It is well known that the 2-skeleton of the clique complex of a chordal graph is
simply connected. Also, we know that any 2-dimensional simply connected simplicial
complex is homotopy equivalent to a wedge of 2-spheres (cf. [4, (9.19)]). The result
therefore follows from Lemma 3.3.
Let G be a graph. The cone over G with apex vertex w, denoted as CwG, is the graph
with V (CwG) = V (G) ⊔ {w} and edge set E(CwG) = E(G) ∪ {(w, v) : v ∈ V (G)}.
Lemma 3.6. Let G be a graph with m triangles. Then ∆L(CwG) is homotopy equivalent
to a wedge of m spheres of dimension 2.
Proof. Let F = {F ∈ (∆(CwG))(2) : |F | = 3 and w /∈ F}. Observe that |F| = m.
Clearly (∆(CwG))(2) \ F ∼= Cw(G) (here G is considered as a 1-dimensional simpli-
cial complex) and therefore (∆(CwG))(2) \ F is contractible. Hence, from [12, Proposi-
tion 7.8], we have that (∆(CwG))(2) ≃ (∆(CwG))(2)/((∆(CwG))(2) \ F). Thus, from
Lemma 3.3, ∆L(CwG) ≃ (∆(CwG))(2) ≃
∨
|F|
S2.
The suspension ΣG of a graph G is the graph with vertex set V (ΣG) = V (G)⊔ {a, b}
and E(ΣG) = E(G) ∪ {(a, v) : v ∈ V (G)} ∪ {(b, v) : v ∈ V (G)}.
Lemma 3.7. Let G be a triangle free connected graph on at least two vertices. Then
∆L(ΣG) ≃ Σ(∆L(G)).
Proof. We know that ∆L(ΣG) ≃ (∆(ΣG))(2). Since G is triangle free, (∆(ΣG))(2) =
Σ((∆(G))(2)) ≃ Σ(∆L(G)).
S. Goyal, S. Shukla and A. Singh: Topology of clique complexes of line graphs 7
3.1 Complete multipartite graphs
Let m1,m2, . . . ,mr be positive integers and let Ai be a set of cardinality mi for 1 ≤ i ≤ r.
A complete multipartite graph Km1,...,mr is a graph on vertex set A1 ⊔ . . . ⊔ Ar and two
vertices are adjacent if and only if they lie in different Ai’s. Using functoriality of ∆L, we
study the homotopy type of ∆L of complete multipartite graphs. The following result can
be obtained from Lemma 3.3.
Corollary 3.8. Let H be a connected induced subgraph of connected graphs G1 and G2.
Then,
∆L(G1
⋃
H
G2) ≃



∆L(G1)
⊔
∆L(G2) if |V(H)| = 0,
∆L(G1)
∨
∆L(G2) if |V(H)| = 1,
∆L(G1)
⋃
∆L(H) ∆L(G2) otherwise.
In particular, for any 1 ≤ s ≤ mr − 1, ∆L(Km1,m2,...,mr ) is homotopy equivalent to the
pushout of
∆L(Km1,m2,...,mr−1,s)  ֓ ∆L(Km1,m2,...,mr−1) !֒ ∆L(Km1,m2,...,mr−1,mr−s).
Proposition 3.9. For m,n, r ∈ N, ∆L(Km,n) ≃
∨
t S
1 and ∆L(Km,n,r) ≃
∨
t(r−1) S
2,
where t = mn− (m+ n− 1).
Proof. For the graph Km,n, recall u(Km,n) = e(Km,n)− e(Tm,n) = mn− (m+n− 1),
where Tm,n denotes a spanning tree of Km,n. Since any bipartite graph is triangle-free,
Corollary 3.4 implies that ∆L(Km,n) is homotopy equivalent to
∨
u(Km,n)
S1.
Further, Km,n,1 is a cone over the triangle-free graph Km,n, and hence ∆L(Km,n,1)
is contractible by Lemma 3.6. Now using Remark 2.4 and Corollary 3.8, ∆L(Km,n,2) ≃
{pt}
∨
{pt}
∨(∨
u(Km,n)
S2
)
. Inductively, we construct Km,n,r as a pushout of
Km,n,r−1  ֓ Km,n !֒ Km,n,1. Then repeating the similar arguments, we get that
∆L(Km,n,r) ≃
∨
u(Km,n)(r−1)
S2.
Note. The homotopy type of ∆L(Km,n) has also been computed by Linusson et al. in
[13, Theorem 1.4] using discrete Morse theory.
Theorem 3.10. For a complete r-partite graph Km1,...,mr , ∆L(Km1,...,mr ) is homotopy
equivalent to a wedge of S2’s for r > 2.
Proof. From Proposition 3.9, ∆L(Km1,m2,m3) is homotopy equivalent to a wedge of 2-
dimensional spheres. So let r > 3. Since Km1,m2,m3,1 is a cone over Km1,m2,m3 ,
Lemma 3.6 gives that ∆L(Km1,m2,m3,1) is homotopy equivalent to a wedge of S
2’s. If
m4 > 1, we inductively construct Km1,m2,m3,m4 as a pushout of
Km1,m2,m3,m4−1  ֓ Km1,m2,m3 !֒ Km1,m2,m3,1.
Note that after applying ∆L to this diagram and using Proposition 3.9, we get the following
pushout diagram
8 Art Discrete Appl. Math. 5 (2022) #P2.06
∨
S2
(∨
S2
)
∨
(∨
B3
)
(∨
S2
)
∨
(∨
B3
)
X ≃ ∆L(Km1,m2,m3,m4)
Figure 1: ∆L(Km1,m2,m3,m4) as a pushout
Here B3 denotes a closed ball of dimension 3. From Figure 1, it is easy to see that
∆L(Km1,m2,m3,m4) is homotopy equivalent to a wedge of spheres of dimension 2 and 3.
However, Theorem 2.2 implies that H̃3(∆L(Km1,m2,m3,m4)) = 0, and hence
∆L(Km1,m2,m3,m4) ≃
∨
S2. We now repeat these arguments for Km1,...,mr to get the
desired result.
4 Wheel-free graphs and 4-regular circulant graphs
For n ≥ 3, a wheel graph on n+ 1 vertices, denoted by Wn, is a graph isomorphic to cone
over a cycle graph Cn. For a wheel graph, we call apex vertex of the cone as the center of
the wheel.
Theorem 4.1. Let G be a connected wheel-free graph. Then ∆L(G) is homotopy equiva-
lent to a wedge of circles.
Proof. In view of Lemma 3.3, we can assume that G does not have any leaf. Observe that
any maximal simplex σ in ∆L(G) is one of the following two types
1. σ = {e1, . . . , et} such that
⋂
i∈[t] ei = {x}, i.e., every edge shares a common vertex.
2. σ = {e1, e2, e3} and e1, e2, e3 forms a triangle in G.
We now show that each facet of ∆L(G) can be collapsed to a 1-dimensional subcomplex.
First we collapse facets of type 1.
Case 1: Let σ = {e1, . . . , et} be a facet and
⋂
i∈[t] ei = {x}. Without loss of generality,
we can assume that t ≥ 3. We partition the set NG(x) = A ⊔B, where
• A =
⋃
i∈[r1]
{ai}, where each ai is an isolated vertex in the induced subgraph
G[NG(x)].
• B =
⊔
j∈[r2]
Bj , where each Bj ⊆ NG(x) such that G[Bj ] is a connected compo-
nent of G[NG(x)] and has more than 1 vertex.
Since G is wheel-free graph, it is easy to see that G[Bj ] is a tree for each j ∈ [r2]. Define,
A = {(x, ai) : i ∈ [r1]} and
B =
⊔
j∈[r2]
{(x, a) : a ∈ Bj}, (4.1)
(cf. Figure 2). For each j ∈ [r2], let Bj = {(x, a) : a ∈ Bj}. It is easy to see that each
{e, e′} is a free face of σ whenever
S. Goyal, S. Shukla and A. Singh: Topology of clique complexes of line graphs 9
(i) e ∈ Bi, e′ ∈ Bj and i ̸= j, or
(ii) e ∈ A and e′ ∈ B, or
(iii) e, e′ ∈ A.
For each j ∈ [r2], let Bj = {e
j
1, e
j
2, . . . , e
j
lj
} and A = {e1, . . . , er1}. By Corol-
lary 2.6(1), σ ↘ ⟨B1, . . . ,Br2 ,A, {e
1
l1
, er1}, . . . , {e
r2
lr2
, er1}⟩.
a2
a1
x
a11
a12
a13
a14
a15
a12
a22
(a) G[{x} ∪NG(x)]
x
a2
a1
(b) A
x e11
e12
e13
e14
e15
e12
e22
a11
a12
a13
a14
a15
a12
a22
(c) B
a11
a12
a13
a14
a15
(d) G[B1]
a12
a13
a14
a15
(e) G[B1 − a11]
Figure 2
Again using Corollary 2.6(2), A ↘ ⟨{e1, er1}, . . . , {er1−1, er1}⟩. We now show that
each Bj can be collapsed onto a 1-dimensional subcomplex of ∆L(G). We prove this by
induction on the number of elements in Bj . If lj = 2 then the simplex Bj itself is of
dimension 1.
Fix j ∈ [r2] and assume that lj ≥ 3. For each 1 ≤ i ≤ lj , let e
j
i = (x, a
j
i ). Without loss
of generality, we can assume that the aj1 is a leaf vertex in tree G[Bj ] and a
j
2 is its parent.
This implies that for any i ∈ {3, . . . , lj}, there exist no triangle in G which contains both
the edges ej1 and e
j
i .
Since ej1 and e
j
3 are not part of any triangle, ({e
j
1, e
j
3},Bj) is a collapsible pair and
therefore Bj ↘ ⟨Bj \ {e
j
1},Bj \ {e
j
3}⟩. If lj = 3, then Bj \ {e
j
1} and Bj \ {e
j
3} are
1-dimensional. Assume lj ≥ 4.
Now ({ej1, e
j
4},Bj \ {e
j
3}) is a collapsible pair and therefore Bj \ {e
j
3} ↘ ⟨Bj \
{ej3, e
j
4},Bj \ {e
j
3, e
j
1}⟩. Since Bj \ {e
j
3, e
j
1} ⊆ Bj \ {e
j
1}, Bj ↘ ⟨Bj \ {e
j
1},Bj \ {e
j
3, e
j
4}⟩.
It is easy to observe that by using collapsible pairs in the following order:
({ej1, e
j
5},Bj \ {e
j
3, e
j
4}), . . . ({e
j
1, e
j
lj
},Bj \ {e
j
3, e
j
4, . . . , e
j
lj−1
})
10 Art Discrete Appl. Math. 5 (2022) #P2.06
and applying the collapses, we get that Bj ↘ ⟨Bj \{e
j
1}, {e
j
1, e
j
2}⟩. Since a
j
1 is a leaf vertex
of G[Bj ], G[Bj − {a
j
1}] is again a tree (cf. Figure 2e). Therefore, from induction, we have
that Bj \ {e
j
1} collapses onto a 1-dimensional subcomplex of ∆L(G). Which implies that,
Bj collapses onto a subcomplex of dimension 1.
Case 2: Once we have collapsed all simplices of type 1 then given any simplex {e1, e2, e3}
of type 2, it is easy to see that ({e1, e2}, {e1, e2, e3}) is always a collapsible pair. Thus we
can collapse all these simplices to a 1-dimensional subcomplex of ∆L(G).
Since G is connected, ∆L(G) is connected. Therefore the result follows from the fact
that any connected one dimensional simplicial complex is homotopy equivalent to a wedge
of circles.
It is an easy observation that any graph which is not K4 and has maximal degree 3 is a
wheel-free graph, and hence the previous result implies the following corollary.
Corollary 4.2. Let G be a connected graph of maximal degree at most 3 and G ≇ K4.
Then ∆L(G) is homotopy equivalent to a wedge of circles.
Let n ≥ 3 be a positive integer and S ⊆ {1, 2, . . . , ⌊n2 ⌋}. The circulant graph Cn(S) is
the graph, whose set of vertices V (Cn(S)) = {0, 1, . . . , n− 1} and any two vertices x and
y adjacent if and only if x−y (mod n) ∈ S∪(−S), where −S = {n−a : a ∈ S}. Circulant
graphs are also Cayley graphs of Zn, the cyclic group on n elements. Since n /∈ S, Cn(S)
is a simple graph, i.e., does not contains any loop. Further, Cn(S) are |S ∪ (−S)|-regular
graphs. We now prove a structural result for 4-regular circulant graphs. This result will
enable us to do the computation of the homotopy type of their ∆L.
Proposition 4.3. Let G be a 4-regular circulant graph. Then each connected component
of G is either wheel-free or isomorphic to K5 or to ΣC4.
Proof. Let {s, t} be the generating set of G such that s < t and V (G) = {0, 1, . . . , n−1}.
Symmetry in the circulant graph implies that connected components of G are isomorphic.
Suppose G is not wheel-free and say it has a subgraph H isomorphic to Wm, a wheel on
m+1 vertices. Since G is 4-regular, 3 ≤ m ≤ 4. Without loss of generality we can assume
that 0 is the center vertex of Wm. Clearly, NG(0) = {s, t, n− s, n− t}.
Case 1: m = 3.
Since W3 ∼= K4, we see that |V (W3) ∩ NG(0)| = 3. Therefore, either s ∼ t or
n− s ∼ n− t in W3. In both the cases we get that t = 2s. Since s < t, n− t < n− s. If
NW3(0) = {s, t, n−t}, then n−t < n−s implies that n−t = 3s, thus n = 3s+2s = 5s.
Similar analysis for any 3 element subset of NG(0) implies that n = 5s. This implies that
