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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P1.01
https://doi.org/10.26493/2590-9770.1260.989
(Also available at http://adam-journal.eu)
Axiomatic characterization of transit functions
of weak hierarchies∗
Manoj Changat
Department of Futures Studies, University of Kerala,
Trivandrum, PIN 695 581, India
Prasanth G. Narasimha-Shenoi
Department of Mathematics, Government College Chittur,
Palakkad, PIN 678 104, India
Peter F. Stadler †
Bioinformatics Group, Department of Computer Science,
Härtelstraße 16-18, D-04107 Leipzig, Germany
Received 17 February 2018, accepted 27 July 2018, published online 8 August 2018
Abstract
Transit functions provide a unified approach to study notions of intervals, convexities,
and betweenness. Recently, their scope has been extended to certain set systems associated
with clustering. We characterize here the class of set systems that correspond to k-ary
monotonic transit functions. Convexities form a subclass and are characterized in terms
of transit functions by two additional axioms. We then focus on axiom systems associated
with weak hierarchies as well as other generalizations of hierarchical set systems.
Keywords: Transit functions, convexities, weak hierarchies, axiom systems.
Math. Subj. Class.: 05C05, 05C99, 52A01
∗This work was supported in part by NBHM DAE, grant no. NBHM/2/48(9)/2014/NBHM (R.P.)/R&D-II/4364
to MC. PFS gratefully acknowledges support by the Erudite scheme of the Kerala State Higher Education Council,
Govt. of Kerala, India.
†PFS holds additional affiliations with the Max Planck Institute for Mathematics in the Sciences in Leipzig,
the Institute for Theoretical Chemistry at the University of Vienna, and the Santa Fe Institute.
E-mail addresses: mchangat@gmail.com (Manoj Changat), prasanthgns@gmail.com (Prasanth G.
Narasimha-Shenoi), studla@bioinf.uni-leipzig.de (Peter F. Stadler)
cb This work is licensed under http://creativecommons.org/licenses/by/3.0/
2 Art Discrete Appl. Math. 2 (2019) #P1.01
1 Introduction
Transit functions have been introduced as unifying approach for results and ideas on inter-
vals, convexities and betweenness in graphs and posets [25]. Formally, a transit function
[25] on a non-empty set V is a function R : V × V → 2V satisfying the three axioms
(t1) u ∈ R(u, v) for all u, v ∈ V .
(t2) R(u, v) = R(v, u) for all u, v ∈ V .
(t3) R(u, u) = {u} for all u ∈ V .
Transit functions capture an abstract notion of “betweenness”, i.e., an element x is consid-
ered to be “between” u and v, if x ∈ R(u, v). Several classes of interesting transit function
associated with connected graphs have been studied from this perspective [15], among them
the interval function [1, 24, 26], induced path function [12, 13, 23], the all paths function
[11], the pre-fiber transit function [25], P3-transit function [18]. Transit function also arose
as models of recombination operators in genetics and evolutionary algorithms [16, 22, 27],
where again, encapsulate the idea that offsprings are genetically “in-between” their parents.
Recently, they have also been applied to set systems naturally arising from clustering
problems [9, 17]. Hierarchical structures play a key role in wide range of applications in the
sciences. In [17] we considered properties of transit functions that are related to hierarchies
and provided a characterization in terms of simple axioms. This begs the question whether
or to what extent related systems of clusters are also within the explanatory range of transit
functions. Weak hierarchies were introduced in [2, 3] as the system of “weak clusters”
satisfying s(a, b) > min{s(a, x), s(b, x)} for all a, b within a cluster and x outside, and in
[5] (with closure under pairwise intersection as additional condition). Subsequently, they
have become a central structure in the theory of hierarchical clustering, see e.g. [8, 9].
They subsume a number of less general constructions such as paired hierarchies [6, 7] and
pyramids [20]. In [4] systems of clusters are considered that are generated by two elements.
These are, as we shall see, closely related to set systems identified by transit functions.
In particular in the context hierarchies, much of the previous work focuses the rela-
tionships of transit functions and convexities [8, 9], see also [4, 15, 25] on convexities in
relation to graph-theoretical constructions. Here we take a somewhat different perspective
and focus on axiom systems on transit functions giving raise to set systems such as those
arising naturally in the context of clustering.
This contribution is organized as follows. We start by characterizing the set systems
that are identified by transit function. Then we proceed to discussing convexities, giving
a characterization of the transit functions that identify them. We then consider the transit
functions arising from some hierarchy-like systems. Here we focus on weak hierarchies as
well as properties discussed in our previous work [17]. We then generalize the results of
Section 2 to k-ary transit functions. Finally we consider k-weak hierarchies. For k > 2
these are not identified by convexities but require the more general setting.
2 Set systems identified by transit functions
Throughout this contribution, V is a finite, non-empty set. Consider an arbitrary set system
X ⊆ 2V and the function
RX (x, y) =
⋂
{A ∈ X | x, y ∈ A}. (2.1)
M. Changat et al.: Axiomatic characterization of transit functions of weak hierarchies 3
We are interested here in those special classes of set systems that are identified by the
function R, that is,
X = {RX (x, y) | x, y ∈ V }. (2.2)
We use the notation ⊂ to mean proper subset, and write ⊆ otherwise.
If p, q ∈ RX (x, y) then by definition in everyA ∈ X that contains x and y also contains
p and q. Thus RX satisfies the monotone axiom [13, 25]
(m) p, q ∈ R(u, v) implies R(p, q) ⊆ R(u, v).
The monotone axiom (m) can be expressed in the following equivalent form:
Lemma 2.1. A transit function R is monotone if and only if, for all u, v ∈ V ,⋂
{R(x, y) | u, v ∈ R(x, y)} = R(u, v). (2.3)
Proof. Condition (m) implies R(u, v) ⊆
⋂
{R(x, y) | u, v ∈ R(x, y)}. By (t1) and
(t2) we have u, v ∈ R(u, v), hence
⋂
{R(x, y) | u, v ∈ R(x, y)} ⊆ R(u, v) and thus
Equation (2.3) holds. Conversely, from Equation (2.3) we immediately obtain R(u, v) ⊆
R(x, y) for all x, y with u, v ∈ R(x, y), i.e., (m) holds.
Lemma 2.2. A set system X is identified by a transit function R if and only if it satisfies
the following axioms:
(KS) {x} ∈ X for all x ∈ V .
(KR) For every C ∈ X there are points p, q ∈ C such that p, q ∈ C ′ implies C ⊆ C ′ for
all C ′ ∈ X .
(KC) For every p, q ∈ V holds
⋂
{C ∈ X | p, q ∈ C} ∈ X .
Proof. Suppose set system X is identified by a transit function R. Then the set system X
satisfies (KS), since R satisfies (t3). By definition of R, C ∈ X implies C = R(u, v) =⋂
{A ∈ X | u, v ∈ A} for some u, v ∈ V . Therefore, for p, q ∈ V such that p, q ∈ A, it
follows that C ⊆ C ′, which proves that X satisfies (KR). Now, for every p, q ∈ C ∈ X ,
R(p, q) =
⋂
{C ∈ X | p, q ∈ C} and hence X satisfies (KC).
Conversely, suppose that a set system X satisfies the axioms (KS), (KR) and (KC). To
show that X is identified by a transit function, define a function R : V × V → 2V as
RX (x, y) =
⋂
{A ∈ X | x, y ∈ A}. Axioms (KS) and (KC) for X imply that this function
R satisfies (t3) and (t1). Since R also satisfies (t2) by definition, we conclude that R is a
transit function. Moreover, R satisfies (m). Now, we show that X = {RX (x, y) | x, y ∈
V }. By axiom (KR),C ∈ X , then there exists, p, q ∈ C such that p, q ∈ C ′ impliesC ⊆ C ′
for all C ′ ∈ X , which implies that C ⊆
⋂
{C ∈ X | p, q ∈ C}, that is, C ⊆ RX (p, q). On
the other hand, p, q ∈ C implies RX (p, q) ⊆ C, Thus every cluster C is a transit set, i.e.
X ⊆ {RX (p, q) | p, q}. By axiom (KC), {RX (x, y) | x, y ∈ V } ⊆ X , which completes
the proof.
In this case R is the canonical transit function of X [25]. Condition (KR) is called
2-arity in [13]. If R is the canonical transit function of X , then (KC) becomes
(m*) For all u, v ∈ V there is p, q ∈ V such that⋂
{R(s, t) | s, t ∈ V and u, v ∈ R(s, t)} = R(p, q).
4 Art Discrete Appl. Math. 2 (2019) #P1.01
This can be seen as a relation of Equation (2.3), which stipulates not just the existence of
p, q ∈ V but insists that {p, q} = {u, v}. The example in Figure 1 shows that axiom (KC)
is much weaker than closure under pairwise intersection. The set systems identified by
transit functions therefore are not convexities in general. We will return to this point in the
next section.
The independence of the axioms (KS), (KR), and (KC) is established by the following
examples.
Example 2.3 ((KS) but not (KR) or (KC)). For V = {a, b, c, d, e} consider
X =
{
{a}, {b}, {c}, {d}, {e}, {a, b, c}, {b, c, d}, {a, b, d}, {a, c, e}
}
.
For every pair of points p, q ∈ {a, b, c} there is another set containing p and q that is not a
superset of {a, b, c}:
a, b ∈ {a, b, d}, a, c ∈ {a, c, e}, and b, c ∈ {b, c, d}.
Thus (KR) does not hold. Since {a, b, c} ∩ {a, b, d} = {a, b} /∈ X , (KC) is not satisfied.
On the other hand, all singletons are in X , i.e., the X satisfies (KS).
Example 2.4 ((KR) but not (KS) or (KC)). For V = {a, b, c, d, e} consider
X =
{
{a, b, c}, {a, b, d}
}
.
We can easily see that X satisfies (KR). Since no singleton is in X , we can see that the
set system fails to satisfy (KS). Now {a, b, c}, {a, b, d} ∈ X , but {a, b, c} ∩ {a, b, d} =
{a, b} /∈ X . Hence X does not satisfy (KC).
Example 2.5 ((KC but not (KS) or (KR)). For V = {a, b, c, d, e} consider
X =
{
{a, b, c}, {b, c, d}, {a, b, d}, {a, c, e}, {a, b}, {b, c}, {a, c}, {b, d}, {a}, {b}, {c}
}
.
We can see that X satisfies (KC). For any pair of elements in {a, b, c}, there are the sets
{b, c, d}, {a, b, d}, {a, c, e} that contain a pair of {a, b, c} but are not supersets of {a, b, c}.
Hence X does not satisfy (KR). Also since no singleton is in X , the set system fails to
satisfy (KS).
Theorem 2.6. There is a 1-1 correspondence between monotone transit functions R on
V and set systems X satisfying (KS), (KR), and (KC) on V . This bijection is given by
X 7→ RX defined in Equation (2.1) and {R(p, q) | p, q ∈ V } 7→ X , respectively.
Proof. Suppose X is a set system satisfying (KS), (KR), and (KC) with transit function R.
By Lemma 2.2, X = {R(u, v) | u, v ∈ V }. The transit function defined by Equation (2.1)
satisfies (m) as argued above.
Now suppose R is a monotone transit function. Then by construction {R(p, q) | p, q ∈
V } satisfies (KS) and (KR). Furthermore (m) implies (m*), which can be rewritten as (KC)
using the fact that X = {R(u, v) | u, v ∈ V }.
M. Changat et al.: Axiomatic characterization of transit functions of weak hierarchies 5
a
x
y z
p
br
q
c
X consists of V , {u} for u ∈ V ,
the nine pairs {a, r}, {a, q}, {b, r}, {b, p}, {c, p},
{c, q}, {a, b}, {b, c}, {a, c} as well and the 18
pairs in {a, b, c, p, q, r} × {x, y, z},
A = {a, r, q, x, y, z}, B = {b, p, r, x, y, z},
C = {c, p, q, x, y, z}, and Z = {x, y, z}.
Figure 1: In addition to the 27 defining pairs we have R(a, p) = R(b, q) = R(c, r) =
V, R(q, r) = A, R(r, p) = B, R(p, q) = C, and R(x, y) = R(x, z) = R(y, z) =
Z, accounting for all pairs in V . Hence it satisfies (KR). For each pair of points i, j,
furthermore, the non-empty intersection of all sets containing them is again a member of
X . This is obvious for all intersections involving two-element sets in X . In the remaining
cases yield the transit sets listed above. The pairwise intersections A ∩ B, A ∩ C, and
B ∩ C, however, are not members of X : any pair in these sets has as intersection of its
containing sets either that pairs or Z.
Definition 2.7. A set system X satisfying (KS),
(K0) ∅ /∈ X , and
(K1) V ∈ X
is called a clustering system [4].
In the terminology of [4], a clustering system is pre-binary if, for every u, v ∈ V the
set system {C ∈ X | u, v ∈ C} has exactly one inclusion-minimal element, i.e., in our
language, if every transit set is a cluster. The proof of Lemma 2.2 shows that this condition
is equivalent to (KR). A cluster system is called binary if it is pre-binary and every cluster
is a transit set. The latter condition is equivalent to (KC). We can summarize this discussion
as
Corollary 2.8. A set system X is a binary clustering system if and only if it satisfies (K0),
(K1), (KS), (KR), and (KC).
Axiom (K1) can trivially be translated into the language of transit functions as
(a’) There is u, v ∈ V such that R(u, v) = V .
Thus we also have
Corollary 2.9. A transit function identifies as a binary clustering system if and only if it
satisfies (m) and (a’).
6 Art Discrete Appl. Math. 2 (2019) #P1.01
3 Transit functions of convexities
A systems of sets K ⊂ 2V that contains V and is closed under intersection is called a
convexity. Usually, one requires the empty set ∅ to be part of a convexity. However, ∅
cannot be a transit set according to (t1). It is convenient in our context, therefore, to use a
slightly modified definition, excluding the empty set and restricting closure to non-empty
intersection. Furthermore, (t3) implies that {x} is a transit set for every x ∈ V . Thus we
are only interested in set systems that contain the singletons.