NG(0) = {s, 2s, 3s, 4s} and therefore G[NG(0) ∪ {0}] is a clique in G. Moreover, 4-
regularity of G implies that G[{0, s, 2s, 3s, 4s}] is a component.
Case 2: m = 4.
Let a ∼ b ∼ c ∼ d ∼ a be the outer cycle of W4. By symmetry, the vertex a is also
a center of a wheel with 5 vertices, say W ′4. Let NG(a) = {b, d, 0, x}. If x = c, then
{a, b, c, 0} forms a K4. By Case 1, we get that {a, b, c, d, 0} forms a K5. Therefore, let
x ̸= c, then x ̸∼ 0 implies that d ∼ x ∼ b ∼ 0 ∼ d is the outer cycle of W ′4. Similarly,
b is the center of some other wheel with 5 vertices, say W ′′4 . Since NG(b) = {a, 0, c, x},
the outer cycle of W ′′4 is given by a ∼ x ∼ c ∼ 0 ∼ a. Therefore, NG(x) = NG(0) =
{a, b, c, d}. Again, the 4-regularity of G implies that G[{0, a, b, c, d, x}] is a component
and is isomorphic to ΣC4.
S. Goyal, S. Shukla and A. Singh: Topology of clique complexes of line graphs 11
Corollary 4.4. Let G be a 4-regular circulant graph. Then each connected component of
∆L(G) is homotopy equivalent to either a wedge of circles or a wedge of 2-spheres.
Proof. We first note that for any cycle graph Cr, ∆L(Cr) ≃ S1 whenever r ≥ 4. From
Proposition 4.3, each connected component of G is either wheel-free or isomorphic to K5
or to ΣC4. Since ∆L(wheel-free) ≃
∨
S1 (cf. Theorem 4.1), ∆L(K5) ≃
∨
S2 (cf.
Theorem 2.1) and ∆L(ΣC4) ≃ Σ(∆L(C4)) ≃ Σ(S1) = S2 (cf. Lemma 3.7), the result
follows.
It is to note here that the computations done in this Section gives the exact homotopy
type of ∆L for all the 2, 3 and 4-regular circulant graphs.
5 Further directions
For any simplicial complex K ≃
(∨
m S
1
)
∨
(∨
n S
2
)
, Corollary 3.8 shows that
∆L(
(∨
m C4
)
∨
(∨
n K4
)
) ≃ K, where the wedge of graphs is taken along a vertex
as 1-dimensional simplicial complexes. It is well known that the clique complex functor ∆
is universal (i.e., given any simplicial complex K, there is a graph G such that ∆(G) ≃ K),
whereas the line graph functor L is not (for example, K1,3 is not a line graph of any graph).
This raises the following natural question.
Question 5.1. Is the functor ∆L universal from the category of graphs to the category of
3-Leray simplicial complexes?
We note here that for all the classes of graphs considered in this article, ∆L has always
been a wedge of equidimensional spheres. Therefore, it would be interesting to know the
following.
Question 5.2. Can the classes of graphs be classified whose ∆L is a wedge of equidi-
mensional spheres? More specifically, can we classify those graphs whose ∆L is simply
connected?
ORCID iDs
Shuchita Goyal https://orcid.org/0000-0003-4958-7706
Samir Shukla https://orcid.org/0000-0002-9929-0524
Anurag Singh https://orcid.org/0000-0003-0497-4841
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P2.07
https://doi.org/10.26493/2590-9770.1417.02a
(Also available at http://adam-journal.eu)
On algebraic structure of the Reed-Muller codes
Harinaivo Andriatahiny
Department of Mathematics and Computer Science, Faculty of Sciences,
University of Antananarivo, Madagascar
Received 20 May 2021, accepted 16 August 2021, published online 20 May 2022
Abstract
It is known that the Reed-Muller codes over a prime field may be described as the
radical powers of a modular group algebra. In this paper, we give a new proof of the same
result in a quotient of a polynomial ring. Special elements in a prime field are studied.
An interpolation polynomial is introduced in order to characterize the coefficients of the
Jennings polynomials.
Keywords: Reed-Muller codes, finite field, interpolation polynomial, Jennings basis.
Math. Subj. Class.: 94B05, 16N40, 12E05, 12E20
1 Introduction
Reed-Muller codes are among the oldest known families of codes. They were discovered
by I.S. Reed and D.E. Muller in 1954. These codes were initially given as binary codes,
but generalizations to q-ary were provided with q a prime power. Reed-Muller codes were
studied by many authors (see, e.g. [3, 5, 7, 8, 9, 11, 12]).
These codes form a class of practically important codes. They have found widespread
applications. A powerful Reed-Muller code was used by Mariner 9 to send back clear
pictures from Mars to Earth in 1972.
A great advantage of the Reed-Muller codes is that they are relatively easy to decode
by using majority logic decoding.
One of the interesting properties of the Reed-Muller codes is that there are several
ways to describe them. They may be described by using finite geometries [2]. Group
algebra approach can be used to characterize the Reed-Muller codes. This approach enables
Berman and Charpin to identify the Reed-Muller codes with the radical powers in a suitable
modular group algebra for the binary and p-ary cases. This is the famous theorem of
Berman for the binary case [4]. The p-ary case was treated by Charpin [6]. Many authors
(see, e.g. [1, 9, 10]) have studied this property of Reed-Muller codes.
E-mail address: hariandriatahiny@gmail.com (Harinaivo Andriatahiny)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P2.07
In this paper, we utilize the quotient algebra B = Fp[X0, . . . , Xm−1]/⟨X
p
0 − 1, . . . ,
X
p
m−1 − 1⟩ as the ambient space of the codes. The radical powers of B are linearly gener-
ated by the Jennings bases. We give some properties of special elements of the finite field
Fp. Then, we obtain the coefficients of the Jennings polynomials by means of an appropri-
ate interpolation function. We can apply this fact to show that the radical powers of B are
the Reed-Muller codes of length pm over Fp.
2 Preliminary results
In this section, we give some properties of the following special elements of the field
Fp = Z/pZ = {0, 1, . . . p− 1}
ai,d :=
i∑
j=0
(−1)i−j
(
i
j
)
jd (2.1)
where i and d are integers such that 0 ≤ i, d ≤ p− 1.
Proposition 2.1. We have
ai,d = i
d−1∑
k=0
(
d− 1
k
)
ai−1,k
for all integers d and i such that 1 ≤ i, d ≤ p− 1.
Proof. We have
i∑
j=0
(−1)i−j
(
i
j
)
jd =
i∑
j=1
(−1)i−j
(
i
j
)
jd =
i∑
j=1
(−1)i−j · j ·
(
i
j
)
jd−1.
Since (
i
j
)
=
i
j
(
i− 1
j − 1
)
,
the last expression become
i
i∑
j=1
(−1)i−1−(j−1)
(
i− 1
j − 1
)
((j − 1) + 1)d−1.
By using the relation
((j − 1) + 1)d−1 =
d−1∑
k=0
(
d− 1
k
)
(j − 1)k,
and introducing J = j − 1, the last expression become
i
d−1∑
k=0
(
d− 1
k
)
(
i−1∑
J=0
(−1)i−1−J
(
i− 1
J
)
Jk).
H. Andriatahiny: On algebraic structure of the Reed-Muller codes 3
Proposition 2.2. Let i be an integer such that 1 ≤ i ≤ p− 1. We have
ai,d = 0
for d = 0, 1, ..., i− 1.
Proof. By induction on i:
– for i = 1: thus d = 0, we have, by convention 00 = 1,
(−1)1 ·
(
1
0
)
· 00 + (−1)0 ·
(
1
1
)
· 10 = −1 + 1 = 0
– suppose the assertion is true for i− 1, i.e.
i−1∑
j=0
(−1)i−1−j
(
i− 1
j
)
jd = 0 , for d = 0, 1, ..., i− 2
and let us prove that it is also true for i.
• For d = 0,
i∑
j=0
(−1)i−j
(
i
j
)
= 0.
• Let d be such that 1 ≤ d ≤ i − 1, thus 0 ≤ d − 1 ≤ i − 2; according to
Proposition 2.1,
i∑
j=0
(−1)i−j
(
i
j
)
jd = i
d−1∑
k=0
(
d− 1
k
)
(
i−1∑
j=0
(−1)i−1−j
(
i− 1
j
)
jk)
= i
d−1∑
k=0
(
d− 1
k
)
· 0 = 0.
Proposition 2.3. Let i be an integer such that 1 ≤ i ≤ p− 1. Thus
ai,i ̸= 0.
Proof. By induction on i.
– For i = 1,
(−1)1
(
1
0
)
01 + (−1)0
(
1
1
)
11 = 0 + 1 ̸= 0
– Assume that the assertion holds for i − 1 with 1 ≤ i − 1 ≤ p − 2 (therefore,
2 ≤ i ≤ p− 1), i.e.
i−1∑
j=0
(−1)i−1−j
(
i− 1
j
)
ji−1 ̸= 0
And we have to show that the assertion is also true for i.
Using Proposition 2.1 and Proposition 2.2, we have
i∑
j=0
(−1)i−j
(
i
j
)
ji = i
i−2∑
k=0
(
i− 1
k
)
· 0 + i
i−1∑
j=0
(−1)i−1−j
(
i− 1
j
)
ji−1 ̸= 0.
4 Art Discrete Appl. Math. 5 (2022) #P2.07
3 Interpolation polynomial
In this section, we study the following interpolation polynomial.
Definition 3.1. Taking account of (2.1), we define
Hi(Y ) = (ai,0 − ai,p−1)−
p−2∑
d=0
ai,d Y
p−1−d ∈ Fp[Y ]. (3.1)
Theorem 3.2. We have
deg(Hi(Y )) = p− 1− i (3.2)
where deg denotes the degree of the polynomial.
Proof. This is clear by using Proposition 2.2 and Proposition 2.3 in (3.1).
Recall the following well known Lemmas.
Lemma 3.3. We have
(
p− 1
d
)
= (−1)d mod p
for d = 0, 1, . . . , p− 1.
Proof. It can be proved by induction on d.
Lemma 3.4. For a ∈ Fp, we have
ap−1 =
{
0 if a = 0,
1 if a ̸= 0.
Proof. It is clear that 0p−1 = 0, because p ≥ 2.
For the second case, note that Fp − {0} is a multiplicative group of order p− 1.
Theorem 3.5. For an integer k such that 0 ≤ k ≤ i, we have
Hi(k) = (−1)
i−k
(
i
k
)
.
H. Andriatahiny: On algebraic structure of the Reed-Muller codes 5
Proof. By using (3.1), (2.1) and the Lemma 3.3, we have
Hi(Y ) = (ai,0 − ai,p−1)−
p−2∑
d=0
ai,d Y
p−1−d
= ai,0 −
p−1∑
d=0
ai,d Y
p−1−d
=
i∑
j=0
(−1)i−j
(
i
j
)
−
p−1∑
d=0
(
i∑
j=0
(−1)i−j
(
i
j
)
jd) Y p−1−d
=
i∑
j=0
(−1)i−j
(
i
j
)
−
i∑
j=0
(−1)i−j
(
i
j
)
[
p−1∑
d=0
jd Y p−1−d]
=
i∑
j=0
(−1)i−j
(
i
j
)
−
i∑
j=0
(−1)i−j
(
i
j
)
[
p−1∑
d=0
(−1)d(−j)d Y p−1−d]
=
i∑
j=0
(−1)i−j
(
i
j
)
−
i∑
j=0
(−1)i−j
(
i
j
)
[
p−1∑
d=0
(
p− 1
d
)
(−j)d Y p−1−d]
=
i∑
j=0
(−1)i−j
(
i
j
)
−
i∑
j=0
(−1)i−j
(
i
j
)
(Y − j)p−1
=
i∑
j=0
(−1)i−j
(
i
j
)
[1− (Y − j)p−1].
And by Lemma 3.4, we have the result.
Remark 3.6. For an integer k such that i < k ≤ p− 1, we have
Hi(k) = 0.
4 Application to Reed-Muller codes
In this section, we give a new proof of the theorem of Berman and Charpin. Recall that
Fp = Z/pZ = {0, 1, . . . , p − 1} is the field of p elements with p a prime number. Let m
be a positive integer.
Definition 4.1. A linear code of length pm over Fp is a linear subspace of the vector space
(Fp)
pm = {(c0, c1, . . . , cpm−1) | ct ∈ Fp, for all t}.
We consider the quotient algebra
B = Fp[X0, . . . , Xm−1]/⟨X
p
0 − 1, . . . , X
p
m−1 − 1⟩ (4.1)
where I = ⟨Xp0 −1, . . . , X
p
m−1−1⟩ is the ideal of the polynomial ring Fp[X0, . . . , Xm−1]
generated by X
p
0 − 1, . . . , X
p
m−1 − 1.
We denote
x0 = X0 + I, . . . , xm−1 = Xm−1 + I. (4.2)
6 Art Discrete Appl. Math. 5 (2022) #P2.07
Remark 4.2. Note that x
p
t = 1 = x
0
t for t = 0, . . . ,m − 1. Then, the exponent it in x
it
t
can be viewed as an integer in Z/pZ = {0, 1, . . . , p− 1}.
We have
B =



p−1∑
i0=0
· · ·
p−1∑
im−1=0
ci0,...,im−1x
i0
0 . . . x
im−1
m−1 | ci0,...,im−1 ∈ Fp


 . (4.3)
Let us fix an order on the set of monomials
{
xi00 . . . x
im−1
m−1 | it ∈ Z/pZ, for all t
}
.
Then, we can consider the isomorphism of vector spaces
Φ : B −→ (Fp)
pm
p−1∑
i0=0
· · ·
p−1∑
im−1=0
ci0,...,im−1x
i0
0 . . . x
im−1
m−1 7−→ (ci0,...,im−1)0≤i0,...,im−1≤p−1.
(4.4)
Therefore, B can be considered as the ambient space for the linear codes of length pm over
Fp, and the polynomial
∑p−1
i0=0
· · ·
∑p−1
im−1=0
ci0,...,im−1x
i0
0 . . . x
im−1
m−1 of B can be identified
with the vector (ci0,...,im−1)0≤i0,...,im−1≤p−1 of (Fp)
pm
and vice-versa.
B is a local ring with maximal ideal R which is the radical of B, i.e.
R = rad(B). (4.5)
Let d be an integer such that 0 ≤ d ≤ m(p − 1). Consider the powers Rd of R. We have
the following sequence of ideals:
{0} ⊂ Rm(p−1) ⊂ · · · ⊂ R2 ⊂ R ⊂ B.
For simplicity, in virtue of (4.2) and Remark 4.2, we use the following notations
x := (x0, ..., xm−1),
Y := (Y0, ..., Ym−1),
i := (i0, . . . , im−1) ∈ (Z/pZ)
m,
j ≤ i if jt ≤ it, for all t, with i, j ∈ (Z/pZ)
m,
x
i := xi00 · . . . · x
im−1
m−1 and
⌊i⌋ := i0 + . . .+ im−1.
Definition 4.3. The Jennings polynomial is defined by
Ji(x) := (x0 − 1)
i0 · . . . · (xm−1 − 1)
im−1
with i := (i0, . . . , im−1) ∈ (Z/pZ)
m.
Remark 4.4. (i) A linear basis of Rd over Fp called the Jennings basis of R
d is
Ed := {Ji(x) | i ∈ (Z/pZ)
m, ⌊i⌋ ≥ d} .
H. Andriatahiny: On algebraic structure of the Reed-Muller codes 7
(ii) We have
dimFp(R
d) = card {i ∈ (Z/pZ)m | ⌊i⌋ ≥ d } (4.6)
where dimFp denotes the dimension of the vector space over Fp and card means the
number of elements in the set.
By taking account of the relation (3.1), we have the following definition.
Definition 4.5. For i := (i0, . . . , im−1) ∈ (Z/pZ)
m, we define the interpolation polyno-
mial
Hi(Y) := Hi0(Y0) · . . . ·Him−1(Ym−1) ∈ Fp[Y0, . . . , Ym−1].
Theorem 4.6. For i ∈ (Z/pZ)m, we have
deg(Hi(Y)) = m(p− 1)− ⌊i⌋. (4.7)
Proof. It is obvious by (3.2).
Theorem 4.7. For i ∈ (Z/pZ)m, we have
Ji(x) =
∑
j≤i
Hi(j)x
j.
Proof. By Theorem 3.5, we have
Ji(x) =
m−1∏
t=0
(xt − 1)
it
=
m−1∏
t=0
(
it∑
jt=0
(−1)it−jt
(
it
jt
)
x
jt
t )
=
m−1∏
t=0
(
it∑
jt=0
Hit(jt)x
jt
t )
=
∑
j≤i
(
m−1∏
t=0
Hit(jt))x
j
=
∑
j≤i
Hi(j)x
j.
Remark 4.8. (i) If there is a t such that jt > it, then Hi(j) = 0.
(ii) The polynomial Ji(x) can be identified with the vector (Hi(j))j∈(Z/pZ)m .
Recall that Y := (Y0, ..., Ym−1). Consider the vector space of the reduced polynomials
in m variables over Fp
P (m, p) :=
{
P (Y) ∈ Fp[Y0, ..., Ym−1] | degYt(P ) ≤ p− 1, for all t
}
where degYt(P ) is the degree of the polynomial P (Y) with respect to the variable Yt.
Let ω be an integer such that 0 ≤ ω ≤ m(p − 1). Consider the subspace of P (m, p)
defined by
Pω(m, p) := {P (Y) ∈ P (m, p) | deg(P ) ≤ ω}
8 Art Discrete Appl. Math. 5 (2022) #P2.07
where deg(P ) is the total degree of the polynomial P (Y).
We have the following isomorphism of vector spaces:
Ψ: P (m, p) −→ B
P (Y) 7−→
∑
j∈(Z/pZ)m
P (j)xj (4.8)
Definition 4.9. The Reed-Muller code of length pm over Fp and of order ω (0 ≤ ω ≤
m(p− 1)) is the subspace of (Fp)
pm defined by
RMFp(m,ω) :=
{
(P (j))j∈(Z/pZ)m ∈ (Fp)
pm | P (Y) ∈ Pω(m, p)
}
. (4.9)
Remark 4.10. (i) According to the isomorphisms (4.4) and (4.8), the Reed-Muller code
RMFp(m,ω) is isomorphic to Pω(m, p).
(ii) We have
dimFp(RMFp(m,ω)) = card
{
m−1∏
t=0
Y ett | 0 ≤ et ≤ p− 1,
m−1∑
t=0
et ≤ ω
}
(4.10)
We know give a new proof of the following theorem
Theorem 4.11 (Berman-Charpin). Let ω be an integer such that 0 ≤ ω ≤ m(p − 1). We
have
RMFp(m,ω) = R
m(p−1)−ω
where R is defined in (4.5).
Proof. For simplicity, let d = m(p − 1) − ω. By (4.7), we have deg(Hi(Y)) ≤ ω, for
⌊i⌋ ≥ d. And it follows from Remark 4.4(i), Remark 4.8(ii) and (4.9) that
Rd ⊆ RMFp(m,ω).
It remains to show that dimFp(RMFp(m,ω)) = dimFp(R
d).
By (4.10), we have
dimFp(RMFp(m,ω)) = card { i ∈ (Z/pZ)
m | ⌊i⌋ ≤ ω }
and by (4.6), we have
dimFp(R
d) = card { i ∈ (Z/pZ)m | ⌊i⌋ ≥ d }.
It is clear that the map
θ : (Z/pZ)m −→ (Z/pZ)m
i = (i0, ..., im−1) 7−→ θ( i ) = (p− 1− i0, ..., p− 1− im−1)
is a bijection with θ−1 = θ. And we obtain
⌊θ( i )⌋ =
m−1∑
t=0
p− 1− it = m(p− 1)− ⌊i⌋.
H. Andriatahiny: On algebraic structure of the Reed-Muller codes 9
It follows that ⌊i⌋ = m(p− 1)− ⌊θ( i )⌋.
Thus, we have the following equivalence
⌊i⌋ ≤ ω ⇐⇒ ⌊θ( i )⌋ ≥ m(p− 1)− ω.
This implies that
card { i ∈ (Z/pZ)m | ⌊i⌋ ≤ ω } = card { i ∈ (Z/pZ)m | ⌊i⌋ ≥ m(p− 1)− ω }.
ORCID iDs
Harinaivo Andriatahiny https://orcid.org/0000-0003-4847-3854
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cuments.pub/document/polynomial-codes-and-finite-geometries-ke
yjkeychapterappdf-polynomial-codes.html?page=1.
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[4] S. D. Berman, On the theory of group codes, Cybernetics 3 (1967), 25–31, doi:10.1007/bf01
072842.
[5] I. F. Blake and R. C. Mullin, The Mathematical Theory of Coding, Academic Press [A sub-
sidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975, doi:10.1016/c2
013-0-10384-4.
[6] P. Charpin, Une generalisation de la construction de berman des codes de reed et muller p-aires,
Commun. Algebra 16 (1988), 2231–2246, doi:10.1080/00927878808823689.
[7] P. Delsarte, J.-M. Goethals and F. J. MacWilliams, On generalized Reed-Muller codes and their
relatives, Information and Control 16 (1970), 403–442, doi:10.1016/s0019-9958(70)90214-7.
[8] T. Kasami, S. Lin and W. Peterson, New generalizations of the Reed-Muller codes–I: Primitive
codes, IEEE Trans. Inf. Theory 14 (1968), 189–199, doi:10.1109/tit.1968.1054127.
[9] E. Kouselo, S. Gonsales, V. T. Markov, K. Martines and A. A. Nechaev, Ideal representations
of Reed-Solomon and Reed-Muller codes, Algebra Logic 51 (2012), 297–320, 414, 417, doi:
10.1007/s10469-012-9183-8.
[10] P. Landrock and O. Manz, Classical codes as ideals in group algebras, Des. Codes Cryptogr. 2
(1992), 273–285, doi:10.1007/bf00141972.
[11] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, North Holland,
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[12] A. Poli, Codes stables sous le groupe des automorphismes isométriques de A =
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A1032.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P2.08
https://doi.org/10.26493/2590-9770.1344.20c
(Also available at http://adam-journal.eu)
Deterministic bootstrap percolation on trees
Robert A. Beeler , Rodney Keaton* , Frederick Norwood ,
Department of Mathematics and Statistics, East Tennessee State University,
Johnson City, TN, USA
Received 19 December 2019, accepted 18 August 2021, published online 20 May 2022
Abstract
In a graph, k-bootstrap percolation is a process by which an “infection” spreads from
an initial set of infected vertices, according to the rule that on each iteration an uninfected
vertex with k infected neighbors becomes infected. This process continues until either
every vertex is infected or every uninfected vertex has fewer than k infected neighbors.
We are particularly interested in the case where every vertex is eventually infected. The
cardinality of a smallest set that results in this is the k-bootstrap percolation number of the
graph. In this paper, we determine the k-bootstrap percolation number for trees of small
diameter, spiders, complete N -ary trees, and caterpillars. For these graph families we also
consider the smallest number of iterations needed for any smallest set to spread to the entire
graph. Finally, we give an upper bound for the k-bootstrap percolation number for general
trees which improves upon previous results.
Keywords: Bootstrap Percolation,Trees
Math. Subj. Class.: 05D99,05C05
1 Introduction
Let G be a graph with vertex set V and edge set E. We define the diameter of a graph to
be the maximum distance between any pair of vertices in V and we define the periphery
of a graph to be the subgraph induced by all vertices in V whose distance to some other
vertex in V is equal to the diameter. Since we are primarily concerned with trees, we note
that any vertex on the periphery of a tree is necessarily a leaf (in other words, a vertex of
degree one).
We begin with A0k(G) ⊆ V (G), a collection of infected vertices. On the t
th iteration
we add newly infected vertices to At−1k (G) if they have at least k neighbors in A
t−1
k (G), to
*Corresponding author.
E-mail addresses: beelerr@etsu.edu (Robert A. Beeler), keatonr@etsu.edu (Rodney Keaton),
norwoodr@etsu.edu (Frederick Norwood)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P2.08
form Atk(G). This process is repeated until vertices not in A
t
k(G) have strictly fewer than
k neighbors in Atk(G) or all vertices of G are in A
t
k(G). The above process is called k-
bootstrap percolation. In particular, for a graph G we are interested in the size of a smallest
A0k(G) so that the entire graph is eventually infected. Throughout the paper, we call such
a set a percolating set and we call the size of this set, denoted bpk(G), the k-bootstrap
percolation number of the graph. Among all smallest percolating sets, there is one that
infects all of the vertices of the graph in the minimum number of iterations. We denote this
minimum value tk(G).
We would now like to give a brief (but far from complete) overview of the history
of bootstrap percolation. In 1968, Bollabás considered an edge coloring of graphs [6]
called “weak saturation,” which later came to be called “graph bootstrap percolation” [2].
Bootstrap percolation on vertices was introduced by Chalupa, Leath, and Reich [8]. Their
study was motivated by a problem in magnetic systems and considered only on a special
class of lattices. In the paper by Chalupa et al. and in most subsequent papers on bootstrap
percolation on lattices, the initial set of infected vertices, A0k(G), is chosen at random.
Bootstrap percolation with A0k(G) being chosen at random has also been considered in
[1, 3, 4, 5, 7, 18, 19, 20]. Alternatively, instead of choosing our initial set randomly, we
choose A0k(G) in order to insure that every vertex in a graph is eventually infected. While
this deterministic approach seems to be less common historically, it has been considered
in [9, 10, 12, 13, 22, 23, 24, 25] and the appendix of [4]. We should also mention that
in addition to the standard bootstrap percolation considered in this paper, there are also
several variants. For example, two-way bootstrap percolation, which has been considered
in [21, 27, 28], and the previously mentioned bootstrap percolation on edges, which has
been recently considered in [14].
In this paper, we will be primarily concerned with bootstrap percolation on trees. One
primary motivation for considering trees is that they are minimally connected graphs. With
this in mind, trees are natural to consider in the context of the extremal values of the k-
bootstrap percolation number due to the fact that every graph has a spanning subtree. We
should mention that our paper can most naturally be considered an extension of the work
of Riedl in [25]. In [25], Riedl finds upper and lower bounds on the k-bootstrap percolation
number for all trees1, and uses these bounds to find the precise k-bootstrap percolation
number for certain N -ary trees. In this paper, we begin by obtaining exact values of the k-
bootstrap percolation number for various commonly occuring families of trees. It should be
noted that we reproduce Riedl’s formula for the k-bootstrap percolation number of certain
N -ary trees, though we use a different and more concrete method. Furthermore, our main
result improves upon the upper bound (See Theorem 5.1) appearing in [25].
In Section 2, we make some basic observations which will be used throughout the
remainder of the paper. In Section 3.1, we consider families of trees of small diameter.
In Section 3.2, we consider families of spiders. In Section 3.3, we consider complete N -
ary trees. This result is also given in [25], since in this case the upper and lower bounds
are tight, so that the ceiling of the lower bound is equal to the floor of the upper bound.
In Section 4, we consider caterpillars. In Section 5, we present a sharp upper bound for
bpk(T ) and compare this bound to the bounds given in [25]. In Section 6, we use this
bound to give bpk(T ) for the trees on eleven vertices or less that do not fall into one of our
families. Finally, we give several open problems for future avenues of research in Section 7.
1We note that the main theorem in [25] is stated incorrectly on page 3. However, it is correctly stated in their
abstract.
R. A. Beeler, R. Keaton and F. Norwood: Deterministic bootstrap percolation on trees 3
2 Basic observations
In this section we state several fundamental results which will be useful for the remainder
of the paper.
We begin with four observations about percolating sets that hold for all graphs.
Observation 2.1. Let G be a graph. (i) All vertices of degree less than k must belong to
any percolating set for G. (ii) In k-bootstrap percolation, if u and v are adjacent vertices
such that deg(u) = deg(v) = k, then at least one of u and v must be in every minimum
k-bootstrap set.
Proof. Let v be a vertex of G with degree less than k. If v is not in a percolating set, then
v can never have at least k infected neighbors. So, v will never be infected.
Likewise, suppose that u and v are adjacent vertices of degree k. If u is not in a
percolating set, then v can never have at least k infected neighbors. Reversing the roles of
u and v yields the result.
Clearly, any percolating set for k + 1 is also a percolating set for k. Ergo, the next
proposition follows immediately from Observation 2.1.
Observation 2.2. If G is a connected graph with maximum degree ∆, then
1 = bp1(G) ≤ bp2(G) ≤ · · · ≤ bp∆+1(G) = |V (G)|.
Observation 2.3. We have bp1(G) is the number of connected components in G.
Based on Observation 2.3, we will only consider the case where k ≥ 2 for the remainder
of this paper.
To obtain an upper bound for the k-bootstrap percolation number we consider the k-
domination number of G. The neighborhood of a vertex x, denoted N(x), is the set of
all vertices adjacent to x. If |N(x)| = 1, then x is a leaf. A k-domination set is a set
S ⊆ V (G) such that for all x ∈ V (G), either x ∈ S or |N(x) ∩ S| ≥ k. If among all
k-domination sets, S has the least number of vertices, then S is a minimum k-domination
set. The cardinality of such a set is the k-domination number of G. This number is denoted
γk(G). The k-domination number was introduced by Fink and Jacobson in 1985 [11]. For
more information on domination and its variations, please refer to [16, 17]. If A0k(G) is
equal to a k-domination set for G, then after a single iteration the entire graph will be
infected. From this, the following bound is immediate.
Observation 2.4. For any graph G, bpk(G) ≤ γk(G).
3 Graph families
In this section, we restrict our attention to certain families of trees for which we can derive
the exact k-bootstrap percolation number.
3.1 Trees of small diameter
We give the k-bootstrap percolation number for all trees with diameter less than or equal to
five.
We begin with diameter two. A tree of diameter two is a star. This graph has a center
vertex x adjacent to n leaves, y1,. . . ,yn. This graph is denoted K1,n.
4 Art Discrete Appl. Math. 5 (2022) #P2.08Art Discrete Appl. Math. x (xxxx) #Pn 5
xy1
y2
y3
y1,1
y1,2
y1,3
y2,1 y2,2
y3,1
x1
x2 x3
x4
Figure 1: The graphs and
• if and only if .
• if and only if .
• if and only if .
Thus is the number of with at least leaves and is the number of with at
most leaves.
Theorem 3.3. Let . For , we have the following:
1. If , then and
if and
if and
if and
if and
2. If , then and
if
if or
Proof. By Observation 2.1, for all , , and , and must be in every percolating
set. Further note that . Thus, ,..., must also be in the
initial set. It follows that . Likewise, ,…, will be
infected after one iteration since they have at least neighbors in the initial set.
Similarly, ,…, will be infected in the iteration after is infected.
Suppose that . If and , then will be infected in
one iteration. Since, , then . Thus, the entire graph
is infected and .
If and , then is infected in one iteration and ,…,
are infected in two iterations. Hence, .
If and , then . Since
but , gets infected one iteration after ,…, are infected. Thus
every vertex is infected after two steps and .
Similarly, if and , then .
Since but , is infected one iteration after ,…,
Figure 1: The graphs K1,3(4; 3, 2, 1) and S4,3(3; 3, 2, 1, 1; 4; 4, 2, 1)
Theorem 3.1. Let k ≥ 2. For the star K1,n, we have the following:
1. If n ≤ k − 1, then bpk(K1,n) = n+ 1 and tk(K1,n) = 0.
2. If n ≥ k, then bpk(K1,n) = n and tk(K1,n) = 1.
Proof. By Observation 2.1, all vertices in {y1, . . . , yn} must be in every percolating set. If
deg(x) = n ≤ k − 1, then x must also be in every percolating set and part 1) follows. If
n ≥ k, then x will get infected after one iteration and part 2) follows.
It follows from the previous theorem that the bound in Observation 2.4 is sharp. In
particular, for the star K1,k we have that bpk(K1,k) = k = γk(K1,k).
A tree of diameter three is a double star. This graph has two adjacent central vertices x
and y. The vertex x is adjacent to r leaves, x1,. . . ,xr. The vertex y is adjacent to s leaves,
y1,. . . ,ys. This graph is denoted Sr,s.
Theorem 3.2. Let k ≥ 2 and r ≥ s ≥ 1. For the double star T = Sr,s, we have the
following:
1. If r ≤ k − 2, then bpk(T ) = r + s+ 2 and tk(T ) = 0.
2. If r ≥ k − 1 and s ≤ k − 2, then bpk(T ) = r + s+ 1 and tk(T ) = 1.
3. If r = s = k − 1, then bpk(T ) = r + s+ 1 and tk(T ) = 1.
4. If r ≥ k and s = k − 1, then bpk(T ) = r + s and tk(T ) = 2.
5. If s ≥ k, then bpk(T ) = r + s and tk(T ) = 1.
Proof. By Observation 2.1, all leaves must be initially infected, so bpk(T ) ≥ r + s. If
r ≤ k − 2, then x and y must both be initially infected, i.e., bpk(T ) = r + s + 2 and
tk(T ) = 0. If r ≥ k − 1 and s ≤ k − 2, then y must be initially infected and x is
infected after one iteration, so bpk(T ) = r + s + 1 and tk(T ) = 1. If r = s = k − 1
then either x or y must be initially infected, and the other is infected after one iteration, so
bpk(T ) = r + s + 1 and tk(T ) = 1. If r ≥ k and s = k − 1, then x is infected after one
iteration and y is infected after two iterations, so bpk(T ) = r+ s and tk(T ) = 2. If s ≥ k,
then both x and y are infected after one iteration, so bpk(T ) = r + s and tk(T ) = 1.
R. A. Beeler, R. Keaton and F. Norwood: Deterministic bootstrap percolation on trees 5
Any tree of diameter four can be obtained by appending leaves to the existing vertices
of K1,n, where n ≥ 2. Suppose that we append c leaves to x, namely x1, . . . , xc and
ai ≥ 1 leaves to yi, namely yi,1, . . . , yi,ai for i = 1, . . . , n. The resulting graph will be
denoted K1,n(c; a1, . . . , an). Without loss of generality, assume that a1 ≥ · · · ≥ an ≥ 1.
An example is shown in Figure 1.
There exist non-negative integers p and q such that the following holds:
• ai ≥ k ≥ 2 if and only if i ≤ p.
• ai = k − 1 if and only if p+ 1 ≤ i ≤ n− q.
• ai ≤ k − 2 if and only if i ≥ n− q + 1.
Thus p is the number of yi with at least k leaves and q is the number of yi with at most
k − 2 leaves.
Theorem 3.3. Let k ≥ 2. For T = K1,n(c; a1, . . . , an), we have the following:
(1) If p+ q + c ≥ k, then bpk(T ) = c+ a1 + · · ·+ an + q and
tk(T ) =