Definition 3.1. A convexity is a set system X ⊆ 2V satisfying (K0), (K1), and
(K2) if A,B ∈ X and A ∩B 6= ∅ then A ∩B ∈ X .
A set system is closed (under pairwise intersection) if it satisfies (K2). A convexity is
called grounded if it satisfies (KS). Thus grounded convexities are the same as closed clus-
tering systems. Closure under non-empty intersections, (K2), immediately implies (KC),
but the converse is not true, as we have seen in Figure 1.
The following result from [14] is now a direct consequence of Lemma 2.2:
Proposition 3.2. A convexity X is identified by a monotone transit function R if and only
if it is grounded (that is, X satisfies (KS)), and satisfies (KR).
It is now easy to characterize the transit functions that identify convexities:
Lemma 3.3. Let R be a monotone transit function. Then the transit sets {R(p, q) | p, q ∈
V } form a convexity if and only R satisfies (a’) and
(m’) For all u, v, x, y ∈ V with R(u, v) ∩ R(x, y) 6= ∅, there is p, q ∈ V such that
R(u, v) ∩R(x, y) = R(p, q).
Proof. Note that by assumption there is a 1-1 correspondence between transit sets and the
sets C ∈ X . Thus (a’) and (m’) directly translate to (K1) and (K2), respectively.
Axiom (m’) was introduced in [17] in the context of hierarchies. It is clear that (m’)
implies (m*), it does not imply (m), however. The example in Figure 1 also shows that
(m*) does not imply (m’).
Figure 2 gives a smaller counterexample. The interval function of a graphGwith vertex
set V , defined by I(x, y) = {z ∈ V | z lies on some shortest path between x and y}, is
not monotone in general [26]. In other words, the collection of intervals is in general not
sufficient to determine the “end points” of the interval.
4 Transit functions of hierarchy-like systems
Consider the following axioms for a set system X ⊆ 2V .
(H) A,B ∈ X implies A ∩B ⊆ {A,B, ∅}.
(PH) A ∈ X properly intersects at most one B ∈ X .
(WH) A,B,C ∈ X implies A ∩B ∩ C ⊆ {A ∩B,A ∩ C,B ∩ C}.
M. Changat et al.: Axiomatic characterization of transit functions of weak hierarchies 7
1
5 4
32
Figure 2: The interval function I of the graph G consists of singletons, a pairs with the
exception of I(1, 3) = {1, 2, 3} (a path), I(1, 4) = {1, 2, 4, 5} (red), and I(3, 5) =
{2, 3, 4, 5}. One easily checks that I satisfies (m). For the transit sets thus (K0), (KS),
(KR), and (KC) are satisfied. The pairwise intersection I(1, 4) ∩ I(3, 5) = {2, 4, 5}, how-
ever is not an interval, and hence (K2) fails. Furthermore, (a’) fails since {1, 2, 3, 4, 5} is
not a transit set.
A set system is called paired hierarchical [7] if it satisfies (K0), (K1), and (PH). If it
satisfies in addition (WH), it is called a weak hierarchy, if (H) holds, it is a hierarchy. It is
well known (and easy to see) that (H) implies (PH) implies (WH). Furthermore (H) implies
(K2). In the following we will sometimes write W instead of X for set systems that are
weak hierarchies.
Among several hierarchy-like set systems considered in the literature, see e.g. [8, 9],
the paired hierarchies are the most restrictive one, while the weak hierarchies are the most
general model. In the following we show that weak hierarchies, and hence also all its more
restrictive subclasses such as the paired hierarchies, are identified by transit functions if
and only if they form a convexity.
Proposition 4.1 ([21]). (WH) implies (KR).
Lemma 4.2. The canonical transit function R of a weak hierarchy X satisfies (m’).
Proof. By construction R(p, q) =
⋂
{C ∈ X | p, q ∈ C} =
⋂h
i=1 Ci with Ci ∈ X
for 1 ≤ i ≤ h and some h ≥ 1. By axiom (WH), the intersection of any three sets Ci
can be replaced by a pair, hence R(p, q) = C ′ ∩ C ′′ for two not necessarily distinct sets
C ′, C ′′ ∈ X .
It was shown in [2] that X is a weak hierarchy if and only if its closure under pair-wise
intersection is a weak hierarchy. Denote this weak hierarchy by X . Since (KR) holds for
X it follows that C ∈ X implies C ∈ X , since every C ∈ X has points r, s that otherwise
appear only in supersets of C, and thus any intersection of sets in X that contains r, s also
contains C. Since X is again a weak hierarchy, all pairwise intersections C ′ ∩ C ′′ ∈ X
also contain points u, v such that every set u, v ∈ D ∈ X satisfies C ′ ∩ C ′′ ⊆ D. Since
all sets D are intersections of elements of X containing u, v, we have C ′ ∩ C ′′ = R(u, v).
Thus all elements of X and their pairwise intersections are transit sets ofR. In other words,
{R(p, q) | p, q ∈ X} = X . Thus R satisfies (m’).
Since R satisfies (m), (m’), and by construction also (a’), Lemma 3.3 and Lemma 4.2
together imply
Corollary 4.3. The canonical transit function of a weak hierarchy is a convexity.
8 Art Discrete Appl. Math. 2 (2019) #P1.01
A similar result was shown with a different approach in [9]. This is in particular also
true of hierarchies [17] and paired hierarchies [9].
Because of the 1-1 correspondence between the clusters of a closed weak hierarchyW
and the transit sets of R, we can directly translate (KW) to the language of transit sets
(w) For any six points p, q, r, s, t, u ∈ V holds
R(p, q) ∩R(r, s) ∩R(t, u) ∈ {R(p, q) ∩R(r, s), R(p, q) ∩R(t, u),
R(r, s) ∩R(t, u)}.
Summarizing our discussion so far we have
Corollary 4.4. R identifies as a closed weak hierarchy if and only if it satisfies (m), (m’),
(a’), and (w).
Using a different route, [9] showed that a convexity is a weak hierarchy if and only if
its transit function satisfies
(w’) There are not three distinct points x1, x2, x3 ∈ V such that for all {h, i, j} =
{1, 2, 3} holds xh /∈ R(xi, xj).
In Section 6 we prove in the more general setting of k-ary transit functions that for a
monotone transit function that (w’) implies (a’) (Lemma 6.4) and (a’) ∧ (w) implies (w)
(Lemma 6.7).
Corollary 4.5. R identifies as a closed weak hierarchy if and only if it satisfies (m), (m’),
and (w’).
The independence of the axioms is established by the following examples.
Example 4.6 ((m), (m’), and (a’), but not (w)). For V = {a, b, c, d, e} consider
R(x, x) = {x} for all x ∈ V, R(a, b) = V, and
R(x, y) = {x, y} for all other pairs.
We can easily see that R satisfies (m), (m’), and (a’). We have
R(b, c) ∩R(c, d) ∩R(b, d) = ∅
but
R(b, c) ∩R(c, d) = {c},
R(b, c) ∩R(b, d) = {b}, and
R(c, d) ∩R(b, d) = {d}.
Therefore (w) does not hold.
Example 4.7 ((m’), (a’), and (w), but not (m)). For V = {a, b, c, d, e}, let
R(x, x) = {x} for all x ∈ V, R(a, b) = {a, b, c, d},
R(b, c) = {b, c, d}, R(b, d) = {b, d},
and for all other pairR(x, y) = V. We observe thatR satisfies (m’), (a’), and (w). However,
(m) fails since c, d ∈ R(a, b) and e ∈ R(c, d), but e /∈ R(a, b).
M. Changat et al.: Axiomatic characterization of transit functions of weak hierarchies 9
Example 4.8 ((m), (a’), and (w), but not (m’)). For V = {a, b, c, d, e} consider
R(a, c) = V, R(a, b) = {a, b},
R(a, d) = V − {c}, R(a, e) = {a, e},
R(b, x) = {b, x} for all x ∈ V, R(c, d) = {c, d},
R(c, e) = V − {a}, and R(d, e) = {d, e}.
It can be verified that R satisfies (m), (a’), and (w). But
R(a, d) ∩R(c, e) = {b, d, e} 6= R(x, y)
for each pair of elements x, y ∈ V. Hence R does not satisfy (m’).
Example 4.9 ((m), (m’), and (w), but not (a’)). For V = {a, b, c, d} consider
R(a, b) = {a, b}, R(a, c) = {a, b, c},
R(a, d) = {a, b, d}, R(b, c) = {b, c},
R(b, d) = {b, d}, and R(c, d) = {c, d}.
We can easily see that R satisfies (m), (m’), (w). However, there is no pair of points
x, y ∈ V such that R(x, y) = V; hence R does not satisfy (a’).
Barthélemy and Brucker [4] call a clustering system strongly binary if it satisfies (KR)
and
(ST) For each S ⊆ V , S 6= ∅, there exist u, v ∈ S such that S ⊆
⋂
{C | u, v ∈ C}.
and show that the strongly binary clustering systems are exactly the closed weak hierar-
chies. We note that (ST) implies (KC) by restricting the sets S to at most two elements.
In [9] it is shown that R is the transit function of paired hierarchy if it satisfies
(ph) For all u, v /∈ R(x, y) holds both x /∈ R(u, y) implies y ∈ R(v, x); and v /∈ R(u, y)
implies x ∈ R(v, y).
We note that in [9] this condition is mistakenly stated for “pairwise distinct u, v, x, y”. It
is also necessary to check the implications for u = v, however. In addition, if x = y, the
first precondition is always false, while the second conditions implies x ∈ R(u, x), which
is always true by (t1). This it suffices to drop the qualifier “pairwise distinct”.
Corollary 4.10. R identifies a closed paired hierarchy if and only if it satisfies (m), (m’),
(a’), and (ph).
4.1 Independence of (m), (m’), (a’) and (ph)
Example 4.11 ((m), (m’), and (a’), but not (ph)). Let V = {a, b, c, d, e} and define R on
V as follows: R(a, b) = V, and for all other pair of elements x, y ∈ V we set R(x, y) =
{x, y}. It is not difficult to check that R satisfies (m), (m’), and (a’). For R(c, d) we have
a, b /∈ R(c, d) but a /∈ R(c, b) and b /∈ R(a, d); d /∈ R(c, b) and a /∈ R(d, b). Thus R does
not satisfy the axiom (ph).
10 Art Discrete Appl. Math. 2 (2019) #P1.01
Example 4.12 ((m), (m’), and (ph), but not (a’)). Let V = {a, b, c, d} and define R on V
as follows:
R(x, x) = {x} for all x ∈ V, R(a, b) = {a, b, c},
R(a, c) = {a, c}, R(b, c) = {b, c},
R(a, d) = {a, c, d} = R(c, d), and R(b, d) = {b, d}.
It is easy to verify that R satisfies (m), (m’) and (ph), but there is no x, y ∈ V such that
R(x, y) = V.
Example 4.13 ((m), (a’), and (ph), but not (m’)). For V = {a, b, c, d, e} define
R : V × V → 2V as follows:
R(x, x) = {x} for all x ∈ V, R(a, b) = {a, b},
R(a, c) = {a, c}, R(a, d) = R(c, d) = {a, b, c, d},
R(a, e) = R(b, e) = R(c, e) = {a, b, c, e}, R(b, c) = {b, c},
R(b, d) = {b, d}, and R(d, e) = V.
We have R(a, d) ∩R(a, e) = {a, b, c} and R(x, y) 6= {a, b, c} for all x, y ∈ V . Therefore
R does not satisfy (m’). The transit sets R(a, b), R(a, c), R(b, c), and R(b, d) are subsets
of R(a, d) = R(c, d). Likewise, R(a, b), R(a, c), and R(b, c) are subsets of R(a, e) =
R(b, e) = R(c, e). Thus R satisfies the axiom (m). Furthermore, R satisfies (a’) because
R(d, e) = V . In order to verify (ph) we observe the following:
c, d /∈ R(a, b): we check that a /∈ R(b, c)⇒ b ∈ R(a, d) and d /∈ R(b, c)⇒ a ∈ R(b, d).
c, e /∈ R(a, b): we check that a /∈ R(b, c)⇒ b ∈ R(a, e) and d /∈ R(b, c)⇒ a ∈ R(b, e).
d, e /∈ R(a, b): we check that a ∈ R(b, d) and e /∈ R(b, d)⇒ a ∈ R(b, e).
b, d /∈ R(a, c): we check that a /∈ R(b, c)⇒ c ∈ R(a, d) and d /∈ R(b, c)⇒ a ∈ R(c, d).
b, e /∈ R(a, c): we check that a /∈ R(b, c)⇒ c ∈ R(a, e) and e /∈ R(b, c)⇒ a ∈ R(c, e).
d, e /∈ R(a, c): we check that a ∈ R(c, d) and e /∈ R(c, d)⇒ c ∈ R(d, e).
Example 4.14 ((m’), (a’), and (ph), but not (m)). Let V = {a, b, c, d},
R(a, c) = {a, b, c}, R(x, x) = {x}
and R(x, y) = V for all other pairs. Since b ∈ R(a, c) and R(a, b) = V * R(a, c), R does
not satisfy axiom (m). We can easily prove that the other axioms hold.
Let us now return to the convexities identified by a transit function. Condition (a’) can
be seen as a weak version of the axiom of antipodality
(a) For every x ∈ V there is x̄ ∈ V such that R(x, x̄) = V.
It is satisfied in particular by hierarchies. Since the restriction of a (weak) hierarchy to one
of its clusters is again a (weak) hierarchy, the following axiom also holds for hierarchies:
(a”) For every u, v ∈ V and x ∈ R(p, q) there is x̄ ∈ V such that R(x, x̄) = R(u, v).