1 if q + c ≥ k and n = p+ q
2 if q + c ≥ k and n ≥ p+ q + 1
2 if q + c ≤ k − 1 and n = p+ q
3 if q + c ≤ k − 1 and n ≥ p+ q + 1.
(2) If p+ q + c ≤ k − 1, then bpk(T ) = c+ a1 + · · ·+ an + q + 1 and
tk(T ) =
{
0 if n = q
1 if p ≥ 1 or n ≥ p+ q + 1.
Proof. By Observation 2.1, for all i, j, and ℓ, yi,j and xℓ must be in every percolating set.
Further note that deg(yi) = ai + 1. Thus, yn−q+1,..., yn must also be in the initial set. It
follows that bpk(T ) ≥ c + a1 + · · · + an + q. Likewise, y1,. . . ,yp will be infected after
one iteration since they have at least k neighbors in the initial set. Similarly, yp+1,. . . ,yn−q
will be infected in the iteration after x is infected.
Suppose that p+ q+ c ≥ k. If q+ c ≥ k and n = p+ q, then x will be infected in one
iteration. Since, n = p+ q, then {yp+1, . . . , yn−q} = ∅. Thus, the entire graph is infected
and tk(T ) = 1.
If q + c ≥ k and n ≥ p + q + 1, then x is infected in one iteration and yp+1,. . . ,yn−q
are infected in two iterations. Hence, tk(T ) = 2.
If q + c ≤ k − 1 and n = p+ q, then {yp+1, . . . , yn−q} = ∅. Since p+ q + c ≥ k but
q + c ≤ k − 1, x gets infected one iteration after y1,. . . ,yp are infected. Thus every vertex
is infected after two steps and tk(T ) = 2.
Similarly, if q + c ≤ k − 1 and n ≥ p + q + 1, then {yp+1, . . . , yn−q} ̸= ∅. Since
p+ q+ c ≥ k but q+ c ≤ k−1, x is infected one iteration after y1,. . . ,yp are infected. The
vertices yp+1,. . . ,yn−q are infected one iteration later. Thus every vertex is infected after
three steps and tk(T ) = 3. This proves part (1).
Now, suppose that p+ q + c ≤ k − 1. As before, y1,...,yp, yn−q+1,..., yn, and x1,.., xc
are either in the initial set, or (in the case of y1,...,yp) infected after one step. Hence, x has
6 Art Discrete Appl. Math. 5 (2022) #P2.08
at most k − 1 neighbors that will eventually be infected. Thus x must be in the initial set.
It follows bpk(T ) ≥ c+a1+ · · ·+an+ q+1. Thus if n = q, then p = 0 and every vertex
must be in the initial set. Hence tk(T ) = 0.
If p ≥ 1 or n ≥ p + q + 1, then y1,. . . ,yn−q are infected after one iteration. Thus
tk(T ) = 1. This proves part (2).
Any tree of diameter five can be obtained by appending leaves to the existing vertices
of the double star. We append c1 leaves to x, namely w1,...,wc1 . We append c2 leaves
to y, namely z1,...,zc2 . Similarly, we append ai leaves to xi, namely xi,1,...,xi,ai , and bj
leaves to yj , namely yj,1,...,yj,bj . A diameter five tree with these parameters is denoted
Sr,s(c1; a1, ..., ar; c2; b1, ..., bs) (see Figure 1). Without loss of generality, assume that
a1 ≥ ... ≥ ar ≥ 1 and b1 ≥ .... ≥ bs ≥ 1.
For convenience of notation, define Xi = {xi,1, ..., xi,ai} and Yj = {yj,1, ..., yj,bj} for
i = 1, ..., r and j = 1, ..., s. Note that the set of leaves is
L = {w1, ..., wc1 , z1, ..., zc2} ∪X1 ∪ · · · ∪Xr ∪ Y1 ∪ · · · ∪ Ys.
and that
|L| = c1 + c2 +
r∑
i=1
ai +
s∑
j=1
bj .
Given k ≥ 2, there exist non-negative integers p1, p2, q1, q2 such that the following
holds:
• ai ≥ k if and only if i ≤ p1.
• bj ≥ k if and only if j ≤ p2.
• ai ≤ k − 2 if and only if i ≥ r − q1 + 1.
• bj ≤ k − 2 if and only if j ≥ s− q2 + 1.
Because our result follows in a very similar manner to the proof of Theorem 3.3, we omit
the details of the proof and only provide the initial sets. In each case, it is straightforward
to verify that the set in question is a minimum percolating set. While we have omitted the
time parameter, this can easily be calculated from these sets.
Theorem 3.4. For a given k, the k-bootstrap percolation number of
T = Sr,s(c1; a1, ..., ar; c2; b1, ..., bs) is as follows:
(i) If p1 + q1 + c1 ≤ k− 2 and p2 + q2 + c2 ≤ k− 2, then bpk(T ) = |L|+ q1 + q2 +2.
(ii) If p1+ q1+ c1 = p2+ q2+ c2 = k−1 or at most one of p1+ q1+ c1 or p2+ q2+ c2
is less than or equal to k − 2, then bpk(T ) = |L|+ q1 + q2 + 1.
(iii) If p1 + q1 + c1 ≥ k − 1 and p2 + q2 + c2 ≥ k − 1, with at most one of p1 + q1 + c1
or p2 + q2 + c2 equaling k − 1, then bpk(T ) = |L|+ q1 + q2.
Proof. (i) Take the set L ∪ {xr−q1+1, ..., xr} ∪ {ys−q2+1, ..., ys} ∪ {x, y}.
(ii) If p1 + q1 + c1 ≤ k − 2, then take the set
L ∪ {xr−q1+1, ..., xr} ∪ {ys−q2+1, ..., ys} ∪ {x}.
R. A. Beeler, R. Keaton and F. Norwood: Deterministic bootstrap percolation on trees 7
If p2 + q2 + c2 ≤ k − 2, then take the set
L ∪ {xr−q1+1, ..., xr} ∪ {ys−q2+1, ..., ys} ∪ {y}.
If p1 + q1 + c1 = p2 + q2 + c2 = k − 1, then we can take
L ∪ {xr−q1+1, ..., xr} ∪ {ys−q2+1, ..., ys}
along with either x or y. While either x or y may be chosen, we give the following proce-
dure for choosing x and y which minimizes the number of iterations needed to completely
infect the graph:
Suppose that r − p1 − q1 = 0. Except for y, every neighbor of x is either in the initial
set or (in the case of x1,...xp1 ) infected after one step. Thus, by including y in the initial
set, we guarantee that every vertex is infected after one step (if p1 = 0) or two steps (if
p1 ≥ 1). As either x or y will not be in the initial set, this gives us the minimum number
of iterations. Using an analogous argument, if s − p2 − q2 = 0, then we include x in the
initial set.
Suppose that r − p1 − q1 ≥ 1 and s − p2 − q2 ≥ 1. Note that this means that
{xp1+1, ..., xr−q1} and {yp2+1, ..., ys−q2} are non-empty sets. These sets are infected one
step after their corresponding center vertex. If p1 = 0, then choosing y guarantees that x
is infected on the first step and every vertex is infected in two. Note that choosing x in
the case where r − p1 − q1 ≥ 1, s − p2 − q2 ≥ 1, p1 = 0, and p2 ≥ 1 will result in
{yp2+1, ..., ys−q2} becoming infected after three iterations. Using an analogous argument,
if r − p1 − q1 ≥ 1, s− p2 − q2 ≥ 1, and p2 = 0, then we choose x for our initial set.
Suppose that r − p1 − q1 ≥ 1, s− p2 − q2 ≥ 1, p1 ≥ 1, and p2 ≥ 1. By choosing x to
be in our initial set, x1,...,xr−q1 , y1,...,yp2 are infected after one step, y is infected on the
second iteration, and yp2+1,...,ys−q2 are infected in three steps. By reversing the roles of x
and y, we see that we do no better by choosing y to be in the initial set.
(iii) Choose L ∪ {xr−q1+1, ..., xr} ∪ {ys−q2+1, ..., ys} as our initial set.
3.2 Spiders
In this section, we consider bootstrap percolation on spiders, which are also commonly
referred to as asters or starlike trees. Let x1, . . . , xe, y1, . . . , yo be positive integers with
xi even for 1 ≤ i ≤ e and yj odd for 1 ≤ j ≤ o. We construct a spider, denoted by
S = S(x1, . . . , xe, y1, . . . , yo), as follows. First, S has a single vertex of degree larger
than 2, which we denote by c. We then add an edge from c to a single leaf from each of the
paths Pxi , Pyj for 1 ≤ i ≤ e, 1 ≤ j ≤ o. Note that for k ≥ 3, the k-bootstrap percolation
number is straightforward to determine using Observation 2.1. However, we include it for
completeness.
Proposition 3.5. Suppose k ≥ 3 and let S be as above. Then,
bpk(S) =
{∑e
i=1 xi +
∑o
j=1 yj + 1 if e+ o ≤ k − 1∑e
i=1 xi +
∑o
j=1 yj if e+ o ≥ k,
and
tk(S) =
{
0 if e+ o ≤ k − 1
1 if e+ o ≥ k.
8 Art Discrete Appl. Math. 5 (2022) #P2.08
We now proceed to determine the 2-bootstrap percolation number for S, which is some-
what more involved than the previous result.
Theorem 3.6. Let S = S(x1, . . . , xe, y1, . . . , yo). Then,
bp2(S) =



∑e
i=1
xi
2 + 1 o = 0∑e
i=1
xi
2 +
y1+1
2 + 1 o = 1∑e
i=1
xi
2 +
∑o
j=1
yj+1
2 o ≥ 2,
and
t2(S) =
{
2 o ≥ 2, e ≥ 1
1 otherwise.
Proof. First, we must initially infect all leaves of S. Second, for each path attached to the
center vertex, we initially infect every other vertex starting from the leaf. Note that this will
completely infect each odd length path after one step.
Case 1: o ≥ 2.
In this case, the initially infected vertices given above also infect the center vertex c
after one step. Note, if e = 0, then we are finished.
Now, suppose that e ≥ 1. Then, after one step each even length path will have one
infected end point and the other end point will be attached to the infected center vertex.
Thus, the entire graph is infected after two steps. Therefore,
∑e
i=1
xi
2 +
∑o
j=1
yj+1
2 vertices
are initially infected. If we initially infect fewer than this number vertices, then there will
be either a leaf or a vertex of degree two which is never infected. This yields the result in
this case.
Case 2: o ≤ 1.
In this case, the initially infected vertices will not infect the center vertex c. Thus, we
must initially infect one additional vertex, and we see that initially infecting c will ensure
that all vertices of the even length paths are infected after one step. Then
∑e
i=1
xi
2 +∑o
j=1
yj+1
2 + 1 are initially infected and this is the smallest possible percolating set. This
yields the result in this case.
3.3 N -ary trees
In this section we consider N -ary trees and give a formula for the k-bootstrap percolation
number of a complete N -ary tree.
For N ≥ 1, we say that a tree T is an N -ary tree of height h if T is a rooted tree in
which each vertex has no more than N children and no child can be further than distance
h from the root. Note that when N = 1, a N -ary tree of height h is simply a path on h+ 1
vertices. For example, the path P2 is a 1-ary tree of height h = 1. With this in mind we
begin with the following result.
Theorem 3.7. For a path on n vertices, Pn, we have
bpk(Pn) =
{
⌈n+12 ⌉ if k = 2
n if k ≥ 3,
and
tk(Pn) =
{
1 if k = 2 and n ≥ 3
0 if k ≥ 3 or n = 1 or n = 2.
R. A. Beeler, R. Keaton and F. Norwood: Deterministic bootstrap percolation on trees 9
Art Discrete Appl. Math. x (xxxx) #Pn 9
Figure 2: The complete binary tree,
Proof. First, suppose that . Then, the degree of each vertex of is less than
. By Observation 2.1, a percolating set for contains every vertex of . So
.
Second, suppose that . Note, if is of size ,
then there must be a vertex in which does not have two neighbors
in . Such a vertex would never be infected. Thus, a percolating set has
cardinality at least .
Now, label the vertices of by . Define the following subset of
,
if
if
Note, . Furthermore, it is clear that after a single iteration, every
vertex of will be infected. Thus, and if .
We now consider the case where . An -ary tree is complete if every
vertex has either 0 or children and all leaves are distance from the root. Note,
for fixed and , there is one complete -ary tree of height . We denote this
graph . A complete 2-ary tree (also called a binary tree) of height three is given
in Figure 2.
Furthermore, we have that the number of vertices of is
and the number of leaves is . For convenience, we will denote the root vertex of
by , the set of children of by , and so on until we have that set of all
leaves of is denoted . We now present the main result of this section.
Theorem 3.8. Let . Then,
if
if
if
if
Figure 2: The complete binary tree, T2,3
Proof. First, suppose that k ≥ 3. Then, the degree of each vertex of Pn is less than k. By
Observation 2.1, a percolating set for Pn contains every vertex of Pn. So bpk(Pn) = n.
, sup ose that k = 2. Note, if A02(Pn) ⊆ V (Pn) is of size ⌈
n+1
2 ⌉−1, then there
must be a vertex in V (Pn)\A02(Pn) which does not have tw neighbors in A
0
2(Pn). Such
a vertex would never be infected. Thus, a percolating set has cardinality at least ⌈n+12 ⌉.
Now, label the vertices of Pn by {v1, . . . , vn}. Define the following subset of V (Pn),
A02(Pn) =
{
{v1, v3, . . . , vn−3, vn−1, vn} if n ≡ 0 (mod 2)
{v1, v3, . . . , vn−2, vn} if n ≡ 1 (mod 2).
Note, |A02(Pn)| = ⌈
n+1
2 ⌉. Furthermore, it is clear that after a single iteration, every vertex
of Pn will be infected. Thus, bpk(Pn) = ⌈
n+1
2 ⌉ and tk(Pn) = 1 if n ≥ 3.
We now consider the case where N > 1. An N -ary tree is complete if every vertex has
either 0 or N children and all leaves are distance h from the root. Note, for fixed N and
h, there is one complete N -ary tree of height h. We denote this graph TN,h. A complete
2-ary tree (also called a binary tree) of height three is given in Figure 2.
Furthermore, we have that the number of vertices of TN,h is
|V (TN,h)| =
h∑
i=0
N i =
Nh+1 − 1
N − 1
,
and the number of leaves is Nh. For convenience, we will denote the root vertex of TN,h
by v0, the set of children of v0 by S1, and so o u til we have that set of all leaves of TN,h
is denoted Sh. We now present the main result of this section.
Theorem 3.8. Let k,N ≥ 2. Then,
bpk(TN,h) =



Nh if k ≤ N
Nh+2−1
N2−1 if k = N + 1, h ≡ 0 (mod 2)
Nh+2−1
N2−1 +
N
N+1 if k = N + 1, h ≡ 1 (mod 2)
Nh+1−1
N−1 if k ≥ N + 2,
10 Art Discrete Appl. Math. 5 (2022) #P2.08
and
tk(TN,h) =



h if k ≤ N
1 if k = N + 1
0 if k ≥ N + 2.
Proof. First, suppose that k ≤ N . By Observation 2.1, each leaf must be in A0k(TN,h), i.e.,
Sh ⊆ A
0
k(TN,h), so bpk(TN,h) ≥ N
h. Furthermore, since each non-leaf has N children,
we see that after one iteration all of the vertices in Sh−1 will be infected, after a second
iteration all of the vertices in Sh−2 will be infected, and the process repeats h times until the
entire tree is infected. Hence, we have only the leaves in A0k(TN,h), so bpk(TN,h) = N
h
and tk(TN,h) = h.
Second, suppose that k ≥ N + 2. Then, every vertex of TN,h has degree strictly less
than k, hence every vertex of TN,h must be in A0k(TN,h). Since |V (TN,h)| =
Nh+1−1
N−1 , the
result follows.
Finally, suppose that k = N + 1. Since every vertex of degree strictly less than k must
be in A0k(TN,h), we have Sh ∪ {v0} ⊆ A
0
k(TN,h). We begin by proving that
bpk(TN,h) ≤
{
Nh+2−1
N2−1 if h ≡ 0 (mod 2)
Nh+2−1
N2−1 +
N
N+1 if h ≡ 1 (mod 2).
Suppose that h is even. Let A0k(TN,h) = Sh ∪ Sh−2 ∪ · · · ∪ S2 ∪ {v0}. Then, after a
single iteration we have that every vertex in TN,h is infected. Thus,
bpk(TN,h) ≤ |A
0
k(TN,h)| =
h
2∑
i=0
|S2i| =
h
2∑
i=0
N2i =
Nh+2 − 1
N2 − 1
.
Suppose that h is odd. Let A0k(TN,h) = Sh ∪ Sh−2 ∪ · · · ∪ S3 ∪ S1 ∪ {v0}. Then, after a
single iteration we have that every vertex in TN,h is infected. Thus,
bpk(TN,h) ≤ 1 +
h−1
2∑
i=0
|S2i+1| = 1 +
h−1
2∑
i=0
N2i+1 =
Nh+2 − 1
N2 − 1
+
N
N + 1
.
We want to show that the set A0k(TN,h) above has the smallest possible size. Every
edge can be used at most once to infect a neighboring vertex, and at least N +1 edges must
be used to infect one vertex. The number of edges in T (N, h) is N
h+1
−1
N−1 − 1 =
Nh+1−N
N−1 ,
therefore at most ⌊N
h+1
−N
N−1
1
N+1⌋ = ⌊
Nh+1−N
N2−1 ⌋ new vertices can be infected. Therefore
the cardinality of the percolating set must be at least N
h+1
−1
N−1 −⌊
Nh+1−N
N2−1 ⌋. If h is even, then
Nh+1−N
N2−1 = N
∑h
2
−1
i=0 N
2i is an integer, and this lower bound on the size of the k-bootstrap
set equals the upper bound above. If h is odd, then N
h+1
−N
N2−1 = N
∑h−3
2
i=0 N
2i+1 + NN+1 is
not an integer. In this case, taking the floor reduces the total by NN+1 , again giving a lower
bound which equals the upper bound above.
Therefore, the given example of a percolating set is minimum.
Recall that in Observation 2.4, we showed that bpk(G) ≤ γk(G). To see that γk(G)−
bpk(G) can be made arbitrarily large, consider Tk,h, where h is sufficiently large. As
R. A. Beeler, R. Keaton and F. Norwood: Deterministic bootstrap percolation on trees 11
Art Discrete Appl. Math. x (xxxx) #Pn 11
v0,1 v0,2 v0,3 v2,1 v2,2
v0
v1 v2 v3
v0,4 v0,5 v0,6 v1,1 v2,3 v2,4 v3,2 v3,3
v3,1
Figure 3: The caterpillar
large. As shown in Theorem 3.8, . However,
To see this, note that a -domination set of minimum size must contain the leaves
of the tree and every vertex that is of even distance from its closest leaf. Hence as
increases, this difference becomes arbitrarily large. It is interesting to note that
when is even and when is
odd. This shows that the difference in Observation 2.2 can be
made arbitrarily large.
4 Caterpillars
In this section, we give a closed formula for the -bootstrap percolation number of
a caterpillar. A caterpillar is obtained from the path on vertices by appending
leaves to the existing vertices of the path. The vertices of the original path, which
are called the spine of the caterpillar, are labeled ,…, in the natural way, and
we call the spine length. We append leaves to for . The caterpillar
with parameters , ,…, will be denoted (see Figure 3). Without
loss of generality, we will assume that for , .
For the caterpillar , our initial percolating set must contain
every vertex of degree less than by Observation 2.1. Thus for , this set must
contain every leaf. Further, it must contain all such that for .
Likewise, if , then is in the set. Similarly, if , then is in
the set. Note that if , then will not be included in our percolating set as
these vertices will be infected after one step.
The above discussion tells us nothing about the following vertices:
• if .
• if .
• if and .
We call these vertices sensitive. We partition the sensitive vertices into two sets,
and , as follows. We let consist of all satisfying and .
We let consist of all satisfying and .
Figure 3: The caterpillar P4(6, 1, 4, 3)
shown in Theorem 3.8, bpk(Tk,h) = k
h. However,
γk(Tk,h) =
⌊h/2⌋∑
i=0
kh−2i =
{
kh+2−1
k2−1 h ≡ 0 (mod 2)
kh+2−1
k2−1 +
k
k+1 − 1 h ≡ 1 (mod 2).
To see this, note that a k-domination set of mini um size must contain the leaves of the
tree and every vertex that is of even distance from its closest leaf. Hence as h increases, this
difference becomes arbitrarily large. It is interesting to note that γk(Tk,h) = bpk+1(Tk,h)
when h is even and γk(Tk,h) + 1 = bpk+1(Tk,h) when h is odd. This shows that the
difference bpk+1(G)− bpk(G) in Observation 2.2 can be made arbitrarily large.
4 Caterpillars
In this section, we give a closed formula for the k-bootstrap percolation number of a cater-
pillar. A caterpillar is obtained from the path on r vertices by a pending leaves to the
existing vertices of the path. The vertices of the original path, which are called the spi e of
the caterpillar, are labeled v1,. . . ,vr in the natural way, and we call r the spine length. We
append xi leaves to vi for 1 ≤ i ≤ r. The caterpillar with parameters r, x1,. . . ,xr will be
denoted Pr(x1, . . . , xr) (see Figure 3). Without loss of generality, we will assume that for
i ∈ {1, r}, xi ≥ 1.
For the caterpillar C = Pr(x1, . . . , xr), our initial percolating set must contain every
vertex of d gree less than k by Observation 2.1. Thus for k ≥ 2, this set t tain
every leaf. Further, it must contai all vi such th t xi ≤ k − 3 for 1 ≤ i ≤ r. Likewise, if
x1 ≤ k − 2, then v1 is in t e set. Similarly, if xr ≤ k − 2, then vr is in the set. Note that if
xi ≥ k, then vi will not be included in our p rcolating set as these vertices will be infected
after one step.
The abov discussion tells us nothing about the following vertices:
• v1 if x1 = k − 1.
• vr if xr = k − 1.
• vi if xi ∈ {k − 2, k − 1} and 2 ≤ i ≤ r − 1.
We call these vertices sensitive. We partition the sensitive vertices into two sets, S1 and S2,
as follows. We let S1 consist of all vi satisfying xi = k − 1 and 1 ≤ i ≤ r. We let S2
consist of all vi satisfying vi = k − 2 and 2 ≤ i ≤ r − 1.
12 Art Discrete Appl. Math. 5 (2022) #P2.08
Consider the subgraph induced by S1 ∪ S2. Label the connected components of this
subgraph L1,. . . ,Lm. We call these connected components sensitive strings. By definition,
two sensitive strings are separated by at least one vertex whose inclusion in the initial set is
decided according to the above discussion. For this reason, we may consider each sensitive
string individually. Our goal for each sensitive string is to determine the minimum number
of vertices to include in our initial set so that the entire string is eventually infected. We
denote this number w(Li) for 1 ≤ i ≤ m.
Lemma 4.1. Let k ≥ 2, let C = Pr(x1, . . . , xr) be a caterpillar, L1, . . . , Lm be the
sensitive strings in C, and w(Li) for 1 ≤ i ≤ m be as above. Then, for 1 ≤ i ≤ m we
have
1. If v1 /∈ V (Li) and vr /∈ V (Li), then
w(Li) =
⌊
|V (Li) ∩ S2|
2
⌋
.
2. If v1 ∈ V (Li) or vr ∈ V (Li) but {v1, vr} ̸⊆ V (Li), then
w(Li) =
⌊
1 + |V (Li) ∩ S2|
2
⌋
.
3. If v1 ∈ V (Li) and vr ∈ V (Li), then
w(Li) =
⌊
2 + |V (Li) ∩ S2|
2
⌋
.
Proof. Let 1 ≤ i ≤ m be fixed and let S = V (Li) ∩ S2 = {s1, ..., st} be a sequence. To
prove part 1), we choose for A0k(C) every other sj beginning with s2. It is necessary to
initially infect every other vertex in S because if two vertices in S are not initially infected,
they must have an infected vertex between them by Observation 2.1. We choose to begin
with s2 because S is flanked by vertices that are either initially infected or will eventu-
ally become infected. Now, consider a connected component of the subgraph induced by
V (Li) ∩ S1. In Case 1, where Li contains no endpoint of the spine, if this component of
S1 lies to the left of s1 or to the right of st it will be eventually infected by the vertex to
its left (right). If, on the other hand, it lies between two connected components of S then
either the sj to its left or the sj to its right will have an even subscript and eventually infect
all of its vertices2. This establishes part 1).
To prove part 2), we assume without loss of generality that v1 ∈ V (Li) and vr /∈
V (Li). Note that Li is adjacent on the right to a vertex w which is either in our initial set or
will be infected eventually. Hence, our result will follow in a similar manner to the proof
of part 1). However, the appropriate set of vertices to include from S is now
S′ =
{
{s1, s3, . . . , st−1} if t ≡ 0 (mod 2)
{s1, s2, s4 . . . , st−1} if t ≡ 1 (mod 2).
2If the size of S is even, then we could just as well have initially infected the sj with odd subscripts. If the
size of S is odd, then initially infecting the si with even subscripts is necessary for the number of initially infected
vertices to be minimum.
R. A. Beeler, R. Keaton and F. Norwood: Deterministic bootstrap percolation on trees 13
Note that this gives us the desired result of
w(Li) =
⌊
1 + |V (Li) ∩ S2|
2
⌋
.
As for part 3), note that if v1 ∈ V (Li) and vr ∈ V (Li), then the entire spine is a
sensitive string. Hence, our result will follow in a similar manner to the proof of part 1).
However, the appropriate set of vertices to include from V (Li) ∩ S2 is now
S′ =
{
{s1, s3, . . . , st−1, st} if t ≡ 0 (mod 2)
{s1, s3, . . . , st} if t ≡ 1 (mod 2).
Note that this gives us the desired result of
w(Li) =
⌊
2 + |V (Li) ∩ S2|
2
⌋
.
The final remaining case is when the entire spine of the caterpillar is in S1, so that
m = 1 and S is empty. In this case, we choose the middle vertex (or one of the two middle
vertices) in S1 to include in the initially infected set. The formula in part 3) gives the
correct weight of the spine as 1.
Combining Observation 2.1 and Lemma 4.1, we obtain the main result of this section.
Note that a caterpillar with spine length one is a star. The k-bootstrap percolation number
of such a caterpillar was given in Theorem 3.1. For this reason, we assume that r ≥ 2. For
convenience of exposition, we let d≤ℓ(C) denote the number of vertices in C of degree less
than or equal to ℓ.
Theorem 4.2. Let k ≥ 2 and let C = Pr(x1, . . . , xr) with r ≥ 2. Let L1,. . . ,Lm be the
sensitive strings in C. For each Li we let w(Li) be as above. The k-bootstrap percolation
number of the caterpillar is
bpk(C) =
m∑
i=1
w(Li) + d≤k−1(C).
Proof. This is a straightforward combination of Observation 2.1 and Lemma 4.1.
Note that a double star is a caterpillar with a spine of length two, so this gives an
alternate proof of Theorem 3.2. Moreover, we can use the above result to show that the
bound from Observation 2.4 is sharp. Consider the caterpillar Pn(t, . . . , t), where k ≥
4 and t ≤ k − 3. Every vertex has degree less than k. Hence, every vertex must be
in a k-domination set and in a percolating set. We also mention that we omit the time
parameter in this setting due to the length and tedium of the required calculation as well as
the complexity of the resulting formula.
5 An upper bound
In this section we present a sharp upper bound for the k-bootstrap percolation number of a
tree. We then compare this bound to other known bounds.
14 Art Discrete Appl. Math. 5 (2022) #P2.08
Before stating the theorem, we define the following notation. Recall that d≤k(T ) is the
number of vertices in T of degree less than or equal to k. Similarly, we let dk(T ) be the
number of degree k vertices in T , and we let d≥k(T ) be the number of vertices in T of
degree greater than or equal to k. Furthermore, for a vertex s ∈ V (T ), we set ℓ(s) to be
the number of leaves adjacent to s.
Theorem 5.1. Let T be a tree and k ≥ 2. Then, bpk(T ) ≤ d≤k−1(T ) +
⌊
dk(T )
2
⌋
.
Proof. We proceed by induction on n, the number of vertices of the tree T .
Up to isomorphism, there is only one tree on two vertices and one tree on three vertices.
The result is easily verified in both cases.
Suppose for induction that the result is true for all trees with at most n vertices.
Let T be a tree with n + 1 vertices and choose a leaf v ∈ V (T ) on the periphery of
T with unique neighbor s. As shown in Theorem 3.1, this result holds for stars. For this
reason, we will assume that T is not a star. Note, since v is on the periphery and T is not a
star, we have that deg(s) = ℓ(s) + 1. We now consider several cases.
Case 1: ℓ(s) ≤ k − 2 or ℓ(s) ≥ k + 1.
In this case, we remove the leaf v from T and denote the resulting tree T ′. Note,
d≤k−1(T
′) = d≤k−1(T ) − 1 and dk(T ′) = dk(T ). By the induction hypothesis, T ′ has a
percolating set, denoted S, of cardinality at most
d≤k−1(T
′) +
⌊
dk(T
′)
2
⌋
= d≤k−1(T )− 1 +
⌊
dk(T )
2
⌋
.
Thus, S ∪{v} eventually infects all of T and has cardinality at most d≤k−1(T )+
⌊
dk(T )
2
⌋
.
Case 2: ℓ(s) = k − 1 and dk(T ) is even.
In this case, we remove the leaf v from T and denote the resulting tree T ′. Note,
d≤k−1(T
′) = d≤k−1(T ) and dk(T ′) = dk(T ) − 1 since the degree of s in T is k and has
been decreased by one in T ′. By the induction hypothesis, T ′ has a percolating set, denoted
S, of cardinality at most
d≤k−1(T
′) +
⌊
dk(T
′)
2
⌋
= d≤k−1(T ) +
⌊
dk(T )− 1
2
⌋
= d≤k−1(T ) +
⌊
dk(T )
2
⌋
− 1,
where we have used that dk(T ) is even in the second equality. Thus, S ∪ {v} eventually
infects all of T and has cardinality at most d≤k−1(T ) +
⌊
dk(T )
2
⌋
.
Case 3: ℓ(s) = k − 1 and dk(T ) is odd.
First, we label the leaves of s by L = {v = v1, v2, . . . , vk−1}. Furthermore, let
t ∈ V (T ) be a non-leaf with st ∈ E(T ), which is possible since T is not a star. We
now remove s and its k − 1 adjacent leaves from T to obtain a tree T ′.
If deg(t) ≤ k − 1 or deg(t) ≥ k + 2 in T , then d≤k−1(T ′) = d≤k−1(T ) − (k − 1)
and dk(T ′) = dk(T )− 1. This follows because we have removed k − 1 leaves, a vertex of
degree k, and decreased the degree of t by one. Then, T ′ has a percolating set, denoted S,
R. A. Beeler, R. Keaton and F. Norwood: Deterministic bootstrap percolation on trees 15
of cardinality at most
d≤k−1(T
′) +
⌊
dk(T
′)
2
⌋
= d≤k−1(T )− (k − 1) +
⌊
dk(T )− 1
2
⌋
= d≤k−1(T )− (k − 1) +
⌊
dk(T )
2
⌋
,
where we have used that dk(T ) is odd in the second equality. Note, t ∈ S, and hence S∪L