Axiom (a), and thus (a”) does not hold for weak hierarchies in general, as the following
counter-example shows. Suppose C1, C2 ∈ W are proper subsets of V ∈ W such that
C1 ∪ C2 = V and C1 ∩ C2 = C3 ∈ W . Choose x ∈ C3, then R(x, y) ⊆ C1 for
M. Changat et al.: Axiomatic characterization of transit functions of weak hierarchies 11
y ∈ C1 and R(x, y) ⊆ C2 for y ∈ C2. Since C1 ∪ C2 = V, there is no x̄ ∈ V such
that R(x, x̄) = V. In fact, the set system W is even a paired hierarchy, since the only
proper intersection is C1 ∩ C2. Hence paired hierarchies, the most restrictive relaxation of
a hierarchy studied in some detail in the literature [7], also do not satisfy (a”).
Axiom (a”) can be translated into conditions of the clusters ofW
Lemma 4.15. Let X be a set system identified by a transit function R. Then R satisfies
(a”) if and only if X satisfies
(KA) For every C ∈ X and every collection QC ⊆ {C ′ ∈ X | C ′ ⊂ C} with non-empty
intersection
⋂
C′∈QC C
′ 6= ∅ holds
⋂
C′∈QC C
′ 6= C.
Proof. Due to the correspondence of transit sets and members of X , (a”) is equivalently
expressed as: For all C ∈ X and all x ∈ C, there is x̄ ∈ C such that {x, x̄} 6⊆ C ′ for any
C ′ ⊂ C. Equivalently, the union of all C ′ ⊂ C that contain any given x is a proper subset
of C, which in turn is equivalent to (KA).
In [17] we considered the axioms
(h’) x ∈ R(u, v)⇒ R(u, x) = R(u, v) or R(x, v) = R(u, v).
(h”) x ∈ R(u, v)⇒ R(u, v) = R(u, x) ∪R(x, v).
Lemma 4.16. (h’) ∧ (m) is equivalent to (h”) ∧ (a”).
Proof. We showed in [17] that (h”) follows from (h’) and monotone axiom. Furthermore,
(h’) implies (a”): for given u, v and x we may choose x̄ = u or x̄ = v.
Conversely, by (a”), for every x ∈ R(u, v) there is x̄ in R(u, v) such that R(x, x̄) =
R(u, v). By (h”), R(u, x) ∪ R(x, v) = R(u, v), hence x̄ ∈ R(u, x) or x̄ ∈ R(x, v), i.e.,
R(u, v) ⊆ R(x, x̄) ⊆ R(u, x) or R(u, v) ⊆ R(x, x̄) ⊆ R(x, v), i.e., (h’) is satisfied. It
was shown in [17] that (h”) implies (m).
The example in Figure 3 shows that (h”) and (a”) together are still insufficient to turn
the transit sets into even a weak hierarchy.
b c
a
d
R(a,b)=R(a,c)=R(a,d)={a,b,c,d}
R(b,c)={b,c}
R(c,d)={c,d}
R(b,d)={b,d}
Figure 3: The transit functionR satisfies (m), (m’), (a”), and (h’). It is not a weak hierarchy
sinceR(b, c)∩R(b, d)∩R(c, d) = ∅ but the three pairwise intersection each contain a point,
and thus also not a hierarchy.
12 Art Discrete Appl. Math. 2 (2019) #P1.01
5 Transit functions with arity > 2
Transit function or 2-ary functions have been generalized to k arguments to generalize
convexities generated by k-ary functions [14]. For n-ary convexities, also refer [28].
Definition 5.1. A function R : V × V . . .× V︸ ︷︷ ︸
k times
→ 2V is a transit function of arity k (or
k-ary transit function) on V if R satisfies the following axioms:
(kt1) u1 ∈ R(u1, u2, . . . , uk);
(kt2) R(u1, u2, . . . , uk) = R(π(u1, u2, . . . , uk)) for all ui ∈ V, where π(u1, u2, . . . , uk)
is any permutation of (u1, u2, . . . , uk);
(kt3) R(u, u, . . . , u) = {u} for all u ∈ V.
For an arbitrary set system X we define the k-ary function RX : V k → 2V by
RX (u1, u2, . . . , uk) =
⋂
{A ∈ X | u1, u2, . . . , uk ∈ A}. (5.1)
As in the case k = 2, the function R is monotone by construction, i.e., it satisfies
(km) For every x1, . . . , xk ∈ R(u1, . . . , uk) holds R(x1, . . . , xk) ⊆ R(u1, . . . , uk),
which in turn implies
(km*) For very T = {t1, . . . , tk} there is Q = {q1, . . . , qk} such that⋂
{R(u1, . . . , uk) | u1, . . . , uk ∈ V and T ∈ R(u1, . . . , uk)} = R(q1, . . . , qk).
We are again interested in the case that the transit sets of R identify the set system X ,
i.e.,
X = {RX (u1, u2, . . . , uk) | u1, u2, . . . , uk ∈ V }. (5.2)
Furthermore we consider the following generalizations of axioms (KR) and (KC):
(kKR) For all C ∈ X there is a set T ⊆ C with |T | ≤ k such that T ⊆ C ′ implies
C ⊆ C ′ for all C ′ ∈ X .
(kKC) For every T ⊆ V with |T | ≤ k holds
⋂
{C ∈ X | T ⊆ C} ∈ X .
Condition (kKR) was introduced in [14]. A necessary condition for R to explain X is that
every set C ∈ X is identified by at most k distinct points. The following statement is a
generalization of Lemma 2.2 and Theorem 2.6.
Theorem 5.2. A set system X ⊆ 2V is identified by a k-ary transit function if and only
if X satisfies (KS), (kKC), and (kKR). Conversely, a k-ary transit function identifies a set
system if and only if R satisfies (km).
Proof. We argue in parallel with the proof of Lemma 2.2. First we note that (kt3) and (KS)
ensure that X and the collection of transit sets both contain all singletons. As shown in
[14], (kKR) is equivalent to X ⊆ {R(u1, . . . , uk) | u1, . . . , uk ∈ V }. Property (km*) and
the definition R together imply that (kKC) is equivalent to {R(u1, . . . , uk) | u1, . . . , uk ∈
V } ⊆ X . The k-ary transit function defined by X is monotone. Conversely, one easily
checks that the transit sets of a monotone k-ary transit function satisfy (KS), and (kKC) (by
rewriting (km*)), and (kKR) holds by construction.
M. Changat et al.: Axiomatic characterization of transit functions of weak hierarchies 13
For a given set system X , the minimal value of k for which (kKR) holds is called the
arity of X [14].
Let X be a convexity on V. For a subset S ⊆ V, the smallest convex set containing S is
the convex hull of S, denoted by
〈S〉X =
⋂
{C ∈ X | S ⊆ C}. (5.3)
Basic concept from the theory of convexities can be generalized to arbitrary set systems:
Definition 5.3. Let X ⊆ 2V be a set system. For every subset S ⊆ V, the closure of S
with respect to a subset T (T -closure of S) is the set
〈S〉T =
⋂
{C ∈ X | S ⊆ C and T ⊆ C}. (5.4)
Note that in general 〈S〉T /∈ X . Interesting set systems arise by requiring 〈S〉T ∈ X
for all S ⊆ V and certain sets T .
Definition 5.4. Let X ⊆ 2V . A set S ⊆ V containing a subset T is XT -independent (T -
independent) if x /∈
⋂
{C ∈ X | T ⊆ (S \ {x}) ⊆ C} holds for all x ∈ S. Otherwise S
is T -dependent. In other words S is T -dependent if x ∈ 〈S \ {x}〉T , for all x ∈ S. The
T -rank r(X )T is the maximum cardinality of an XT -independent set S in V.
We can generalize the definition of well-known convexity invariants such as Carathéo-
dory number with respect to the T -closure in a set system X identified with a k-ary transit
function as follows.
Definition 5.5. The T -Carathéodory number c of a set system X is the smallest integer c
(if it exists) such that for any finite subset F of V containing T , we have
〈F 〉XT =
⋃
{〈S〉X | T ⊆ S ⊆ F, |S| ≤ c} . (5.5)
The usual definition of the Carathéodory number is recovered by dropping the restric-
tions on T .
The arity of a convexity [14] is the smallest integer (which exists since V is finite in our
setting) such that
X = {C ⊆ V | F ⊂ C, |F | ≤ c implies 〈F 〉C ⊆ C}. (5.6)
By axiom (K2), 〈S〉X ∈ X . Furthermore, (K2) implies (kKC) and every set system satisfies
(kKR) for sufficiently large k. Every grounded convexity X is therefore identified by a
transit function with sufficient arity. More precisely we have
Proposition 5.6 ([14]). X is a grounded convexity on V of arity k if and only if X is
identified by a k-ary transit function R on V .
Lemma 5.7. The k-ary transit function R identifies a convexity if and only if it satisfies
(km) and the following two conditions:
(ka’) There are vertices u1, u2, . . . , uk such that R(u1, u2, . . . , uk) = V .
(km’) If R(u1, u2, . . . , uk) ∩ R(v1, v2, . . . , vk) 6= ∅ then there is x1, x2, . . . , xk such
that R(u1, u2, . . . , uk) ∩R(v1, v2, . . . , vk) = R(x1, x2, . . . , xk).
Proof. Monotonicity follows directly from the construction of R in Equation (5.1). Condi-
tion (km’) states closure w.r.t. to intersection, i.e., is equivalent to (K2), and axiom (ka’) is
equivalent to (K1).
14 Art Discrete Appl. Math. 2 (2019) #P1.01
6 k-weak hierarchies
We now turn to k-weak hierarchies [3, 10, 19] as a generalization of weak hierarchies.
Definition 6.1. A k-weak hierarchy on V is a set system X ∈ 2V so that (K0), (K1), (KS)
and one of the two equivalent conditions
(kWH) A1, A2, A3, . . . , Ak+1 ∈ W implies
k+1⋂
i=1
Ai ⊆

k⋂
i=1,i6=j
Ai
∣∣∣ 1 ≤ j ≤ k + 1
.
(kWH’) There are no k + 1 elements x1, . . . , xk+1 such that xi ∈ Aj iff i 6= j.
is satisfied.
The equivalence of (kWH) and (kWH’) is established in [3, 19]. A k-weak hierarchy
is closed (w.r.t. intersection) if in addition (K2) is satisfied. It is clear that a closed k-weak
hierarchy is a convexity.
In the following we write x1, . . . , x̂i, . . . , xk+1 for sequence that leaves out xi, i.e.,
x1, . . . , xi−1, xi+1, . . . , xk+1.
Lemma 6.2. If the set system X is a closed k-weak hierarchy, then its rank satisfies
r(X ) ≤ k.
Proof. Let x1, . . . , xk+1 be distinct k + 1 convexly independent points. Hence xi /∈
〈x1, x2, . . . , x̂i, . . . , xk+1〉X . Let Ci = 〈x1, x2, . . . , x̂i, . . . , xk+1〉X . Then xi /∈ ∩Cj
for i = 1, . . . , k + 1. But some xi ∈
⋂k+1
i 6=j Cj . This is a contradiction to X being a closed
k-weak hierarchy.
It can be verified that if X is a closed k-weak hierarchy, then the Helly number and
Carathéodory number of X is at most k.
Without recourse to the theory of convexities we can prove directly
Lemma 6.3. A k-weak hierarchy satisfies (kKR).
Proof. First consider an arbitrary set system X ⊆ 2V and fix a set C ∈ X . Let T ⊆ C
be a set of minimum cardinality with the property that T ⊆ C ′ implies C ⊆ C ′ for all
C ′ ∈ X . Then for each a ∈ T there is a set Ca such that a /∈ Ca and T \ {a} ⊆ Ca. This
statement is a simple consequence of the minimality of T : For every a ∈ T , there must
be sets C ′ ∈ X not containing C that do not contain a. Otherwise a could be removed
from T , contradicting minimality. Now consider the set Xa of clusters that do not contain
a. Suppose none of them contain T \ {a}. Then C ′ ∈ Xa also lacks some other element
of a′ ∈ T and hence is recognizable as not containing C by virtue of a′. Thus a can be
removed from T , contradicting minimality of T .
Now let X be a k-weak hierarchy.
Suppose there is a C ∈ X such that the minimal set T has cardinality |T | ≥ k + 1.
Then there are at least k + 1 distinct points ai and corresponding clusters Ci ∈ W with
ai /∈ Ci and T \ {ai} ⊆ Ci. The intersection
⋂k+1
i=1 Ci contains none of the ai. By Axiom
(kWH), however, this intersection can be written as the intersection of at most k of these
clusters, and thus must contain at least one of the ai, a contradiction. Thus the cardinality
of T is at most k and the lemma follows.
M. Changat et al.: Axiomatic characterization of transit functions of weak hierarchies 15
The set system defined by the transit sets is not necessarily a convexity. We can con-
clude, however, that every transit set of a k-weak hierarchy is the intersection of at most k
others. We can generalize the notion of convex hulls and convex independence using the
concept of a weak closure, we can show that for k-weak hierarchies, the axiom (kw’) can
be extended.
Now consider the following axioms:
(kw) For any x1,x2, . . . ,xk+1 ∈ V k holds
k+1⋂
i=1
R(xi) ⊆

k+1⋂
i=1,i6=j
R(xi)
∣∣∣ 1 ≤ j ≤ k + 1
 .
(kw’) For every set of k + 1 distinct points x1, . . . , xk+1 ∈ V holds
xi ∈ R(x1, x2, . . . x̂i, . . . , xk+1).
Lemma 6.4. Axioms (kw’) and (km) imply (a’).
Proof. Consider a set V with at least k + 1 elements and let R be any k-ary transit func-
tion on V. If we have R(x1, . . . , xk) = V, we are done. Otherwise, there exists an
element xk+1 ∈ V that is not in R(x1, . . . , xk). By axiom (kw’) there exists i such
that xi ∈ R(x1, . . . , x̂i, . . . , xk+1). Since R satisfies (km), we have R(x1, . . . , xk) ⊆
R(x1, . . . , x̂i, . . . , xk+1). If R(x1, . . . , x̂i, . . . , xk+1) = V, we are done. Otherwise we
can find an element, say xk+2 ∈ V \R(x1, . . . , x̂i, . . . , xk+1). Again by the axiom (kw’),
there is some xj ∈ R(x1, . . . , x̂i, . . . , x̂j , . . . , xk+1, xk+2), and (km) implies
xj ∈ R(x1, . . . , x̂i, . . . , xj , . . . , xk+1, xk+2)
⊆ R(x1, . . . , x̂i, . . . , x̂j , . . . , xk+1, xk+2).