eventually infects all of T and has cardinality at most d≤k−1(T ) +
⌊
dk(T )
2
⌋
.
If deg(t) = k, then d≤k−1(T ′) = d≤k−1(T )− (k − 2) and dk(T ′) = dk(T )− 2. This
follows because we have removed k − 1 leaves, a vertex of degree k, and decreased the
degree of t by one. Then, T ′ has a percolating set, denoted S, of cardinality at most
d≤k−1(T
′) +
⌊
dk(T
′)
2
⌋
= d≤k−1(T )− (k − 2) +
⌊
dk(T )− 2
2
⌋
= d≤k−1(T )− (k − 1) +
⌊
dk(T )
2
⌋
.
Note, t ∈ S, and hence S ∪ L eventually infects all of T and has cardinality at most
d≤k−1(T ) +
⌊
dk(T )
2
⌋
.
If deg(t) = k + 1, then d≤k−1(T ′) = d≤k−1(T )− (k − 1) and dk(T ′) = dk(T ). This
follows because we have removed k − 1 leaves, a vertex of degree k, and decreased the
degree of t by one. Then, T ′ has a percolating set, denoted S, of cardinality at most
d≤k−1(T
′) +
⌊
dk(T
′)
2
⌋
= d≤k−1(T )− (k − 1) +
⌊
dk(T )
2
⌋
.
Note, as S eventually infects all of T ′, we have that t will eventually be infected, and hence
S ∪ L eventually infects all of T and has cardinality at most d≤k−1(T ) +
⌊
dk(T )
2
⌋
.
Case 4: ℓ(s) = k and dk(T ) is even.
In this case, we remove the leaf v from T and denote the resulting tree T ′. Note,
d≤k−1(T
′) = d≤k−1(T )− 1 and dk(T ′) = dk(T ) + 1 since the degree of s in T is k + 1
and has been decreased by one in T ′. By the induction hypothesis, T ′ has a percolating set,
denoted S, of cardinality at most
d≤k−1(T
′) +
⌊
dk(T
′)
2
⌋
= d≤k−1(T )− 1 +
⌊
dk(T ) + 1
2
⌋
= d≤k−1(T )− 1 +
⌊
dk(T )
2
⌋
,
where we have used that dk(T ) is even in the second equality. Thus, S ∪ {v} eventually
infects all of T and has cardinality at most d≤k−1(T ) +
⌊
dk(T )
2
⌋
.
Case 5: ℓ(s) = k and dk(T ) is odd.
In this case, we remove two leaves from T , say v and w, which are both supported by s,
and denote the resulting tree T ′. Note, d≤k−1(T ′) = d≤k−1(T ) − 1 and dk(T ′) = dk(T )
16 Art Discrete Appl. Math. 5 (2022) #P2.08
since the degree of s in T is k + 1 and has been decreased by two in T ′. By the induction
hypothesis, T ′ has a percolating set, denoted S, of cardinality at most
d≤k−1(T
′) +
⌊
dk(T
′)
2
⌋
= d≤k−1(T )− 1 +
⌊
dk(T )
2
⌋
.
Note, s ∈ S since deg(s) = k − 1 in T ′. Thus, (S ∪ {v, w})\{s} eventually infects all of
T and has cardinality at most d≤k−1(T ) +
⌊
dk(T )
2
⌋
.
In all cases we have shown that T has a percolating set of cardinality at most d≤k−1(T )+⌊
dk(T )
2
⌋
, and hence bpk(T ) ≤ d≤k−1(T )+
⌊
dk(T )
2
⌋
. The proof follows by induction.
We now make a few observations concerning the above bound.
1. The above bound is sharp for paths when k = 2 and for the family of caterpillars of
the form Pn(k − 2, k − 2, . . . , k − 2, k − 2), where k ≥ 3.
2. We can make the difference d≤k−1(T ) +
⌊
dk(T )
2
⌋
− bpk(T ) arbitrarily large using
the family of caterpillars Pn(k, k − 2, k, k − 2, . . . , k − 2, k).
3. For a connected graph G, we can remove edges from G to obtain a spanning tree T
of G. Then the inequality bpk(G) ≤ bpk(T ) combined with the above upper bound
gives an upper bound for bpk(G).
We conclude this section by comparing our above result with the bounds obtained by
Riedl in [25]. The upper and lower bounds for bpk(T ) given by Riedl can be found in
Proposition 3 (lower bound) and Theorem 4 (upper bound) of [25] and are given by
(k − 1)n+ 1
k
≤ bpk(T ) ≤
kn+ d≤k−1(T )
k + 1
, (5.1)
where n is the order of the tree and d≤k−1(T ) is defined before the statement of Theo-
rem 5.1. It should be noted that our quantity bpk(T ) is denoted in [25] as m(T, k). More-
over, the upper bound given in [25] is actually an upper bound for a different, but larger,
quantity than bpk(T ).
We first mention that following the statement of Proposition 3 in [25], Riedl mentions
that for k = 2 his bound is sharp for odd length paths, and for k > 2 his lower bound
is sharp for complete k-ary trees and complete k − 1-ary trees. Note, this is precisely the
cases of Theorem 3.8 with k = N,N + 1.
With regards to the bound in Theorem 5.1, we have that this is equal to the lower
bound in Equation 5.1 for paths of odd length when k = 2. Moreover, by writing n =
d≤k−1(T ) + d≥k(T ), we can rewrite the upper bound in Equation 5.1 as
bpk(T ) ≤ d≤k−1(T ) +
kd≥k(T )
k + 1
.
As d≥k(T ) ≥ dk(T ) and k > 1 we have
d≤k−1(T ) +
kd≥k(T )
k + 1
≥ d≤k−1(T ) +
kdk(T )
k + 1
≥ d≤k−1(T ) +
⌊
dk(T )
2
⌋
,
which is precisely the upper bound in Theorem 5.1. Hence, Theorem 5.1 gives an improve-
ment upon the upper bound in [25].
R. A. Beeler, R. Keaton and F. Norwood: Deterministic bootstrap percolation on trees 17
6 Trees of small order
In this section, we use the above results to complete the characterization of trees on eleven
vertices or less. Throughout, we denote such a tree by T .
There are 201 non-isomorphic trees on ten vertices or less (see Harary [15] or Stein-
bach’s “Field Guide to Simple Graphs” [26]). All but seven of these can be classified as
spiders, caterpillars, or trees of diameter at most five. These seven trees all have degree
sequence [3, 3, 2, 2, 2, 2, 1, 1, 1, 1]. Riedl’s lower bound (Equation 5.1) shows that the 2-
bootstrap percolation number satisfies bp2(T ) ≥ 6. The bound given in Theorem 5.1
shows that bp2(T ) ≤ 6. Therefore, bp2(T ) = 6 and t2(T ) ≤ 2 in these cases. Except
for the four cases in which the two vertices of degree three are adjacent, we have that
bp3(T ) = 8 and t3(T ) = 1. In the four cases in which the two vertices of degree three are
adjacent, we have that bp3(T ) = 9 and t3(T ) = 1 by Observation 2.1.
As for the 235 non-isomorphic trees on eleven vertices, all but 42 of these can be
classified as spiders, caterpillars, or trees of diameter at most five. Note that Riedl’s lower
bound guarantees that bp2(T ) ≥ 6 and bp3(T ) ≥ 8.
Fifteen of these have degree sequence [3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1]. In these cases, the
bound given in Theorem 5.1 shows that bp2(T ) ≤ 6. Hence bp2(T ) = 6. The trivial
lower bound given by Observation 2.1 shows that bp3(T ) ≥ 9 while the bound given in
Theorem 5.1 shows that bp3(T ) ≤ 10. It is straightforward to check that bp3(T ) = 9 for
these trees if and only if their two vertices of degree three are not adjacent. Otherwise, we
have that bp3(T ) = 10.
Thirteen of these have degree sequence [3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1]. Again, the bound
given in Theorem 5.1 shows that bp2(T ) ≤ 6. Combining this with Riedl’s lower bound
yields bp2(T ) = 6. Further, due to Riedl and Theorem 5.1, we have that 8 ≤ bp3(T ) ≤ 9.
Of these, only one has no two vertices of degree three adjacent. Therefore, bp3(T ) = 8 in
this case. For the remaining twelve, bp3(T ) = 9 due to Observation 2.1.
The fourteen remaining trees have degree sequence [4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1]. Com-
bining Riedl’s bound with Theorem 5.1 yields 6 ≤ bp2(T ) ≤ 7. It is straightforward to
check that eight of these have bp2(T ) = 6 and the remaining six have bp2(T ) = 7. Note
that the trivial lower bound and Theorem 5.1 guarantee that bp3(T ) = 9. Observation 2.1
shows that bp4(T ) = 10 in all of these cases.
7 Open problems
In this section, we give open problems related to this study as possible avenues for future
research.
Suppose that we want every vertex to be infected within t iterations. Among all k-
bootstrap sets that will infect the graph within t iterations, choose one with minimum car-
dinality. What is the cardinality of such a set?
Suppose that we limit the size of the initial set. What is the maximum number of
vertices that can be infected? How is this maximum changed if we also limit the number
of iterations?
ORCID iDs
Robert A. Beeler https://orcid.org/0000-0001-9818-5020
Rodney Keaton https://orcid.org/0000-0002-7837-7276
18 Art Discrete Appl. Math. 5 (2022) #P2.08
Frederick Norwood https://orcid.org/0000-0003-1966-6307
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P2.09
https://doi.org/10.26493/2590-9770.1393.2be
(Also available at http://adam-journal.eu)
On distance signless Laplacian spectra of power
graphs of the integer modulo group*
Bilal A. Rather , Shariefuddin Pirzada†
Department of Mathematics, University of Kashmir, Srinagar 190006, India
Tariq A. Naikoo
Islamia College of Science and Commerce, Srinagar 190001, Kashmir, India
Received 12 November 2020, accepted 20 December 2021, published online 19 July 2022
Abstract
For a finite group G, the power graph P(G) is a simple connected graph whose vertex
set is the set of elements of G and two distinct vertices are adjacent if and only if one is a
power of the other. In this article, we obtain the distance signless Laplacian spectrum of
power graphs of the integer modulo groups Zn. We characterize the values of n, for which
power graphs of Zn is distance signless Laplacian integral.
Keywords: Signless Laplacian matrix, distance signless Laplacian matrix, finite groups, power graphs.
Math. Subj. Class.: 05C50, 05C25, 15A18
1 Introduction
A graph G = G(V (G), E(G)) consists of the vertex set V (G) = {v1, v2, . . . , vn} and an
edge set E(G). The cardinalities of V (G) and E(G) are called the order and the size of
G and are taken as n and m. The set of vertices incident on v ∈ V (G), denoted by N(v),
is the neighborhood of v. The degree of v, denoted by dv , is the cardinality of N(v). A
graph G is said to be regular if degree of each vertex is same. We assume all our graphs are
connected and simple. Our notations are standard and are taken from [16].
The adjacency matrix A = (aij) of G is an n× n matrix whose (i, j)-entry is equal to
1, if vi is adjacent to vj and equal to 0, otherwise. Let D(G) = diag(d1, d2, . . . , dn) be the
*We are thankful to the anonymous referee for his valuable suggestions.
†Corresponding author. The author is supported by the SERB-DST research project number
MTR/2017/000084.
E-mail address: bilalahmadrr@gmail.com (Bilal A. Rather), pirzadasd@kashmiruniversity.ac.in
(Shariefuddin Pirzada), tariqnaikoo@rediffmail.com (Tariq A. Naikoo)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P2.09
diagonal matrix of vertex degrees di = dvi , i = 1, 2, . . . , n of G. The positive semi-definite
matrices L(G) = D(G)−A(G) and Q(G) = D(G)+A(G) are respectively the Laplacian
and the signless Laplacian matrices and their multiset of eigenvalues are respectively the
Laplacian spectrum and signless Laplacian spectrum of G. More about these matrices can
be seen in [6].
In a graph G, the distance between two vertices u, v ∈ V (G), denoted by d(u, v), is
defined as the length of a shortest path between u and v. The diameter of G is the maximum
distance between any two vertices of G. The distance matrix of G, denoted by D(G), is
defined as D(G) = (duv), where duv = d(u, v) and duu = 0. A complete survey of the
matrix D(G) is given in [3]. The transmission of the vertex v (or transmission degree),
denoted by Tr(v), is defined to be the sum of the distances from v to all other vertices in
G, that is, Tr(v) =
∑
u∈V (G) d(u, v). We observe that transmission of vi is same as the i
th
row sum of the matrix D(G).
Let Tr(G) = diag(Tr1, T r2, . . . , T rn) be the diagonal matrix of vertex transmissions
of G. Aouchiche and Hansen [2] introduced the distance Laplacian L(G) = Tr(G)−D(G)
and the distance signless Laplacian Q(G) = Tr(G) + D(G) for the distance matrix of
a connected graph. These matrices are real symmetric positive and semi-definite. Our
interest is in the matrix Q(G), we denote its eigenvalues by ρi’s, order them as ρn ≤
ρn−1 ≤ · · · ≤ ρ1, where ρ1 is known as the distance signless Laplacian spectral radius G.
Further information about the matrix Q(G) can be seen in [2, 4].
Kelarev and Quinn [12] defined the directed power graph of a semigroup S as a directed
graph with vertex set S in which two vertices x, y ∈ S are joined by an arc from x to y if
and only if x ̸= y and yi = x for some positive integer i. Chakrabarty et al. [8] defined
the undirected power graph P(G) of a group G as an undirected graph with vertex set as G
and two vertices x, y ∈ G are adjacent if and only if xi = y or yj = x, for 2 ≤ i, j ≤ n.
Let Zn = {0, 1, 2, . . . , n− 1} be the cyclic group of integers modulo n. Then a and b
in P(Zn) are adjacent if there exists a positive integer m such that either a = mb or
b = ma. Such graphs have valuable applications and are related to automata theory [11],
besides being useful in characterizing finite groups. More on power graphs can be seen in
[1, 7, 8, 14]. Laplacian spectrum of power graphs of finite cyclic and dihedral groups have
been investigated in [9]. In [9], it is shown that Laplacian spectral radius of power graph
of any finite group coincides with the order of group G. Spectral properties of adjacency
matrix of P(G) were investigated in [15]. Other spectral results of power graphs can be
seen in [5, 18, 19, 20, 21].
The identity of the group G is denoted by e. The proper power graph of P(G), denoted
by P(G∗) = P(G \ {e}), is obtained by removing the vertex e. By Un, we denote the set
{a ∈ Zn | 1 ≤ a < n, gcd(a, n) = 1} and U
∗
n = Un ∪ {0}. Mn(F) denotes the set of
n × n matrices with entries from field F. Also, Kn, K1,n−1 and Pn respectively denote
the complete graph, the star and the path. For other undefined notations and terminology,
the readers are referred to [6, 10, 16].
The rest of the paper is organized as follows. In Section 2, we find the distance signless
Laplacian spectrum of the power graph P(Zn) in terms of the adjacency eigenvalues and
the eigenvalues of the quotient matrix. We end up with some comments for further work.
B. A. Rather et al.: On distance signless Laplacian spectra of power graphs of the integer . . . 3
2 Distance signless Laplacian spectra of power graphs of finite cyclic
group Zn
Let n be a positive integer with canonical decomposition n = pn11 p
n2
2 . . . p
nr
r and let
τ(n) [13] denotes the number of positive factors of n. Then
τ(n) = (n1 + 1)(n2 + 1) . . . (nr + 1). (2.1)
The Euler’s totient function φ(n) [13] denotes the number of positive integers less or
equal to n and relatively prime to n. If n be a positive integer, then
∑
d|n
φ(d) = n. (2.2)
A divisor d of n is a proper divisor of n, if 1 < d < n. Let d1, d2, . . . , dt be the
distinct proper divisors of n. Let Gn be a simple graph with vertex set {d1, d2, . . . , dt} in
which two distinct vertices are adjacent if and only if di|dj , for 1 ≤ i < j ≤ t. If n is
in canonical decomposition, then by Equation (2.1), the size of Gn is given by |V (Gn)| =∏r
i=1(ni + 1)− 2. If n = pq, where p < q are primes, then p does not divide q. So Gn is
disconnected when n is a product of two distinct primes.
Let A be m×m matrix
A =