If R(x1, . . . , x̂i, . . . , x̂j , . . . , xk+1, xk+2) = V then we are done. Otherwise we can repeat
the argument. Since V contains only a finite number of elements, we eventually find a set
of k elements, say u1, . . . , uk from V so that R(u1, u2, . . . , uk) = V, which proves the
axiom (a’).
Lemma 6.5. Let X be a k-weak hierarchy identified by a k-ary transit function R. Then
the T -rank, r(XT ) ≤ k.
Proof. Let S = {x1, . . . , xk+1} be a T -independent set, i.e., for every xi ∈ S we have
xi /∈
⋂
{C ∈ X | T ⊆ (S \ {xi}) ⊆ C}. Let Ci ∈ X contains T and the set S \ {xi}
for every i. Then xi /∈ ∩Cj for i = 1, . . . , k + 1. But some xi ∈
⋂k+1
i 6=j Cj . This is a
contradiction to X being a k-weak hierarchy.
Remark 6.6. If the set system X identified with a k-ary transit function is a k-weak hier-
archy, then it can be shown easily that the T -Carathéodory number of X is at most k, as
any set S with |F | > k + 1 is T -dependent and hence any x ∈ 〈F 〉XT belongs to 〈Fi〉XT ,
where Fi is subset of F with |Fi| at most k.
Lemma 6.7. Let R be a k-ary transit function satisfying (km) and (a’). Then the axioms
(kw) and (kw’) are equivalent.
16 Art Discrete Appl. Math. 2 (2019) #P1.01
Proof. Let R satisfies (kw). Then the set system X identified by R is a k-weak hierarchy.
By definition of R, R(x1, . . . , xk) =
⋂
{Ci ∈ X | x1, . . . , xk ∈ Ci} = 〈x1, . . . , xk〉X .
Now by Lemma 6.5, any distinct k + 1 points, x1, . . . , xk+1 are dependent with respect to
some subset T contained in S. That is, xi ∈ R(x1, . . . , x̂i, . . . , xk+1) for some i. Hence
R satisfies (kw’).
Suppose R satisfies (kw’). Proving that R satisfies (kw) is equivalent to showing that
X is a k-weak hierarchy. Suppose this is not the case. Then we can find k + 1 distinct
elements x1, . . . , xk+1 ∈ S and C1, . . . , Ck+1 ∈ X such that xi ∈ Cj if and only if
i 6= j. By axiom (kw’), xi ∈ R(x1, . . . , x̂i, . . . , xk) for some xi, say xk+1. Then xk+1 ∈
Ci for i = 1, . . . , k and xk+1 /∈ Ck+1. Since x1, . . . , xk ∈ Ck+1, by definition of R,
R(x1, . . . , xk) ⊆ Ck+1. But xk+1 /∈ Ck+1 and xk+1 ∈ R(x1, . . . , xk), a contradiction.
Hence R satisfies (kw).
Theorem 6.8. The set system X induced by the k-ary transit function R on V is a closed
k-weak hierarchy if and only R satisfies the axioms (m), (m’), and (kw’).
Proof. Suppose X be a closed k-weak hierarchy induced by the k-ary transit function R
on V. Since a closed k weak hierarchy is a convexity, R satisfies (m), (m’). By Lemma 6.7
R satisfies (kw’).
Conversely suppose R satisfies (m), (m’), and (kw’). Since R satisfies (m) and (kw’),
by Lemma 6.4, R satisfies (a′). Hence the transit sets form a convexity X . By Lemma 6.7,
R satisfies (kw). Hence X is a closed k-weak hierarchy.
Remark 6.9. Axiom (kw) or (kw’) is a direct translation of (kWH). The condition is nec-
essary and sufficient given that (m), (m’), (a’) are equivalent to the transit sets forming a
convexity.
We briefly discuss the mutual dependencies between the axioms (km), (km’), (kw),
(kw’), and (a’). We already know that (km) and (kw’) implies (a’). Furthermore, if (km)
and (a’) are satisfied, then (kw) and (kw’) are equivalent.
Example 6.10 ((km), (a’), and (km’), but not (kw) and (kw’)). Let V = {x1, x2, . . . , xk+1}.
Define R on V as follows:
R(a1, a2, . . . , ak) = V,
for all other k-tuples
R(x1, x2, . . . , xk) = {x1, x2, . . . , xk}.
Example 6.11 ((km), (a’), (kw), and (kw’), but not (km’)). Let V = {x1, x2, . . . , xk+1}.
Let
R(x1, x3, x3, . . . , x3) = V,
R(x1, x4, x4, . . . , x4) = V − {x3},
R(x3, x5, x5, . . . , x5) = V − {x1},
R(x1, x2, x2, . . . , x2) = {x1, x2},
R(x1, x5, x5, . . . , x5) = {x1, x5},
R(x2, x2, x2, . . . , y) = {x2, y} for all y ∈ V
M. Changat et al.: Axiomatic characterization of transit functions of weak hierarchies 17
and set R(x1, x2, x3, . . . , xk) = V for all other k-tuples. We can see that
R(x1, x4, x4, . . . , x4) ∩R(x3, x5, x5, . . . , x5) = {x2, x4, . . . , xk+1}
but there is no k-tuple whose R-image is {x2, x4, . . . , xk+1}.
Example 6.12 ((km’), (a’), (kw), and (kw’), but not (km)). Consider V = {x1, x2, . . . ,
xk+1} and set
R(x1, x2, . . . , x2) = {x1, x2, x3, x4},
R(x2, x3, . . . , x3) = {x2, x3, x4},
R(x2, x4, . . . , x4) = {x2, x4},
and R(x1, x2, . . . xk) = V for all other k-tuples. Since R(x3, x4, . . . x4) = V we can see
that R does not satisfy the axiom (km).
Example 6.13 ((km), (km’), and (kw), but not (a’)). V = {x1, x2, . . . , xk+1}, consider R
on V by
R(x1, x2, . . . , x2) = {x1, x2},
R(x1, x3, . . . , x3) = {x1, x2, x3},
R(x1, x4, . . . , x4) = {x1, x2, x4},
R(x2, x3, . . . , x3) = {x2, x3},
R(x2, x4, . . . , x4) = {x2, x4},
R(x3, x4, . . . , x4) = {x3, x4},
and set R(u1, u2, . . . uk) = {u1, u2, . . . uk, x3} for all other k-tuples (u1, u2, . . . uk).
References
[1] K. Balakrishnan, M. Changat, A. K. Lakshmikuttyamma, J. Mathew, H. M. Mulder, P. G.
Narasimha-Shenoi and N. Narayanan, Axiomatic characterization of the interval function of a
block graph, Discrete Math. 338 (2015), 885–894, doi:10.1016/j.disc.2015.01.004.
[2] H.-J. Bandelt and A. W. M. Dress, Weak hierarchies associated with similarity measures—
an additive clustering technique, Bull. Math. Biol. 51 (1989), 133–166, doi:10.1016/
s0092-8240(89)80053-9.
[3] H.-J. Bandelt and A. W. M. Dress, An order theoretic framework for overlapping clustering,
Discrete Math. 136 (1994), 21–37, doi:10.1016/0012-365x(94)00105-r.
[4] J.-P. Barthélemy and F. Brucker, Binary clustering, Discrete Appl. Math. 156 (2008), 1237–
1250, doi:10.1016/j.dam.2007.05.024.
[5] A. Batbedat, Les isomorphismes HTS et HTE (après la bijection de Benzécri-Johnson), Metron
46 (1988), 47–59.
[6] P. Bertrand, Systems of sets such that each set properly intersects at most one other set—
application to cluster analysis, Discrete Appl. Math. 156 (2008), 1220–1236, doi:10.1016/j.
dam.2007.05.023.
[7] P. Bertrand and F. Brucker, On lower-maximal paired-ultrametrics, in: P. Brito, G. Cucumel,
P. Bertrand and F. de Carvalho (eds.), Selected Contributions in Data Analysis and Classifi-
cation, Springer, Berlin, Heidelberg, Studies in Classification, Data Analysis, and Knowledge
Organization, pp. 455–464, 2007, doi:10.1007/978-3-540-73560-1 42.
18 Art Discrete Appl. Math. 2 (2019) #P1.01
[8] P. Bertrand and J. Diatta, Weak hierarchies: a central clustering structure, in: F. Aleskerov,
B. Goldengorin and P. M. Pardalos (eds.), Clusters, Orders, and Trees: Methods and Appli-
cations, Springer, New York, NY, volume 92 of Springer Optimization and Its Applications,
2014, doi:10.1007/978-1-4939-0742-7 14, papers from the International Workshop held at the
National Research University Higher School of Economics (NRU HSE), Moscow, December
12 – 13, 2012.
[9] P. Bertrand and J. Diatta, Multilevel clustering models and interval convexities, Discrete Appl.
Math. 222 (2017), 54–66, doi:10.1016/j.dam.2016.12.019.
[10] P. Bertrand and M. F. Janowitz, The k-weak hierarchical representations: an extension
of the indexed closed weak hierarchies, Discrete Appl. Math. 127 (2003), 199–220, doi:
10.1016/s0166-218x(02)00206-8.
[11] M. Changat, S. Klavžar and H. M. Mulder, The all-paths transit function of a graph, Czechoslo-
vak Math. J. 51 (2001), 439–448, doi:10.1023/a:1013715518448.
[12] M. Changat, J. Mathew and H. M. Mulder, The induced path function, monotonicity and be-
tweenness, Discrete Appl. Math. 158 (2010), 426–433, doi:10.1016/j.dam.2009.10.004.
[13] M. Changat and J. Mathews, Induced path transit function, monotone and Peano axioms, Dis-
crete Math. 286 (2004), 185–194, doi:10.1016/j.disc.2004.02.017.
[14] M. Changat, J. Mathews, I. Peterin and P. G. Narasimha-Shenoi, n-ary transit functions in
graphs, Discuss. Math. Graph Theory 30 (2010), 671–685, doi:10.7151/dmgt.1522.
[15] M. Changat, H. M. Mulder and G. Sierksma, Convexities related to path properties on graphs,
Discrete Math. 290 (2005), 117–131, doi:10.1016/j.disc.2003.07.014.
[16] M. Changat, P. G. Narasimha-Shenoi, M. Kovše, F. Hosseinnezhad, S. Mohandas, A. Ra-
machandran and P. F. Stadler, Topological representation of the transit sets of k-point crossover
operators, 2017, arXiv:1712.09022 [math.CO].
[17] M. Changat, F. H. Nezhad and P. F. Stadler, Axiomatic characterization of transit functions
of hierarchies, Ars Math. Contemp. 14 (2018), 117–128, https://amc-journal.eu/
index.php/amc/article/view/831.
[18] M. Changat, G. N. Prasanth and J. Mathews, Triangle path transit functions, betweenness and
pseudo-modular graphs, Discrete Math. 309 (2009), 1575–1583, doi:10.1016/j.disc.2008.02.
043.
[19] J. Diatta, Dissimilarités multivoies et généralisations d’hypergraphes sans triangles, Math. In-
form. Sci. Humaines 138 (1997), 57–73, http://www.numdam.org/item?id=MSH_
1997__138__57_0.
[20] E. Diday, Orders and overlapping clusters in pyramids, in: J. de Leeuw, W. Heiser, J. Meul-
man and F. Critchley (eds.), Multidimensional Data Analysis, DSWO Press, Leiden, volume 7
of M&T Series, pp. 201–234, 1986, proceedings of a workshop held at Pembroke College,
Cambridge University, England, June 30 – July 2, 1985.
[21] A. W. M. Dress, Towards a theory of holistic clustering, in: B. Mirkin, F. R. McMorris, F. S.
Roberts and A. Rzhetsky (eds.), Mathematical Hierarchies and Biology, American Mathe-
matical Society, Providence, RI, volume 37 of DIMACS Series in Discrete Mathematics and
Theoretical Computer Science, pp. 271–290, 1997, papers from the DIMACS Workshop held
in Piscataway, NJ, November 13 – 15, 1996.
[22] P. Gitchoff and G. P. Wagner, Recombination induced hypergraphs: a new approach
to mutation-recombination isomorphism, Complexity 2 (1996), 37–43, doi:10.1002/(sici)
1099-0526(199609/10)2:1〈37::aid-cplx9〉3.0.co;2-c.
[23] M. A. Morgana and H. M. Mulder, The induced path convexity, betweenness, and svelte graphs,
Discrete Math. 254 (2002), 349–370, doi:10.1016/s0012-365x(01)00296-5.
M. Changat et al.: Axiomatic characterization of transit functions of weak hierarchies 19
[24] H. M. Mulder, The Interval Function of a Graph, volume 132 of Mathematical Centre Tracts,
Vrij University, Amsterdam, 1981.
[25] H. M. Mulder, Transit functions on graphs (and posets), in: M. Changat, S. Klavžar, H. M.
Mulder and A. Vijayakumar (eds.), Convexity in Discrete Structures, Ramanujan Mathematical
Society, Mysore, volume 5 of Ramanujan Mathematical Society Lecture Notes Series, 2008 pp.
117–130, joint proceedings of the International Instructional Workshop held in Thiruvanantha-
puram, March 22 – April 2, 2006 and the International Workshop on Metric and Convex Graph
Theory held in Barcelona, June 12 – 16, 2006.
[26] H. M. Mulder and L. Nebeský, Axiomatic characterization of the interval function of a graph,
European J. Combin. 30 (2009), 1172–1185, doi:10.1016/j.ejc.2008.09.007.
[27] P. F. Stadler and G. P. Wagner, Algebraic theory of recombination spaces, Evol. Comput. 5
(1997), 241–275, doi:10.1162/evco.1997.5.3.241.
[28] M. L. J. van de Vel, Theory of Convex Structures, volume 50 of North-Holland Mathematical
Library, Elsevier, Amsterdam, 1993.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P1.02
https://doi.org/10.26493/2590-9770.1261.9a0
(Also available at http://adam-journal.eu)
On König-Egerváry collections of maximum
critical independent sets∗
Vadim E. Levit
Department of Computer Science, Ariel University, Israel
Eugen Mandrescu
Department of Computer Science, Holon Institute of Technology, Israel
Received 20 April 2017, accepted 13 May 2018, published online 8 August 2018
Abstract
Let G be a simple graph with vertex set V (G). A set S ⊆ V (G) is independent if no
two vertices from S are adjacent. Let Ind(G) denote the family of all independent sets.
The graph G is said to be König-Egerváry if α (G) + µ (G) = |V (G)|, where α (G)
denotes the size of a maximum independent set and µ (G) is the cardinality of a maximum
matching. A family Γ ⊆ Ind(G) is a König-Egerváry collection if |
⋃
Γ|+ |
⋂
Γ| = 2α(G).
The number d (X) = |X| − |N(X)| is the difference of X ⊆ V (G), and a set A ∈
Ind(G) is critical if d(A) = max{d (I) : I ∈ Ind(G)}.
In this paper, we show that if the family of all maximum critical independent sets of
a graph G is a König-Egerváry collection, then G is a König-Egerváry graph. This result
generalizes one of our conjectures verified by Short in 2016.
Keywords: Maximum independent set, critical set, kernel, nucleus, core, corona, diadem, König-
Egerváry graph.
Math. Subj. Class.: 05C69, 05C70, 05A20
1 Introduction
Throughout this paper G is a finite simple graph with vertex set V (G) and edge set E(G).
If X ⊆ V (G), then G[X] is the subgraph of G induced by X . By G−W we mean either
the subgraph G[V (G)−W ], if W ⊆ V (G), or the subgraph obtained by deleting the edge
set W , for W ⊆ E(G). In either case, we use G− w, whenever W = {w}.
∗We would like to thank the anonymous reviewers for their constructive comments, which helped us to improve
the presentation of the paper.
E-mail addresses: levitv@ariel.ac.il (Vadim E. Levit), eugen−m@hit.ac.il (Eugen Mandrescu)
cb This work is licensed under http://creativecommons.org/licenses/by/3.0/
2 Art Discrete Appl. Math. 2 (2019) #P1.02
The neighborhood N(v) of a vertex v ∈ V (G) is the set {w : w ∈ V (G) and vw ∈
E (G)}. The neighborhood N(A) of A ⊆ V (G) is {v ∈ V (G) : N(v) ∩ A 6= ∅}, and
N [A] = N(A) ∪A.
A set S ⊆ V (G) is independent if no two vertices from S are adjacent, and by Ind(G)
we mean the family of all the independent sets of G. An independent set of maximum size
is a maximum independent set of G, and α(G) = max{|S| : S ∈ Ind(G)}.
Let Ω(G) denote the family of all maximum independent sets,
core(G) =
⋂
{S : S ∈ Ω(G)} [12], and corona(G) =
⋃
{S : S ∈ Ω(G)} [3].
If A ∈ Ω(G[N [A]]), then A is a local maximum independent set of G [13].
A matching is a set M of pairwise non-incident edges of G. A matching of maximum
cardinality, denoted µ(G), is a maximum matching.
For X ⊆ V (G), the number |X| − |N(X)| is the difference of X , denoted d(X).
The critical difference d(G) is max{d(X) : X ⊆ V (G)}. The number max{d(I) : I ∈
Ind(G)} is the critical independence difference of G, denoted id(G). Clearly, d(G) ≥
id(G). It was shown in [23] that d(G) = id(G) holds for every graph G. If A is an
independent set in G with d (X) = id(G), then A is a critical independent set [23].
Theorem 1.1 ([20]). Every local maximum independent set is a subset of a maximum
independent set.
Proposition 1.2 ([15]). Each critical independent set is a local maximum independent set.
Combining Theorem 1.1 and Proposition 1.2 one can conclude with the following.
Corollary 1.3 ([4]). Every critical independent set can be enlarged to a maximum inde-
pendent set.
For a graph G, let us denote
ker(G) =
⋂
{A : A is a critical independent set} [16],
MaxCritIndep(G) = {S : S is a maximum critical independent set} ,
nucleus(G) =
⋂
MaxCritIndep(G) [8], and
diadem(G) =
⋃
MaxCritIndep(G) [18].
Clearly, ker(G) ⊆ nucleus(G) and, by Corollary 1.3, the inclusion diadem(G) ⊆
corona(G) is true for each graph G.
Theorem 1.4 ([16]). For a graph G, the following assertions are true:
(i) ker(G) ⊆ core(G);
(ii) if A and B are critical in G, then A ∪B and A ∩B are critical as well.
Let us consider the graphs G1 and G2 from Figure 1: core(G1) = {a, b, c, d} and it is
a critical set, while core(G2) = {x, y, z, w} and it is not critical.
Moreover,
ker(G1) = {a, b, c} ⊂ core(G1) ⊂ {a, b, c, d, g} = nucleus(G1),
as MaxCritIndep(G1) = {{a, b, c, d, e, g} , {a, b, c, d, f, g}}.
In addition, notice that diadem(G1) ( corona(G1).
V. E. Levit and E. Mandrescu: On König-Egerváry collections of maximum critical . . . 3
w w w w w w w
w w w w w w
@
@
@
@
@
@
@
@
@
a
b
c e f g
d
G1
w w w w w w
w w w w w
@
@
@ 

 @
@
@
x
y
z
w
G2
Figure 1: Both G1 and G2 are not König-Egerváry graphs.
Theorem 1.5 ([7]). Let ∅ 6= Γ ⊆ Ω(G). If
⋃
Γ is critical, then
⋂
Γ is critical as well.
It is well known that α(G) + µ(G) ≤ |V (G)| holds for every graph G. Recall that if
α(G) + µ(G) = |V (G)|, then G is a König-Egerváry graph [5, 22]. For example, each
bipartite graph is a König-Egerváry graph. Various properties of König-Egerváry graphs
can be found in [2, 6, 9, 14, 17].
Theorem 1.6 ([11, 15]). For a graph G, the following assertions are equivalent:
(i) G is a König-Egerváry graph;
(ii) there exists some maximum independent set which is critical;
(iii) each of its maximum independent sets is critical.
If Γ,Γ′ are two collections of sets, we write Γ′ C Γ if
⋃
Γ′ ⊆
⋃
Γ and
⋂
Γ ⊆
⋂
Γ′
[8]. Clearly, the relation C is a preorder. The following theorem extends and generalizes
some findings from [19].
Theorem 1.7 ([8]). Let ∅ 6= Γ ⊆ Ω(G).
(i) If Γ′ ⊆ Ind(G) is such that Γ′ C Γ, then
∣∣∣⋂Γ′∣∣∣+ ∣∣∣⋃Γ′∣∣∣ ≤ ∣∣∣⋂Γ∣∣∣+ ∣∣∣⋃Γ∣∣∣.
(ii) 2α(G) ≤
∣∣∣⋂Γ∣∣∣+ ∣∣∣⋃Γ∣∣∣.
(iii) If, in addition, G is a König-Egerváry graph, then
∣∣∣⋂Γ∣∣∣+ ∣∣∣⋃Γ∣∣∣ = 2α(G), and, in
particular, |corona(G)|+ |core(G)| = 2α(G).
Let us notice that if S ∈ Ind(G), then G[N [S]] is not necessarily a König-Egerváry
graph. For instance, consider the graphG1 from Figure 1, and S1 = {d, g}, S2 = {d, e, g}.
Then, G1[N [S1]] is a König-Egerváry graph, while G1[N [S2]] is not a König-Egerváry
graph.
Theorem 1.8 ([11]). For every graph G, there is some X ⊆ V (G), such that:
(i) X = N [S] for every S ∈ MaxCritIndep(G);
(ii) G[X] is a König-Egerváry graph.
In other words, Theorem 1.8(i) claims that X = N [S] does not depend on the choice
of S ∈ MaxCritIndep(G). The critical independence number of G is defined as α′(G) =
max{|S| : S ∈ MaxCritIndep(G)} [11].
Recently, the following conjectures were validated in [21].
4 Art Discrete Appl. Math. 2 (2019) #P1.02
Conjecture 1.9 ([8]). If |nucleus(G)| + |diadem(G)| = 2α(G), then G is a König-
Egerváry graph.
Conjecture 1.10 ([7]). If |diadem(G)| = |corona(G)|, then G is a König-Egerváry
graph.
Conjecture 1.11 ([7]). |ker(G)|+ |diadem(G)| ≤ 2α(G) for every graph G.
An alternative proof of the inequality |ker(G)|+ |diadem(G)| ≤ 2α(G) may be found
in [1].
In this paper, we involve these findings in a more general framework, where they appear
as corollaries.
2 Results
Lemma 2.1. If S ∈ MaxCritIndep(G) and X = N [S], then
MaxCritIndep(G) C Ω (G[X]) .
Proof. By Proposition 1.2, we get that α (G[X]) = |S|. Since, in accordance with
Theorem 1.8(i), X = N [A] for each A ∈ MaxCritIndep(G), we may conclude that
MaxCritIndep(G) ⊆ Ω (G[X]). Hence, MaxCritIndep(G) C Ω (G[X]).
There is a graphGwith MaxCritIndep(G) ( Ω (G[X]), S ∈ MaxCritIndep(G), and
X = N [S]. For instance, the graph G from Figure 2 has
MaxCritIndep(G) = {{a, b, c, d, e, g} , {a, b, c, d, f, g}} , X = N [{a, b, c, d, e, g}],
while {a, b, c, d, e, k} ∈ Ω (G[X])−MaxCritIndep(G).
w w w w w w w
w w w w w w
@
@
@
@
@
@
@
@
@
a
b
c e f g
kdG
Figure 2: d({a, b, c, d, e, k}) = 1 < 2 = d(G).
Corollary 2.2 ([21]). If S ∈ MaxCritIndep(G) and X = N [S], then
diadem(G) ⊆ diadem(G[X]) and nucleus(G[X]) ⊆ nucleus(G).
Proof. In accordance with Theorem 1.8(ii), G[X] is a König-Egerváry graph. Hence, The-
orem 1.6(iii) implies that MaxCritIndep(G[X]) = Ω (G[X]). Therefore, Lemma 2.1
ensures that MaxCritIndep(G) C MaxCritIndep(G[X]), which, by definition, means
diadem(G) ⊆ diadem(G[X]) and nucleus(G[X]) ⊆ nucleus(G).
Lemma 2.3. If ∅ 6= Γ′ ⊆ MaxCritIndep(G) and ∅ 6= Γ ⊆ Ω(G), then∣∣∣⋂Γ′∣∣∣+ ∣∣∣⋃Γ′∣∣∣ ≤ 2α′(G) ≤ 2α(G) ≤ ∣∣∣⋂Γ∣∣∣+ ∣∣∣⋃Γ∣∣∣ .
V. E. Levit and E. Mandrescu: On König-Egerváry collections of maximum critical . . . 5
Proof. Let S ∈ MaxCritIndep(G) and X = N [S]. Since Γ′ ⊆ MaxCritIndep(G), and,
by Lemma 2.1, MaxCritIndep(G) C Ω (G[X]), we get Γ′ C Ω (G[X]). According to
Theorem 1.8(ii), G[X] is a König-Egerváry graph. Now, using Theorem 1.7(ii)–(iii), we
obtain ∣∣∣⋂Γ′∣∣∣+ ∣∣∣⋃Γ′∣∣∣ ≤ |core(G[X])|+ |corona(G[X])|
= 2α(G[X]) = 2α′(G) ≤ 2α(G) ≤
∣∣∣⋂Γ∣∣∣+ ∣∣∣⋃Γ∣∣∣ ,
as claimed.
If Γ′ = MaxCritIndep(G) and Γ = Ω(G), Lemma 2.3 immediately implies the fol-
lowing.
Corollary 2.4 ([21]). |nucleus(G)|+ |diadem(G)| ≤ 2α (G) for every graph G.
Since ker(G) ⊆ nucleus(G), Corollary 2.4 validates Conjecture 1.11.
Let us recall that a family of independent sets Γ is a König-Egerváry collection if∣∣∣⋂Γ∣∣∣+ ∣∣∣⋃Γ∣∣∣ = 2α(G) [8].
If there exists a König-Egerváry collection Γ ⊆ Ω(G), this does not oblige G to be a
König-Egerváry graph. For instance, the graph G from Figure 3 satisfies |corona(G)| +
|core(G)| = 2α(G), i.e., Ω(G) is a König-Egerváry collection, while G is not a König-
Egerváry graph.
w w w w w
w w w w
@
@
@
H
HHH
HH






a b c
d
e
f
G
Figure 3: core(G) = {a, b} and corona(G) = {a, b, c, d, e, f}.
Theorem 2.5. For a graph G, the following assertions are equivalent:
(i) G is a König-Egerváry graph;
(ii) every non-empty family of maximum critical independent sets ofG is a König-Egerváry
collection;
(iii) there is a König-Egerváry collection of maximum critical independent sets of G.
Proof. (i) =⇒ (ii): By Theorem 1.6, we obtain MaxCritIndep(G) = Ω (G). Further, in
accordance with Theorem 1.7(iii), each ∅ 6= Γ′ ⊆ MaxCritIndep(G) is a König-Egerváry
collection.
(ii) =⇒ (iii): Clear.
(iii) =⇒ (i): Let Γ′ ⊆ MaxCritIndep(G) be a König-Egerváry collection, S ∈ Γ′
and X = N [S]. Since, by Lemma 2.1, MaxCritIndep(G) C Ω (G[X]), we arrive at the
conclusion that Γ′ C Ω (G[X]), and hence,∣∣∣⋂Γ′∣∣∣+ ∣∣∣⋃Γ′∣∣∣ ≤ |nucleus(G[X])|+ |diadem(G[X])| .