A1,1 A1,2 · · · A1,m
A2,1 A2,2 · · · A2,m
...
...
. . .
...
Am,1 Am,2 · · · Am,m

 ,
whose rows and columns are partitioned according to a partition P = {P1, P2, . . . , Pm}
of X = {1, 2, . . . , n}. The quotient matrix Q is the m × m matrix whose entries are the
average row sums of the blocks Ai,j of A. The partition P is called equitable if each block
Ai,j of A has constant row (and column) sum and in such case the matrix Q is known as
equitable quotient matrix. In general, the eigenvalues of Q interlace the eigenvalues of A.
If the partition is equitable, then each eigenvalue of Q [6] is an eigenvalue of A.
Let Gi = Gi(Vi, Ei) be graphs of order ni, where i = 1, . . . , n. The joined union
[22] of graphs G1, G2, . . . , Gn, denoted by G[G1, G2, . . . , Gn], is defined as the graph
H(W,F ) with
W =
n⋃
i=1
Vi and F =
n⋃
i=1
Ei ∪
⋃
{vi,vj}∈E
Vi × Vj .
In other words, if G has n vertices labelled as {1, 2, . . . , n}, then vi in Gi and vj in Gj
are adjacent in the joined union if i and j are adjacent in G. Thus, the usual join of two
graphs G1 and G2 is a special case of the joined union K2[G1, G2] = G1▽G2 where K2
is the complete graph of order 2.
The following result [17] gives the distance signless Laplacian spectrum of the join of
two regular graphs.
Theorem 2.1. Let G1 and G2 be r1 and r2 regular graphs of order n1 and n2, respectively.
Let λ1 = r1, λ2, . . . , λn1 and µ1 = r2, µ2, . . . , µn2 be the adjacency eigenvalues of G1
4 Art Discrete Appl. Math. 5 (2022) #P2.09
and G2, respectively. Then the distance signless Laplacian spectrum of G1▽G2 with order
n = n1+n2 consists of n1−1 eigenvalues of the type 2n−n2−r1−4−λi, (2 ≤ i ≤ n1),
and n2 − 1 eigenvalues of the type 2n− n1 − r2 − 4− µj , (2 ≤ j ≤ n2) together with the
two eigenvalues of the following matrix
(
n+ 3n1 − 2r1 − 4 n2
n1 n+ 3n2 − 2r2 − 4
)
, (2.3)
where 2 ≤ i ≤ n1 and 2 ≤ j ≤ n2.
The next result gives the distance signless Laplacian spectrum of the joined union of
regular graphs G1, . . . , Gn, in terms of adjacency spectrum of the graphs G1, G2, . . . , Gn
and the eigenvalues of quotient matrix.
Theorem 2.2 ([17]). Let G be a graph of order n having vertex set V (G) = {v1, . . . , vn}.
Let Gi be ri regular graphs of order ni having adjacency eigenvalues λi1 = ri ≥ λi2 ≥
. . . ≥ λini , where i = 1, 2, . . . , n. The distance signless Laplacian spectrum of the joined
union graph G[G1, . . . , Gn] of order N =
∑n
i=1 ni consists of the eigenvalues 2ni +n
′
i −
ri − λik − 4 for i = 1, . . . , n and k = 2, 3, . . . , ni, where n
′
i =
∑n
k=1,k ̸=i nkdG(vi, vk).
The remaining n eigenvalues are given by the equitable quotient matrix
Q =


4n1 + n
′
1 − 2r1 − 4 n2dG(v1, v2) . . . nndG(v1, vn)
n1dG(v2, v1) 4n2 + n
′
2 − 2r2 − 4 . . . nndG(v2, vn)
...
...
. . .
...
n1dG(vn, v1) n2dG(vn, v2) . . . 4nn + n
′
n − 2rn − 4

 .
The following result says that n− 2 is always a distance signless Laplacian eigenvalue
of the power graph.
Theorem 2.3. Let G be a finite group of order n ≥ 3. Then n − 2 is a distance signless
Laplacian eigenvalues of P(G) with multiplicity at least b − 1, where b is the number of
elements of G which generate all elements of group G.
Proof. Let B(G) be the set of vertices of G consisting of the identity e and those elements
of G which generate all elements of G. Thus, by the definition of power graph, the induced
subgraph P(B(G)) is the complete graph Kb, where b is the cardinality of B(G). To avoid
triviality, we assume that G \ B(G) ̸= {}, so that we obtain, P(G) ∼= Kb▽P(G \ B(G)).
By using Theorem 2.1, we get the distance signless Laplacian eigenvalue
2n− n2 − r1 − λ1k − 4 = 2n− n+ b− b+ 1 + 1− 4 = n− 2,
with at least multiplicity b− 1, since n− 2 can also be the eigenvalue of matrix (2.3).
Taking in particular G = Zn, a finite cyclic group of order n, we have the following
observation.
Corollary 2.4. Let Zn be a finite cyclic group of order n ≥ 3. Then n − 2 is a distance
signless Laplacian eigenvalue of P(Zn) with multiplicity at least φ(n), where φ is Euler’s
totient function.
B. A. Rather et al.: On distance signless Laplacian spectra of power graphs of the integer . . . 5
Proof. Since identity 0 and the invertible elements of the group Zn, which are φ(n) in
number, generate all the elements of group Zn, therefore P(Zn) = Kφ(n)+1▽P(Zn \U
∗
n),
where U∗n = Un ∪ {0} and Zn \ U
∗
n ̸= {}, since power graph of the empty set is empty.
Thus, by Theorem 2.3, n − 2 is a distance signless Laplacian eigenvalue of P(Zn) with
multiplicity at least φ(n).
It will be interesting to characterize the power graphs for which equality holds in The-
orem 2.3 and Corollary 2.4. Therefore, we have the following problem.
Problem 2.5. Characterize the power graphs P(G), where G is a finite group of order
n ≥ 3 having n − 2 as the distance signless Laplacian eigenvalue with multiplicity b − 1,
where b is the number of group G which generate all the elements of G. Also, characterize
the power graphs P(Zn) having n − 2 as the distance signless Laplacian eigenvalue with
multiplicity exactly φ(n).
From Theorem 2.3, we observe that if P(G\B(G)) is known, then the distance signless
Laplacian spectrum of P(G) can be completely determined. So, it will be interesting to
study the structure of P(G \B(G)), and looking for graph parameters related to it.
If G is a finite group, then P(G) [8] is complete if and only if G is cyclic group of prime
power order.
The following result [14] shows that the power graph of a cyclic group Zn can be
written as the joined union of complete graphs.
Theorem 2.6. If Zn is a finite cyclic group, then the power graph has the following form:
P(Zn) = Kφ(n)+1▽Gn[Kφ(d1),Kφ(d2), . . . ,Kφ(dt)]
= H[Kφ(n)+1,Kφ(d1),Kφ(d2), . . . ,Kφ(dt)],
where H = Kφ(n)+1▽Gn with the vertex set {v1, . . . , vt+1} and t is the number of proper
divisors of n.
We note that if n = p, then Gn is empty graph and we get P(Zp) = Kp. By applying
Theorem 2.2, we can find the distance signless Laplacian spectrum of P(G) in terms of
adjacency eigenvalues of Kω and the eigenvalues of the quotient matrix. It is well known
that the adjacency eigenvalues of Kω are {ω, (−1)
[ω−1]}. As t is the number of divisors of
n, so by Theorem 2.2, n− t out of the n distance signless Laplacian eigenvalues are known
to be non negative integers and the remaining t distance signless Laplacian eigenvalues are
the eigenvalues of the quotient matrix Q.
In the following result, we obtain the distance signless Laplacian eigenvalues of the
power graph of Zn.
Theorem 2.7. The distance signless Laplacian spectrum of P(Zn) consists of the eigen-
values
{
(n− 2)[φ(n)],(φ(d1) + n
′
2 − 2)
[φ(d1)−1], (φ(d2) + n
′
3 − 2)
[φ(d2)−1],
. . . , (φ(dt) + n
′
t+1)
[φ(dt)−1]
}
,
where di, 1 ≤ i ≤ t are the proper divisors of n and n
′
i =
t+1∑
k=2,k ̸=i
φ(dk)dP(Zn)(vi, vk).
The remaining t+1 distance signless Laplacian eigenvalues of P(Zn) are the eigenvalues
6 Art Discrete Appl. Math. 5 (2022) #P2.09
of the following quotient matrix
Q =


n+ φ(n)− 1 φ(d1) . . . φ(dn)
(φ(n) + 1)d(v2, v1) d
′
2 . . . φ(dt)d(v2, vt+1)
...
...
. . .
...
(φ(n) + 1)d(vt+1, v1) (φ(d1))d(vt+1, v2) . . . d
′
t+1

 , (2.4)
where d′i = 2ni + n
′
i − 2, for i = 2, . . . , t + 1 and n1 = φ(n) + 1, nj = φ(dj),
j = 2, 3, . . . , n.
Proof. Let Zn be a finite cyclic group of order n. Then it is well known that the identity
0 and φ(n) elements of Zn together generate every other element of Zn. Thus, by the
definition power graph, φ(n) + 1 vertices are connected to every vertex of P(Zn). So, by
Theorem 2.6, we have
P(Zn) = H[Kφ(n)+1,Kφ(d1),Kφ(d2), . . . ,Kφ(dt)].
Clearly, n1 = φ(n) + 1 and ni = φ(di−1), for i = 2, . . . , t + 1. Now, using the fact that∑
1,n ̸=d|n
φ(d) = n − φ(n) − 1 and Theorem 2.3, we see that n − 2 is the distance signless
Laplacian eigenvalue of P(Zn) with multiplicity φ(n). Again, using Theorem 2.2 and the
adjacency eigenvalues of Kω , we get
2n2 + n
′
2 − r2 − λ2k − 4 = 2n2 + n
′
2 − n2 +1+ 1− 4 = n2 + n
′
2 − 2 = φ(d1) + n
′
2 − 4
as the distance signless Laplacian eigenvalue of P(Zn) with multiplicity φ(d1)− 1. Simi-
larly other distance signless Laplacian eigenvalue of P(Zn) are φ(di) + n
′
i − 4 with mul-
tiplicities φ(di) − 1, for i = 3, 4, . . . , t + 1. In this way, we have obtained n − t distance
signless Laplacian eigenvalues of P(Zn) and the remaining distance signless Laplacian
eigenvalues are given by matrix (2.4).
The following are consequences of Theorem 2.7.
Corollary 2.8. If n = pm1 , where p is a prime and m1 is a non negative integer, then the
distance signless Laplacian spectrum of P(Zn) is {2n− 2, (n− 2)
[n−1]}.
Proof. If n = pm1 , where p is prime and m1 is a non negative integer, then as shown in [8],
P(Zn) ∼= Kn and distance signless Laplacian spectrum, of Kn is {2n − 2,
(n− 2)[n−1]}.
Corollary 2.9. If n = pq, where p and q (p < q) are primes, then the distance signless
Laplacian spectrum of P(Zn) is
{
(n− 2)[φ(n)], (n+ φ(q)− 2)[φ(p)−1], (n+ φ(p)− 2)[φ(q)−1]
}
and the zeros of the following cubic polynomial
x3−(1 + 2p+ 2q + 2pq)x2 +
(
−40 + 4p+ 3p2 + 8q + 8pq + p2q + q2 + pq2 + p2q2
)
x
− 76 + 16p− p2 − 2p3 + 24q + 70pq − 6p2q − p3q − 5q2 − 16pq2 − 13p2q2
+ p3q2 + pq3 + p2q3.
B. A. Rather et al.: On distance signless Laplacian spectra of power graphs of the integer . . . 7
Proof. As p and q are the proper divisor of n, so Gn is 2K1 and K1▽2K1 is the path P3
on three vertices. Thus, by Theorem 2.6, the power graph of P(Zn) is
P(Zn) = P3[Kp−1,Kφ(pq)+1,Kq+1].
Also, (n′1, n
′
2, n
′
3) = (φ(pq)+1+2φ(q), φ(p)+φ(q), 2φ(p)+φ(pq)+1). Now, by Corol-
lary 2.4, n − 2 is the distance signless Laplacian eigenvalues of P(Zn) with multiplicity
φ(n). Again, by using the Theorem 2.7, n1 + n
′
1 − 2 = n + φ(q) − 2 and n + φ(p) − 2
are the distance signless Laplacian eigenvalues of P(Zn) with multiplicities φ(p) − 1 and
φ(q) − 1, respectively. The remaining distance signless Laplacian eigenvalues of P(Zn)
are given by the following matrix


pq + p+ q − 4 pq − p− q + 2 2(q − 1)
p− 1 2pq − p− q q − 1
2(p− 1) pq − p− q + 2 pq + p+ q − 4