6 Art Discrete Appl. Math. 2 (2019) #P1.02
According to Theorem 1.8(ii),G[X] is a König-Egerváry graph. Using Theorem 1.7(iii),
we infer that
2α(G) =
∣∣∣⋂Γ′∣∣∣+ ∣∣∣⋃Γ′∣∣∣
≤ |nucleus(G[X])|+ |diadem(G[X])| = 2α(G[X]) ≤ 2α(G).
Consequently, we obtain α(G[X]) = α(G), which ensures that G is a König-Egerváry
graph.
Since |nucleus(G)| + |diadem(G)| = 2α(G) means that MaxCritIndep(G) is a
König-Egerváry collection, Theorem 2.5 immediately implies the following.
Corollary 2.6 ([21]). If |nucleus(G)| + |diadem(G)| = 2α(G), then G is a König-
Egerváry graph.
It is worth mentioning that Corollary 2.6 validates Conjecture 1.9.
If ∅ 6= Γ ⊆ Ω(G), then none of
⋂
Γ and
⋃
Γ is necessarily critical. For instance,
consider the graph G from Figure 3, and Γ = {{a, b, c, e} , {a, b, c, f}} ⊆ Ω(G).
Lemma 2.7. Let Γ ⊆ Ω(G) and ∅ 6= Γ′ ⊆ MaxCritIndep(G), be such that for every
A ∈ Γ′ there exists S ∈ Γ enjoying A ⊆ S. If
⋂
Γ is a critical set, then the following
assertions are true:
(i)
⋂
Γ ⊆
⋂
Γ′;
(ii) Γ′ C Γ;
(iii)
∣∣∣⋂Γ′∣∣∣+ ∣∣∣⋃Γ′∣∣∣ ≤ ∣∣∣⋂Γ∣∣∣+ ∣∣∣⋃Γ∣∣∣;
(iv)
⋂
Γ′ =
⋂
Γ, if, in addition,
⋃
Γ′ =
⋃
Γ.
Proof.
(i) Let A ∈ Γ′ and S ∈ Γ, such that A ⊆ S. Since
⋂
Γ ⊆ S, it follows that
A ∪
⋂
Γ ⊆ S, and hence, A ∪
⋂
Γ is independent. By Theorem 1.4, we get
that A∪
⋂
Γ is a critical independent set. Since A ⊆ A∪
⋂
Γ and A is a maximum
critical independent set, we infer that
⋂
Γ ⊆ A. Thus,
⋂
Γ ⊆ A for every A ∈ Γ′.
Therefore,
⋂
Γ ⊆
⋂
Γ′.
(ii) By Part (i), we know that
⋂
Γ ⊆
⋂
Γ′. According to the hypothesis, every element
of Γ′ is included in some element of Γ. Hence, we deduce that
⋃
Γ′ ⊆
⋃
Γ.
(iii) The inequality follows from Part (ii) and Theorem 1.7(i).
(iv) Part (iii) implies
∣∣∣⋂Γ′∣∣∣ ≤ ∣∣∣⋂Γ∣∣∣, and using Part (i), we obtain⋂Γ = ⋂Γ′.
Proposition 2.8. Let Γ ⊆ Ω(G) and ∅ 6= Γ′ ⊆ MaxCritIndep(G) be such that for every
A ∈ Γ′ there exists S ∈ Γ such that A ⊆ S. If
⋃
Γ′ =
⋃
Γ, then G is a König-Egerváry
graph.
V. E. Levit and E. Mandrescu: On König-Egerváry collections of maximum critical . . . 7
Proof. Since, by Theorem 1.4(ii),
⋃
Γ′ is critical, we get that
⋃
Γ is critical. Hence,
according to Theorem 1.5, we infer that
⋂
Γ is critical. Applying Lemma 2.7, we obtain⋂
Γ =
⋂
Γ′. Further, we have
2α(G) ≤
∣∣∣⋂Γ∣∣∣+ ∣∣∣⋃Γ∣∣∣ = ∣∣∣⋂Γ′∣∣∣+ ∣∣∣⋃Γ′∣∣∣
≤ |core(G[X])|+ |corona(G[X])| = 2α(G[X]) ≤ 2α(G).
Consequently,
∣∣∣⋂Γ′∣∣∣ + ∣∣∣⋃Γ′∣∣∣ = 2α(G), which ensures, by Theorem 2.5, that G is a
König-Egerváry graph.
If Γ′ = MaxCritIndep(G) and Γ = Ω(G), Proposition 2.8 immediately implies the
following.
Corollary 2.9 ([21]). If diadem(G) = corona(G), then G is a König-Egerváry graph.
It is worth mentioning that Corollary 2.9 validates Conjecture 1.10.
3 Conclusions
In this paper we focus on interconnections between unions and intersections of maximum
critical independents sets of a graph. In [21], the question arises about polynomial-time
complexity of computing the following lower bound for the independence number
|nucleus(G)|+ |diadem(G)| ≤ 2α (G) .
Actually, Lemma 2.3 tells us that α′(G) is a better lower bound, since
|nucleus(G)|+ |diadem(G)| ≤ 2α′(G) ≤ 2α (G) ,
while α′(G) is polynomially computable [10]. It seems promising to pursue upper bounds
for α (G) in terms of α′(G). Let us call G a k-bounded graph if α (G) ≤ k · α′(G). For
instance, König-Egerváry graphs are 1-bounded, in accordance with Theorem 1.6. It is
worth emphasizing that the independence number can be computed in polynomial time for
König-Egerváry graphs, since in this case α(G) = α′(G).
Proposition 3.1. If S ∈ MaxCritIndep(G), then 2α′(G) = d (ker(G)) + |N [S]|.
Proof. Since ker(G) and S are critical sets of the graph G, we obtain
d (ker(G)) + |N [S]| = |ker(G)| − |N (ker(G))|+ |S|+ |N (S)|
= |S| − |N (S)|+ |S|+ |N (S)| = 2 |S| = 2α′(G),
which completes the proof.
By Theorem 1.6 and Proposition 3.1, if G is a König-Egerváry graph and
S ∈ MaxCritIndep(G), then we get
2α(G) = d (ker(G)) + |N [S]| ,
because α′(G) = α(G), and consequently,
2α (G) ≤ |ker(G)|+ |N [S]| .
8 Art Discrete Appl. Math. 2 (2019) #P1.02
Proposition 3.2 ([10]). There is a matching from N (S) into S for every critical indepen-
dent set S.
Proposition 3.3. If 2α (G) ≤ |ker(G)|+ |N [S]|, where S ∈ MaxCritIndep(G), then G
is 32 -bounded. More precisely,
α′(G) ≤ α (G) ≤ α′(G) + |N (ker(G))|
2
≤ 3
2
α′(G).
Proof. According to Proposition 3.2, there is a matching from N (S) into S, since S is
critical. Hence, |N [S]| ≤ 2α′(G). Therefore, taking account that ker(G) is a critical
independent set, we obtain
2α (G) ≤ |ker(G)|+ |N [S]| ≤ |ker(G)|+ 2α′(G) ≤ 3α′(G).
In accordance with Proposition 3.1, we get
|ker(G)|+ |N [S]| − |N (ker(G))| = 2α′(G) ≤ 2α (G) .
Thus
|ker(G)|+ |N [S]|
2
− |N (ker(G))|
2
= α′(G) ≤ α (G)
≤ |ker(G)|+ |N [S]|
2
= α′(G) +
|N (ker(G))|
2
≤ 3
2
α′(G),
since, ker(G) is a critical set and, by Proposition 3.2, there exists a matching from
N (ker(G)) into ker(G).
Let us emphasize that the bound α (G) ≤ α′(G) + |N(ker(G))|2 is of polynomial-time
complexity, since ker(G) [16] and α′(G) [10] can be computed polynomially.
References
[1] A. Bhattacharya, A. Mondal and T. S. Murthy, Problems on matchings and independent sets of
a graph, Discrete Math. 341 (2018), 1561–1572, doi:10.1016/j.disc.2018.02.021.
[2] F. Bonomo, M. C. Dourado, G. Durán, L. Faria, L. N. Grippo and M. D. Safe, Forbidden
subgraphs and the König-Egerváry property, Discrete Appl. Math. 161 (2013), 2380–2388,
doi:10.1016/j.dam.2013.04.020.
[3] E. Boros, M. C. Golumbic and V. E. Levit, On the number of vertices belonging to all maximum
stable sets of a graph, Discrete Appl. Math. 124 (2002), 17–25, doi:10.1016/s0166-218x(01)
00327-4.
[4] S. Butenko and S. Trukhanov, Using critical sets to solve the maximum independent set prob-
lem, Oper. Res. Lett. 35 (2007), 519–524, doi:10.1016/j.orl.2006.07.004.
[5] R. W. Deming, Independence numbers of graphs – an extension of the Koenig-Egervary theo-
rem, Discrete Math. 27 (1979), 23–33, doi:10.1016/0012-365x(79)90066-9.
[6] A. Jarden, V. E. Levit and E. Mandrescu, Two more characterizations of König-Egerváry
graphs, Discrete Appl. Math. 231 (2017), 175–180, doi:10.1016/j.dam.2016.05.012.
V. E. Levit and E. Mandrescu: On König-Egerváry collections of maximum critical . . . 9
[7] A. Jarden, V. E. Levit and E. Mandrescu, Critical and maximum independent sets of a graph,
Discrete Appl. Math. (2018), doi:10.1016/j.dam.2018.03.058, in press.
[8] A. Jarden, V. E. Levit and E. Mandrescu, Monotonic properties of collections of maximum
independent sets of a graph, Order (2018), doi:10.1007/s11083-018-9461-8, in press.
[9] E. Korach, T. Nguyen and B. Peis, Subgraph characterization of red/blue-split graph and Kőnig
Egerváry graphs, in: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Dis-
crete Algorithms, ACM, New York, 2006 pp. 842–850, doi:10.1145/1109557.1109650, held in
Miami, FL, January 22 – 24, 2006.
[10] C. E. Larson, A note on critical independence reductions, Bull. Inst. Combin. Appl. 51 (2007),
34–46.
[11] C. E. Larson, The critical independence number and an independence decomposition, European
J. Combin. 32 (2011), 294–300, doi:10.1016/j.ejc.2010.10.004.
[12] V. E. Levit and E. Mandrescu, Combinatorial properties of the family of maximum stable sets
of a graph, Discrete Appl. Math. 117 (2002), 149–161, doi:10.1016/s0166-218x(01)00183-4.
[13] V. E. Levit and E. Mandrescu, A new Greedoid: the family of local maximum stable sets of a
forest, Discrete Appl. Math. 124 (2002), 91–101, doi:10.1016/s0166-218x(01)00332-8.
[14] V. E. Levit and E. Mandrescu, On α+-stable König-Egerváry graphs, Discrete Math. 263
(2003), 179–190, doi:10.1016/s0012-365x(02)00528-9.
[15] V. E. Levit and E. Mandrescu, Critical independent sets and König-Egerváry graphs, Graphs
Combin. 28 (2012), 243–250, doi:10.1007/s00373-011-1037-y.
[16] V. E. Levit and E. Mandrescu, Vertices belonging to all critical sets of a graph, SIAM J. Discrete
Math. 26 (2012), 399–403, doi:10.1137/110823560.
[17] V. E. Levit and E. Mandrescu, On maximum matchings in König-Egerváry graphs, Discrete
Appl. Math. 161 (2013), 1635–1638, doi:10.1016/j.dam.2013.01.005.
[18] V. E. Levit and E. Mandrescu, Critical independent sets of a graph, 2014, arXiv:1407.7368
[cs.DM].
[19] V. E. Levit and E. Mandrescu, A set and collection lemma, Electron. J. Combin.
21 (2014), #P1.40, http://www.combinatorics.org/ojs/index.php/eljc/
article/view/v21i1p40.
[20] G. L. Nemhauser and L. E. Trotter, Jr., Vertex packings: structural properties and algorithms,
Math. Programming 8 (1975), 232–248, doi:10.1007/bf01580444.
[21] T. Short, On some conjectures concerning critical independent sets of a graph, Electron. J.
Combin. 23 (2016), #P2.43, http://www.combinatorics.org/ojs/index.php/
eljc/article/view/v23i2p43.
[22] F. Sterboul, A characterization of the graphs in which the transversal number equals the match-
ing number, J. Comb. Theory Ser. B 27 (1979), 228–229, doi:10.1016/0095-8956(79)90085-6.
[23] C. Q. Zhang, Finding critical independent sets and critical vertex subsets are polynomial prob-
lems, SIAM J. Discrete Math. 3 (1990), 431–438, doi:10.1137/0403037.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P1.03
https://doi.org/10.26493/2590-9770.1262.32a
(Also available at http://adam-journal.eu)
Maximizing general first Zagreb and
sum-connectivity indices for unicyclic graphs
with given independence number
Ioan Tomescu
Faculty of Mathematics and Computer Science, University of Bucharest,
Str. Academiei, 14, 010014 Bucharest, Romania
Received 27 November 2017, accepted 15 May 2018, published online 8 August 2018
Abstract
In this paper it is shown that in the class of unicyclic graphs of order n and independence
number s, the spider graph S∆(n, s) is the unique graph maximizing general first Zagreb
index 0Rα(G) for α > 1 and general sum-connectivity index χα(G) for α ≥ 1.
Keywords: Unicyclic graph, independence number, general first Zagreb index, general sum-connecti-
vity number, spider graph, Jensen inequality.
Math. Subj. Class.: 05C35, 05C69
1 Introduction
Let G be a simple graph having vertex set V (G) and edge set E(G). For a vertex u ∈
V (G), d(u) denotes the degree of u and N(u) the set of vertices adjacent with u. The
maximum vertex degree of G is denoted by ∆(G). K1,n−1 and Cn will denote, respec-
tively, the star and the cycle on n vertices. The distance between vertices u and v of a
connected graph, denoted by d(u, v), is the length of a shortest path between them. For
x ∈ V (G) and A ⊂ V (G), the distance d(x,A) between x and A is miny∈A d(x, y). If
x ∈ V (G), G−x denotes the subgraph of G obtained by deleting x and its incident edges.
Similar notations are G−xy and G+xy, where xy ∈ E(G) and xy /∈ E(G), respectively.
Given a graph G, a subset S of V (G) is said to be an independent set of G if every two
vertices of S are not adjacent. The maximum number of vertices in an independent set of
G is called the independence number ofG and is denoted by α(G). A unicyclic graphG of
order n is connected, has n edges and it consists of a cycle Cr, where 3 ≤ r ≤ n and some
vertex-disjoint trees having each a vertex common with Cr. It is not difficult to see that if
E-mail address: ioan@fmi.unibuc.ro (Ioan Tomescu)
cb This work is licensed under http://creativecommons.org/licenses/by/3.0/
2 Art Discrete Appl. Math. 2 (2019) #P1.03
G is a unicyclic graph of order n ≥ 3, then bn/2c ≤ α(G) ≤ n− 2. The lower bound can
be deduced since a unicyclic graph can be obtained from a tree, which is a bipartite graph,
by adding a new edge. The validity of the upper bound follows from the property that if
3 ≤ r ≤ n then α(Cr) ≤ r − 2 (equality holds only for r = 3 and r = 4).
For every n ≥ 3 and bn/2c ≤ s ≤ n − 2, the spider graph denoted by S∆(n, s)
is a unicyclic graph of order n consisting of 2s − n + 1 edges and n − s − 2 paths of
length 2 having a common endvertex with a triangleK3; in other words, it is obtained from
K1,s+1 + e by attaching a pendant edge to n− s− 2 pendant vertices of K1,s+1 + e. We
have α(S∆(n, s)) = s.
The graph, denoted by Hn, is defined as follows: for n = 2k it consists of a cycle Ck
and k pendant vertices adjacent each to a single vertex of Ck such that each vertex of Ck
has degree three. For n = 2k + 1, Hn is composed from Ck+1 and k pendant vertices
adjacent each to a single vertex of Ck+1 such that k vertices of Ck have degree three and
one vertex has degree two.
For other notations in graph theory, we refer [16].
The Randić index R(G) [12], one of the most used molecular descriptors in structure-
property and structure-activity relationship studies [5, 6, 7, 11, 13, 14], was defined as
R(G) =
∑
uv∈E(G)
(d(u)d(v))−1/2.
The general Randić connectivity index (or general product-connectivity index) of G,
denoted by Rα, was defined by Bollobás and Erdős [1] as
Rα = Rα(G) =
∑
uv∈E(G)
(d(u)d(v))α,
where α is a real number. Then R−1/2 is the classical Randić connectivity index and for
α = 1 it is also known as second Zagreb index and denoted by M2(G).
This concept was extended to the general sum-connectivity index χα(G) in [20], which
is defined by
χα(G) =
∑
uv∈E(G)
(d(u) + d(v))α,
where α is a real number. The sum-connectivity index χ−1/2(G) was proposed in [19].
The general first Zagreb index (sometimes referred as “zeroth-order general Randić
index”), denoted by 0Rα(G) was defined as
0Rα(G) =
∑
u∈V (G)
d(u)α,
where α is a real number. For α = −1/2 this index was defined in [9] and [10] and
for α = 2 it is also known as first Zagreb index and denoted by M1(G). Notice that
χ1(G) =
0R2(G) = M1(G).
Thus, the general Randić connectivity index generalizes both the ordinary Randić con-
nectivity index and the second Zagreb index, while the general sum-connectivity index
generalizes both the ordinary sum-connectivity index and the first Zagreb index [20].
Several extremal properties of the sum-connectivity and general sum-connectivity in-
dices for trees, unicyclic graphs and general graphs were given in [3, 4, 19, 20].
I. Tomescu: Maximizing general first Zagreb and sum-connectivity indices for unicyclic graphs . . . 3
Das, Xu and Gutman [2] proved that in the class of trees of order n and independence
number s, the spur Sn,s maximizes both first and second Zagreb indices and this graph is
unique with these properties. Tomescu and Jamil [15] showed that in the same class of
trees T , Sn,s is the unique graph maximizing general first Zagreb index 0Rα(T ) for α > 1
and general sum-connectivity index χα(T ) for α ≥ 1.
In this paper, we show that the spider graph S∆(n, s) is the unique graph maximizing
general first Zagreb index 0Rα(G) for α > 1 and general sum-connectivity index χα(G)
for α ≥ 1 in the set of unicyclic graphs of order n and independence number s (bn/2c ≤
s ≤ n− 2).
2 Preliminary results
The following inequality may be deduced in a straightforward way:
Lemma 2.1. Let x > 0. If β < 0 then (1 + x)β > 1 + βx.
The general first Zagreb index and general sum-connectivity index of S∆(n, s) are
given by:
0Rα(S∆(n, s)) = (s+ 1)
α + s(1− 2α) + 2αn− 1;
χα(S∆(n, s)) = (n− s)(s+ 3)α + (2s− n+ 1)(s+ 2)α + (n− s− 2)3α + 4α.
The cycle Cn has independence number equal to bn/2c.
Lemma 2.2. Let n ≥ 5. Then (2.1) holds for α > 1 and (2.2) holds for α ≥ 1:
0Rα(S∆(n, bn/2c)) > 0Rα(Cn) (2.1)
χα(S∆(n, bn/2c)) > χα(Cn). (2.2)
Proof. We get 0Rα(Cn) = n2α and χα(Cn) = n4α. If n is even, n = 2k, (2.1) can be
written as
(k + 1)α − 2αk + k − 1 > 0, (2.3)
where k ≥ 3 and α > 1. Consider the function ϕ(x) = (x + 1)α − 2αx + x − 1,
where x ≥ 3. We get ϕ′(x) = α(x + 1)α−1 − 2α + 1 ≥ α4α−1 − 2α + 1. By letting
ψ(y) = y4y−1− 2y + 1, where y > 1, we have ψ′(y) = 4y−1(1 + y ln 4)− ln 2 · 2y . Since
2y > 2 we deduce
ψ′(y) > 2y
(
1 + y ln 4
2
− ln 2
)
> 2y
(
1 + ln 4
2
− ln 2
)
= 2y−1 > 0.
Because ψ(1) = 0 we have ψ(y) > 0 for y > 1, thus ϕ(x) is strictly increasing for
x ≥ 3 and α > 1. It follows that it is sufficient to prove (2.3) for k = 3. For k = 3 (2.3)
becomes
4α − 3 · 2α + 2 > 0, (2.4)
or (2α − 1)(2α − 2) > 0, which is true for α > 1.
If n = 2k + 1, where k ≥ 2, (2.1) becomes (2.3) in which k ≥ 2. For k = 2 (2.3)
yields 3α − 2 · 2α + 1 > 0, which holds by Jensen inequality since function xα is strictly
convex for α > 1.
4 Art Discrete Appl. Math. 2 (2019) #P1.03
In order to prove (2.2) consider first the case n even, n = 2k. In this case (2.2) is
k(k + 3)α + (k + 2)α + (k − 2)3α − (2k − 1)4α > 0, (2.5)
where k ≥ 3 and α ≥ 1. For k = 3 (2.5) becomes 3 · 6α + 5α + 3α − 5 · 4α > 0, which is
true since 5α + 3α ≥ 2 · 4α by Jensen inequality and 3 · 6α > 3 · 4α.
Consider the function ξ(x) = x(x+ 3)α + (x+ 2)α + (x− 2)3α− (2x− 1)4α, where
x ≥ 3. We get ξ′(x) = (x+ 3)α + αx(x+ 3)α−1 + α(x+ 2)α−1 + 3α − 2 · 4α. We have
(x+ 3)α + 3α − 2 · 4α ≥ 6α + 3α − 2 · 4α ≥ 2 · 4.5α − 2 · 4α > 0 by Jensen inequality.
This implies that ξ′(x) > 0, hence ξ(x) is strictly increasing. Thus (2.5) is valid since it
holds for k = 3. If n = 2k + 1, where k ≥ 2, the proof is similar, using in the same way
Jensen inequality.
Lemma 2.3. If n ≥ 5 and α ≥ 1, χα(S∆(n, s)) is strictly increasing in s for bn/2c ≤
s ≤ n− 2.
Proof. Let
f(x) = (n− x)(x+ 3)α + (2x− n+ 1)(x+ 2)α + (n− x)3α.
We have χα(S∆(n, s)) = f(s)− 2 · 3α + 4α. We will show that f(x) is strictly increasing
in x for n ≥ 5 and 2 ≤ (n− 1)/2 ≤ x ≤ n− 2. We have
f ′(x) = (x+ 3)α−1(α(n− x)− x− 3) + (x+ 2)α−1(2x+ 4 + 2αx− αn+ α)− 3α.
If the coefficient of (x + 3)α−1 is greater than or equal to zero, then f ′(x) > 0 since
2αx− αn+ α ≥ 0, which implies
(x+ 2)α−1(2x+ 4 + 2αx− αn+ α)− 3α ≥ 2(x+ 2)α − 3α ≥ 2 · 4α − 3α > 0.
The coefficient of (x+ 3)α−1 is
x(−α− 1) + αn− 3 ≥ (n− 2)(−α− 1) + αn− 3 = −n+ 2α− 1 ≥ 0
for α ≥ (n+ 1)/2.
Suppose that 1 ≤ α < (n + 1)/2. We will also prove that f ′(x) > 0 in this case. We
can write f ′(x) = (x+ 3)α−1E(n, x, α), where
E(n, x, α) = α(n−x)−x−3+
(
1 +
1
x+ 2
)1−α
(2x+4+2αx−αn+α)− 3
α
(x+ 3)α−1
.
Lemma 2.1 yields (
1 +
1
x+ 2
)1−α
> 1 +
1− α
x+ 2
,
which implies
E(n, x, α) > (x+ 1)(α+ 1) + 2− 2α2 + α(α− 1)(n+ 3)
x+ 2
− 3
α
(x+ 3)α−1
. (2.6)
I. Tomescu: Maximizing general first Zagreb and sum-connectivity indices for unicyclic graphs . . . 5
Since α−1 < n−12 and x ≥
n−1
2 it follows that x > α−1, which implies (x+1)(α+1) >
α2 + α. Since x+ 2 < n+ 3 we get α(α−1)(n+3)x+2 ≥ α(α− 1) and from (2.6) we obtain
E(n, x, α) > 2− 3
α
(x+ 3)α−1
.
If α ≥ 2 then maxx≥2 3
α
(x+3)α−1 = 5(
3
5 )
α ≤ 95 , which implies E(n, x, α) > 0. The same
conclusion holds if 1 ≤ α < 2 since in this case we have
x ≥ 2 > α, (x+ 1)(α+ 1) > (α+ 1)2, 3
α
(x+ 3)α−1
= 3
(
3
x+ 3
)α−1
≤ 3
and (2.6) yields E(n, x, α) > (α+ 1)2 + 2− 2α2 + α(α− 1)− 3 = α ≥ 1.
The following observation will be useful.
Lemma 2.4. Let G be a graph and x ∈ V (G), which is adjacent to pendant vertices
v1, . . . , vr. If r ≥ 2 then any maximum independent subset of V (G) contains v1, . . . , vr.
Lemma 2.5. The function
h(x) = (x− 2)((x+ a)α − (x+ a− 1)α)
is strictly increasing for x ≥ 2, a ≥ 1 and α ≥ 1.
Proof. We get
h′(x) = (x+ a)α − (x+ a− 1)α + α(x− 2)((x+ a)α−1 − (x+ a− 1)α−1) > 0
for x ≥ 2, a ≥ 1 and α ≥ 1.
3 Main results
By simple inspection we can see that for n = 6 spider graph S∆(6, s) is the unique extremal
graph G of order six and independence number s, 3 ≤ s ≤ 4, having maximum 0Rα(G)
unless s = 3 and 1 < α < 2, when 0Rα(S∆(6, 3)) < 0Rα(H6) (note that H6 consists
of a triangle K3 and three pendant vertices adjacent to different vertices of K3). For n =
6, s = 3 and α ∈ {1, 2} both graphs H6 and S∆(6, 3) are extremal. The case n ≥ 7 is
settled below.
Theorem 3.1. Let n ≥ 7, bn/2c ≤ s ≤ n − 2 and G be a unicyclic graph of order n
with independence number s. Then for every α > 1, 0Rα(G) is maximum if and only if
G = S∆(n, s).
Proof. The proof is by induction on n. For n = 7 the proof is by inspection, using Jensen
inequality or mathematical software [17]; there are 4 graphs with s = 3, 15 graphs with
s = 4 and 5 graphs having s = 5.
Let n ≥ 8 and suppose that the property is true for all unicyclic graphs of order n− 1.
LetG be a unicyclic graph of order n and independence number s having maximum general
first Zagreb index. By Lemma 2.2 0Rα(Cn) cannot be maximum; it follows that ∆(G) ≥
3. Its independence number verifies s ≥ 4. Denote by C the unique cycle of G, whose
length is at most n− 1. G has at least one pendant vertex. Let x1 be a pendant vertex such
that the distance d(x1, C) is maximum. We shall consider two cases:
6 Art Discrete Appl. Math. 2 (2019) #P1.03
• Case 1: d(x1, C) ≥ 2 and
• Case 2: d(x1, C) = 1.
Case 1. Let x1, x2, . . . , xp, where p ≥ 3 and xp ∈ C be the unique path from x1 to C. By
letting d(x2) = d2, since for every vertex u in N(u) at most two vertices are adjacent, we
obtain s ≥ ∆(G)− 1 ≥ d2 − 1, or d2 ≤ s+ 1. Other two subcases may hold:
• Subcase 1.1: α(G− x1) = α(G)− 1 and
• Subcase 1.2: α(G− x1) = α(G).
Subcase 1.1. By the induction hypothesis we can write
0Rα(G) =
0Rα(G− x1) + 1 + dα2 − (d2 − 1)α
≤ 0Rα(S∆(n− 1, s− 1)) + 1 + dα2 − (d2 − 1)α
= sα + 2α(n− s) + s− 2 + 1 + dα2 − (d2 − 1)α.