 .
Proceeding as in Corollary 2.9, the distance signless Laplacian spectrum of P(Zn) can
be discussed, for n = pqr, where p, q and r, (p < q < r) are primes.
Corollary 2.10. If n = pqr, where p, q and r (p < q < r) are primes, then the distance
signless Laplacian eigenvalues of P(Zn) are
{
(n− 2)[φ(n)], (n+ n3 + n4 + n7 − 2)
[n2−1], (n+ n2 + n4 + n6 − 2)
[n3−1],
(n+ n2 + n3 + n5 − 2)
[n4−1], (n+ n4 + n6 + n7 − 2)
[n5−1],
(n+ n3 + n5 + n7 − 2)
[n6−1](n+ n2 + n5 + n6 − 2)
[n7−1]
}
,
where n2 = φ(p), n3 = φ(q), n4 = φ(r), n5 = φ(pq), n6 = φ(pr) and n7 = φ(qr).
The remaining 7 distance signless Laplacian eigenvalues of P(Zn) are the eigenvalues of
the following matrix


n+ φ(n)− 4 φ(p) φ(q) φ(r) φ(pq) φ(pr) φ(qr)
φ(n) + 1 d2 2φ(q) 2φ(r) φ(pq) φ(pr) 2φ(qr)
φ(n) + 1 2φ(p) d3 2φ(r) φ(pq) 2φ(pr) φ(qr)
φ(n) + 1 2φ(p) 2φ(q) d4 2φ(pq) φ(pr) φ(qr)
φ(n) + 1 φ(p) φ(q) 2φ(r) d5 2φ(pr) 2φ(qr)
φ(n) + 1 φ(p) 2φ(q) φ(r) 2φ(pq) d6 2φ(qr)
φ(n) + 1 2φ(p) φ(q) φ(r) 2φ(pq) 2φ(pr) d7


,
where d2 = n + n2 + n3 + n4 + n7 − 2, d3 = n + n2 + n3 + n4 + n6 − 2,
d4 = n+n2+n3+n4+n5−2, d5 = n+n4+n5+n6+n7−2, d6 = n+n3+n5+n6+n7−2,
and d7 = n+ n2 + n5 + n6 + n7 − 2.
We recall that the proper power graph P(G∗) of a group G is the power graph of G\{e}.
The proper power graph P(Z∗n) is connected, as Zn is a cycle group and there exists at least
one element say 1 connected to all the other vertices of P(Z∗n). Thus the distance signless
Laplacian matrix makes sense on P(Z∗n). Analogues of Theorems 2.3 and 2.2 can be
proved on the proper power graph P(Z∗n). We state them without proofs.
Theorem 2.11. Let Zn be a finite cyclic group of order n ≥ 3. Then n− 2 is the distance
signless Laplacian eigenvalues of P(Z∗n) with multiplicity at least φ(n)− 1.
8 Art Discrete Appl. Math. 5 (2022) #P2.09
Theorem 2.12. The distance signless Laplacian spectrum of P(Z∗n) is
{
(n− 2)[φ(n)−1],(φ(d1) + n
′
2 − 2)
[φ(d1)−1], (φ(d2) + n
′
3 − 2)
[φ(d2)−1],
. . . , (φ(dt) + n
′
t+1)
[φ(dt)−1]
}
together with the eigenvalues of the following quotient matrix
Q =


n+ φ(n)− 2 φ(d1) . . . φ(dn)
φ(n)d(v2, v1) d
′
2 . . . φ(dt)d(v2, vt+1)
...
...
. . .
...
φ(n)d(vt+1, v1) (φ(d1))d(vt+1, v2) . . . d
′
t+1

 ,
where d′i = 2ni + n
′
i − 2, n
′
i =
∑t+1
k=2,k ̸=i φ(dk)dP(Z∗n)(vi, vk), for i = 2, . . . , t + 1,
n1 = φ(n), nj = φ(dj), j = 2, 3, . . . , t.
The following Lemma can be found in [4].
Lemma 2.13. Let G be a connected graph on n vertices. If n − 2 is a distance signless
Laplacian eigenvalue with multiplicity µ, then the complement G of G contains at least µ
components, each of which is bipartite or an isolated vertex.
Now, we discuss Problem 2.5 for the power group of the group Zn.
Proposition 2.14. Let P(Zn) be the power group of order n ≥ 2. Then n − 2 is the
distance signless Laplacian eigenvalue of P(Zn) with multiplicity φ(n) if and only if n is
prime power or product of two distinct primes.
Proof. By Corollaries 2.8 and 2.9, we see that n − 2 is the distance signless Lapalcian
eigenvalue of P(Zn) with multiplicity exactly φ(n). Also, complement of P(Zn) are not
necessarily bipartite or an isolated vertex when n is other than prime power or product of
two distinct primes. Thus, by Lemma 2.13, we see that P(Zpm1 ) and P(Zpq) are the only
two candidates of P(Zn), where n − 2 is the distance signless Laplacian eigenvalue with
multiplicity φ(n).
The following Lemma [2] states that upon edge deletion, the distance signless Laplacian
eigenvalues increase.
Lemma 2.15. Let G be a connected graph of order n and size m, where m ≥ n and let
G′ = G − e be a connected graph obtained from G by deleting an edge. Let ρL1 (G) ≥
ρL2 (G) ≥ · · · ≥ ρ
L
n(G) and ρ
L
1 (G
′) ≥ ρL2 (G
′) ≥ · · · ≥ ρLn(G
′) respectively be the distance
Laplacian eigenvalues of G and G′. Then ρLi (G
′) ≥ ρLi (G) holds for all 1 ≤ i ≤ n.
Since P(Zpm1 ) is the complete graph, so the following consequence is immediate from
the above lemma.
Proposition 2.16. Let P(G) be the power graph of order n = pm1 , where p is prime and
m1 is positive integer. Then
ρi(P(G)) ≥ ρi(P(Zpm1 )),
equality holds if and only G ∼= Zpm1 .
B. A. Rather et al.: On distance signless Laplacian spectra of power graphs of the integer . . . 9
From Proposition 2.16, it follows that P(Zn) has minimal spectrum among all power
graphs whose order is a prime power. It will be an interesting to characterize the extremal
power graph with some given spectral graph invariant.
A matrix M ∈ Mn(R) is said to be integral, if its spectrum consists of only integers.
Likewise, a graph G is signless Laplacian integral if and only if the matrix Q(G) is integral.
In case of the joined union, G[G1.G2, . . . , Gn] is integral if and only if each of Gi and its
associated equitable quotient matrix is integral. Thus for the power graphs, we have the
following result.
Proposition 2.17. The power graph P(Zn) is signless Laplacian integral if and only if n
is prime power.
ORCID iDs
Bilal A. Rather https://orcid.org/0000-0003-1381-0291
Shariefuddin Pirzada https://orcid.org/0000-0002-1137-517X
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P2.10
https://doi.org/10.26493/2590-9770.1511.af5
(Also available at http://adam-journal.eu)
The dominated chromatic number
of middle graphs
Kijung Kim*
Department of Mathematics, Pusan National University, Busan 46241, Republic of Korea
Received 29 November 2021, accepted 24 April 2022, published online 7 June 2022
Abstract
A dominated coloring of a graph is a proper vertex coloring such that every color class is
dominated with at least one vertex. The minimum number of colors needed for a dominated
coloring of a graph G is the dominated chromatic number of G. The middle graph M(G)
of a graph G is the graph obtained by subdividing each edge of G exactly once and joining
all these newly introduced vertices of adjacent edges of G. For a graph G without isolated
vertices, the dominated chromatic number of M(G) is completely determined.
Keywords: Dominated coloring, dominator coloring, dominated chromatic number, middle graph.
Math. Subj. Class.: 05C15, 05C69
1 Introduction
Let G = (V,E) be a finite, undirected and simple graph with the vertex set V = V (G)
and edge set E = E(G). The open neighborhood of v ∈ V (G) is the set NG(v) =
{u ∈ V (G) | uv ∈ E(G)}. For a subset S of V (G), the subgraph obtained from G by
deleting all vertices in S and all edges incident with S is denoted by G− S.
In [6], Hamada and Yoshimura defined the middle graph of a graph. The middle graph
M(G) of a graph G is the graph obtained by subdividing each edge of G exactly once and
joining all these newly introduced vertices of adjacent edges of G. The precise definition
of M(G) is as follows. The vertex set V (M(G)) is V (G) ∪ E(G). Two vertices v, w ∈
V (M(G)) are adjacent in M(G) if
(i) v, w ∈ E(G) and v, w are adjacent in G or
(ii) v ∈ V (G), w ∈ E(G) and v, w are incident in G.
*This research was supported by Basic Science Research Program through the National Research Foundation
of Korea funded by the Ministry of Education (2020R1I1A1A01055403).
E-mail address: knukkj@pusan.ac.kr (Kijung Kim)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P2.10
Definition 1.1. A dominated coloring of a graph G is a proper (vertex) k-coloring
{C1, C2, . . . , Ck} where for each i ∈ {1, 2, . . . , k}, there exists a vertex u ∈ V (G) such
that Ci ⊆ NG(u). We say that u dominates Ci or Ci is dominated by u. The minimum
number of colors needed for a dominated coloring of a graph G is the dominated chromatic
number of G and is denoted by χdom(G).
Figure 1: The middle graph M(P4) of P4.
Example 1.2. In Figure 1, {{u1, u2}, {v1, u3}, {v2, u4}, {v3}} is a dominated coloring of
M(P4). Indeed, the dominated chromatic number of M(P4) is four.
A dominated coloring of G using χdom(G) colors is called a χdom(G)-coloring of G.
Merouane et al. introduced the concept of a dominated coloring in [9]. They adopted
algorithmic approach for dominated coloring problems and proved that if G is a triangle-
free graph, then χdom(G) is equal to its total domination number. In [3], Chen provided an
application of dominated coloring in social networks. In [8], Klažar and Tavakoli proposed
an application of dominated coloring in genetic networks.
As a concept closely related to dominated colorings, a dominator coloring of a graph is
a proper coloring in which every vertex dominates every vertex of at least one color class.
This concept was introduced for the first by Gera et al. in [5]. There have been several
follow-up studies in [1,2,4,10]. Recently, Klažar and Tavakoli showed that although dom-
inated colorings and dominator colorings appear quite similar, they are strikingly different
on corona products (see [8]). In [7], Kazemnejad et al. considered total dominator coloring
in middle graphs and gave the total dominator chromatic number of middle graph of several
known families of graphs. In contrast, in this note we completely determine the dominated
chromatic number of all middle graphs. We shall prove the following:
Theorem 1.3. Let G be a graph without isolated vertices. Then
χdom(M(G)) =
⌈
|V (G)|+ |E(G)|
2
⌉
.
As an application of Theorem 1.3, for a star K1,n we obtain a χdom(M(K1,n))-coloring
as follows: Let V (K1,n) = {v, v1, v2, . . . , vn}, where v is the central vertex. If n is even,
then {{vi, vvi+1} | i = 2k − 1, 1 ≤ k ≤
n
2 } ∪ {{vj , vvj−1} | j = 2k, 1 ≤ k ≤
n
2 } ∪ {{v}} is a χdom(M(K1,n))-coloring. If n is odd, then {{vi, vvi+1} | i = 2k − 1,
1 ≤ k ≤ n−12 } ∪ {{vj , vvj−1} | j = 2k, 1 ≤ k ≤
n−1
2 } ∪ {{v, vn}, {vvn}} is a
χdom(M(K1,n))-coloring.
K. Kim: The dominated chromatic number of middle graphs 3
2 Proof of main theorem
In this section, we prove our main theorem.
Proof of Theorem 1.3. Since χdom(G) =
∑t
i=1 χdom(Gi) for a graph G with components
G1, ..., Gt, it suffices to consider the case of a connected graph G. We proceed by proving
two claims.
Claim 1.
⌈
|V (G)|+|E(G)|
2
⌉
≤ χdom(M(G)).
Let {V1, . . . , Vk} be a dominated coloring of M(G). Let v ∈ V (G) belong to Vi, and
let e = vw ∈ E(G) satisfy NM(G)(e) ⊇ Vi. Then |Vi ∩NM(G)[w]| ≤ 1. This implies that
|Vi| ≤ 2.
Let e ∈ E(G) belong to Vi, and let x ∈ V (M(G)) satisfy NM(G)(x) ⊇ Vi. If
x ∈ V (G), then |Vi| = 1 since every edge of NM(G)(x)\{e} is adjacent to e. If x ∈ E(G),
then |Vi ∩ (NM(G)(x) \ {e})| ≤ 1. This implies that |Vi| ≤ 2.
Thus, every dominated coloring of M(G) needs at least
⌈
|V (G)|+|E(G)|
2
⌉
colors.
Claim 2.
⌈
|V (G)|+|E(G)|
2
⌉
= χdom(M(G)).
We prove the equality by induction on the order of G. For a graph G of order n ≤ 3,
it is easy to see that
⌈
|V (G)|+|E(G)|
2
⌉
= χdom(M(G)). Let G be a graph of order n ≥ 4.
Suppose that every graph G′ of order n′(< n) has
⌈
|V (G′)|+|E(G′)|
2
⌉
= χdom(M(G
′)).
Now we fix a vertex v ∈ V (G) and consider G′ := G − {v}. Denote the set of edges
between v and G′ by E(v,G′). Let C = {V1, . . . , Vk} be a χdom(M(G
′))-coloring of
M(G′), and let l := |E(v,G′)| and E(v,G′) = {vv1, vv2, . . . , vvl}. It follows from the
proof of Claim 1 that at most two vertices in {v1, v2, . . . , vl} belong to the same color class.
We divide our consideration into two cases.
Case 1: l is odd.
Without loss of generality, we can assume that vi ∈ Vi for each i ∈ {1, . . . ,
l+1
2 }. We
show that C is extended to a χdom(M(G))-coloring of M(G) by adding
l+1
2 color classes.
Set
V ′i := (Vi \ {vi}) ∪ {vvi}
for 1 ≤ i ≤ l+12 ,
Vk+i := {vi, vv l+1
2
+i}
for 1 ≤ i ≤ l+12 − 1,
Vk+ l+1
2
= {v, v l+1
2
}.
Then {V ′i | 1 ≤ i ≤
l+1
2 }∪{V l+1
2
+1, . . . , Vk}∪{Vk+i | 1 ≤ i ≤
l+1
2 −1}∪{Vk+ l+1
2
}
is a χdom(M(G))-coloring of M(G).
Case 2: l is even.
Without loss of generality, we can assume that vi ∈ Vi for each i ∈ {1, . . . ,
l
2}. We
show that C is extended to a χdom(M(G))-coloring of M(G) by adding
l
2 or
l
2 + 1 color
classes. We consider two subcases depending on |V (G′)|+ |E(G′)|.
4 Art Discrete Appl. Math. 5 (2022) #P2.10
Subcase 2.1: |V (G′)|+ |E(G′)| is even. Note that |V (G′)|+ |E(G′)| = 2k.
Set
V ′i := (Vi \ {vi}) ∪ {vvi}
for 1 ≤ i ≤ l2 ,
Vk+i := {vi, vv l
2
+i}
for 1 ≤ i ≤ l2 ,
Vk+ l
2
+1 = {v}.
Then {V ′i | 1 ≤ i ≤
l
2} ∪ {V l
2
+1, . . . , Vk} ∪ {Vk+i | 1 ≤ i ≤
l
2} ∪ {Vk+ l
2
+1} is a
χdom(M(G))-coloring of M(G).
Subcase 2.2: |V (G′)|+ |E(G′)| is odd. Note that there is only a singleton in C, which
is denoted by Vk.
Set
V ′i := (Vi \ {vi}) ∪ {vvi}
for 1 ≤ i ≤ l2 ,
Vk+i := {vi, vv l
2
+i}
for 1 ≤ i ≤ l2 ,
V ′k = Vk ∪ {v}.
Then {V ′i | 1 ≤ i ≤
l
2} ∪ {V l
2
+1, . . . , Vk−1} ∪ {V
′
k} ∪ {Vk+i | 1 ≤ i ≤
l
2} is a
χdom(M(G))-coloring of M(G).
ORCID iDs
Kijung Kim https://orcid.org/0000-0002-6623-5123
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