Since the function xα − (x − 1)α is strictly increasing for x ≥ 1 and α > 1, it follows
that dα2 − (d2 − 1)α ≤ (s+ 1)α − sα, which implies 0Rα(G) ≤ 0Rα(S∆(n, s)), equality
holding if and only if d2 = s + 1. But this equality is not possible. If d2 = s + 1 holds,
then two vertices in N(x2) are adjacent since otherwise we would have s ≥ d2. In this
case, since x2 /∈ C, G would have at least two cycles, a contradiction.
Consequently, 0Rα(G) < (s+ 1)α + 2α(n− s) + s− 1 = 0Rα(S∆(n, s)), a contra-
diction.
Subcase 1.2. Next we assume that α(G − x1) = α(G). If x2 would be adjacent to a
vertex w 6= x1, x3, the degree of w cannot be greater than one, since in this case the path
x1, . . . , xp would not have maximum length. It follows that d(w) = 1 and by Lemma 2.4
every maximum independent set of vertices of G includes both x1 and w. This implies
α(G− x1) = α(G)− 1, which contradicts the hypothesis. It follows that d2 = 2. We can
write
0Rα(G) =
0Rα(G− x1) + 2α
≤ 0Rα(S∆(n− 1, s)) + 2α
= (s+ 1)α + 2α(n− 1− s) + s− 1 + 2α
= 0Rα(S∆(n, s)).
The equality holds if and only ifG−x1 = S∆(n−1, s) and pendant vertex x1 is adjacent to
a pendant vertex of S∆(n−1, s). Let u be the vertex of degree s+1 of S∆(n−1, s). If x1 is
adjacent to a pendant vertex v2 of S∆(n− 1, s) such that d(v2, u) = 2, the resulting graph
G has α(G) = s+ 1, which contradicts the hypothesis. We deduce that x1 is adjacent to a
pendant vertex which is adjacent to u in S∆(n− 1, s), which implies that G = S∆(n, s).
Case 2. In this case we shall also consider two subcases:
• Subcase 2.1: There exists a pendant vertex x1 such that d(x1, C) = 1 and
α(G− x1) = α(G)− 1; and
• Subcase 2.2: For all pendant vertices x we have d(x,C) = 1 and α(G−x) = α(G).
I. Tomescu: Maximizing general first Zagreb and sum-connectivity indices for unicyclic graphs . . . 7
Subcase 2.1. As for Subcase 1.1 we get d2 = d(x2) ≤ s + 1 and by the same arguments
0Rα(G) ≤ (s+1)α+2α(n−s)+s−1 = 0Rα(S∆(n, s)) holds, with equality if and only
if d(x2) = s+1 andG−x1 = S∆(n−1, s−1). It follows that x1 is adjacent to the vertex
of degree s in S∆(n − 1, s − 1), i.e., G = S∆(n, s). Since d(x1, C) = max{d(x,C) :
d(x) = 1} = 1, this equality is possible only for s = n− 2.
Subcase 2.2. In this case a vertex of C may be adjacent to a single pendant vertex x, since
otherwise we would have α(G−x) = α(G)−1 by Lemma 2.4. We deduce thatG consists
of C and some pendant vertices adjacent to vertices of C such that each vertex y ∈ C has
its degree d(y) ∈ {2, 3}. We shall prove that in this case 0Rα(G) < 0Rα(S∆(n, s)), a
contradiction.
Suppose that on C there exist four consecutive vertices x, u, v, y such that d(u) =
d(v) = 2. In this case we shall define a new unicyclic graph G1 of order n by G1 =
G−vy+uy. We deduce 0Rα(G1)− 0Rα(G) = 3α+1α−2 ·2α > 0 by Jensen inequality
since α > 1. If on C there exist six vertices x, r, y, p, s, q (y may coincide with p) such
that d(x) = d(y) = d(p) = d(q) = 3 and d(r) = d(s) = 2, we define a new unicyclic
graphG2 with the same vertex set as follows: G2 = G−{xr, ry}+{xy, rs}. By the same
argument we obtain 0Rα(G2) > 0Rα(G). If G 6= Hn, by applying step by step this type
of transformations we get Hn, such that 0Rα(Hn) > 0Rα(G).
We have 0Rα(Hn) = k · 3α + k for n = 2k and k · 3α + 2α + k for n = 2k + 1 and
0Rα(S∆(2k, k)) = (k + 1)
α + k2α + k − 1 and
0Rα(S∆(2k + 1, k) = (k + 1)
α + (k + 1)2α + k − 1.
In both cases, n = 2k or n = 2k + 1 the inequalities 0Rα(S∆(n, bn/2c)) > 0Rα(Hn)
coincide with
(k + 1)α − k(3α − 2α)− 1 > 0 (3.1)
for every k ≥ 4 and α > 1. Let g(x) = (x+ 1)α − x(3α − 2α)− 1. We have
g(4) = 5α − 4 · 3α + 4 · 2α − 1 > 0 for α > 1 [17] and
g′(x) = α(x+ 1)α−1 − 3α + 2α.
g′(x) is strictly increasing and g′(4) = α5α−1−3α+2α > 0 for α > 1 [17]. It follows that
g′(x) > 0, hence g(x) is strictly increasing for x ≥ 4 and α > 1 and (3.1) is proved. Con-
sequently, we can write 0Rα(G) ≤ 0Rα(Hn) < 0Rα(S∆(n, bn/2c)) ≤ 0Rα(S∆(n, s))
since the last term is strictly increasing in s, a contradiction.
Since the function 0Rα(S∆(n, s)) is strictly increasing in s, bn/2c ≤ s ≤ n − 2, we
deduce:
Corollary 3.2 ([8, 18]). Let G be a unicyclic graph of order n ≥ 7. Then for every α > 1,
0Rα(G) is maximum if and only if G = S∆(n, n− 2) = K1,n−1 + e.
A similar result holds for general sum-connectivity index.
Theorem 3.3. Let n ≥ 3, bn/2c ≤ s ≤ n − 2 and G be a unicyclic graph of order n
with independence number s. Then for every α ≥ 1, χα(G) is maximum if and only if
G = S∆(n, s). For n = 6 and α = 1 there exists another extremal graph, H6.
8 Art Discrete Appl. Math. 2 (2019) #P1.03
Proof. We shall use induction on n in the same way as in the proof of Theorem 3.1. For
n = 3 there is a unique unicyclic graph on three vertices, S∆(3, 1) = K3. For n = 4 there
are two unicyclic graphs, C4 and K1,3 + e = S∆(4, 2) and the theorem is verified.
Let n ≥ 5 and suppose that the theorem is true for all unicyclic graphs of order n− 1.
Let G be a unicyclic graph of order n and independence number s having maximum gen-
eral sum-connectivity index. By Lemma 2.2 χα(Cn) cannot be maximum; it follows that
∆(G) ≥ 3. Denote by C the unique cycle of G, whose length is at most n− 1. Let x1 be a
pendant vertex such that the distance d(x1, C) is maximum. We shall consider four cases:
• Case 1.1: d(x1, C) ≥ 2 and α(G− x1) = α(G)− 1;
• Case 1.2: d(x1, C) ≥ 2 and α(G− x1) = α(G);
• Case 2.1: max{d(x,C) | d(x) = 1} = 1 and there exists a pendant vertex x1 such
that α(G− x1) = α(G)− 1;
• Case 2.2: max{d(x,C) | d(x) = 1} = 1 and for all pendant vertices x we have
α(G− x) = α(G).
Case 1.1. Let x1, x2, x3, . . . be the path between x1 and C. Since this path has maximum
length, it follows that x3 is the unique vertex in N(x2) such that d3 = d(x3) ≥ 2. As in
the proof of Theorem 3.1 we deduce d2 = d(x2) ≤ s+ 1.
We have
χα(G) = χα(G− x1) + (d2 + 1)α + (d2 − 2)((d2 + 1)α − dα2 ) + (d2 + d3)α
− (d2 + d3 − 1)α.
x2 being adjacent to d2−1 pendant vertices and inG−x2x3 the degree of x3 being d3−1,
it follows that d2− 1 + d3− 2 ≤ s, or d2 + d3 ≤ s+ 3. We get (d2 + 1)α ≤ (s+ 2)α with
equality if and only if d2 = s+1 and (d2+d3)α−(d2+d3−1)α ≤ (s+3)α−(s+2)α with
equality only if d2 +d3 = s+ 3. Since by Lemma 2.5 the function (x−2)((x+ 1)α−xα)
is strictly increasing in x for x ≥ 2 and α ≥ 1, by the induction hypothesis we obtain
χα(G) ≤ χα(S∆(n− 1, s− 1)) + (s+ 2)α + (s− 1)((s+ 2)α − (s+ 1)α)
+ (s+ 3)α − (s+ 2)α
= (n− s)(s+ 2)α + (2s− n)(s+ 1)α + (n− s− 2)3α + 4α
+ (s− 1)((s+ 2)α − (s+ 1)α) + (s+ 3)α.
By denoting the last expression by F (n, s, α), we have F (n, s, α) ≤ χα(S∆(n, s)) if and
only if
(n− s− 1)(s+ 3)α + (n− s− 1)(s+ 1)α ≥ 2(n− s− 1)(s+ 2)α. (3.2)
Since n − s − 1 ≥ 1, (3.2) is equivalent to (s + 3)α + (s + 1)α ≥ 2(s + 2)α, which
is true by Jensen inequality, with equality only for α = 1. If the inequality is strict, G
cannot be extremal, a contradiction. For α = 1 we have equality only for d2 = s + 1 and
d2 +d3 = s+3, which implies d3 = 2 andG−x1 = S∆(n−1, s−1), x2 being the vertex
of degree s in S∆(n− 1, s− 1). In this case we have d(x1, C) = 1, which contradicts the
hypothesis.
I. Tomescu: Maximizing general first Zagreb and sum-connectivity indices for unicyclic graphs . . . 9
Case 1.2. As in the proof of Theorem 3.1 we obtain x2 = d(x2) = 2 and d3 = d(x3) ≤
∆(G) ≤ s+ 1. By the induction hypothesis we get
χα(G) = χα(G− x1) + 3α + (d3 + 2)α − (d3 + 1)α
≤ χα(S∆(n− 1, s)) + 3α + (s+ 3)α − (s+ 2)α
= χα(S∆(n, s)),
with equality if and only if G − x1 = S∆(n − 1, s), d2 = 2 and d3 = s + 1, i.e.,
G = S∆(n, s).
Case 2.1. In this case x1 is adjacent to x2 ∈ C. Let x3 and x4 be the vertices adjacent to
x2 on C and denote d(x2) = d2 ≥ 3, d(x3) = d3 and d(x4) = d4. We deduce
χα(G) = χα(G− x1) + (d2 + 1)α + (d2 − 3)((d2 + 1)α − dα2 ) + (d2 + d3)α
− (d2 + d3 − 1)α + (d2 + d4)α − (d2 + d4 − 1)α.
x2 is adjacent with d2 − 2 pendant vertices and in G− x2x3 the degree of x3 is d3 − 1. It
follows that d2 − 2 + d3 − 1 ≤ s, or d2 + d3 ≤ s + 3. Similarly, d2 + d4 ≤ s + 3. One
obtains
(d2 + d3)
α − (d2 + d3 − 1)α ≤ (s+ 3)α − (s+ 2)α;
(d2 + d4)
α − (d2 + d4 − 1)α ≤ (s+ 3)α − (s+ 2)α.
Since d2 ≤ s+ 1, by Lemma 2.5 we deduce
(d2 − 3)((d2 + 1)α − dα2 ) ≤ (s− 2)((s+ 2)α − (s+ 1)α).
By the induction hypothesis we get
χα(G) ≤ χα(S∆(n− 1, s− 1)) + (s+ 2)α + (s− 2)((s+ 2)α − (s+ 1)α)
+ 2(s+ 3)α − 2(s+ 2)α
= (n− s)(s+ 2)α + (2s− n)(s+ 1)α + (n− s− 2)3α + 4α
− (s+ 2)α + (s− 2)((s+ 2)α − (s+ 1)α) + 2(s+ 3)α.
This upper bound is less than or equal to χα(S∆(n, s)) if and only if
(n− s− 2)(s+ 3)α + (n− s− 2)(s+ 1)α ≥ 2(n− s− 2)(s+ 2)α. (3.3)
If s = n− 2 then (3.3) is an equality, S∆(n− 1, s− 1) has no pendant path of length 2, it
coincides withK1,n−2 +e, d2 = s+1, d3 = d4 = 2 and all inequalities become equalities.
In this case G = S∆(n, s). If s < n− 2 then χα(G− x1) < χα(S∆(n− 1, s− 1)) since
S∆(n − 1, s − 1) has pendant paths of length 2 and G − x1 does not have by hypothesis.
If s < n− 2 then (3.3) is valid by Jensen inequality (for α = 1 (3.3) is an equality), but in
this case we have χα(G) < χα(S∆(n, s)), a contradiction.
Case 2.2. As in the proof of Theorem 3.1 we deduce thatG consists ofC and some pendant
vertices adjacent to vertices of C such that each vertex y ∈ C has its degree d(y) ∈ {2, 3}.
We shall prove that in this case χα(G) < χα(S∆(n, s)) unless α = 1 and G = H6, a
contradiction.
10 Art Discrete Appl. Math. 2 (2019) #P1.03
Suppose that on C there exist four consecutive vertices x, u, v, y such that d(u) =
d(v) = 2. In this case we shall define a new unicyclic graph G1 of order n by G1 =
G− vy + uy. We deduce
χα(G1)− χα(G) = (d(x) + 3)α + (d(y) + 3)α − (d(x) + 2)α − (d(y) + 2)α > 0.
If on C there exist six vertices x, r, y, p, s, q (y may coincide with p) such that d(x) =
d(y) = d(p) = d(q) = 3 and d(r) = d(s) = 2, we define a new unicyclic graph G2 with
the same vertex set as follows: G2 = G− {xr, ry}+ {xy, rs}. We obtain
χα(G2)− χα(G) = 3 · 6α + 4α − 4 · 5α > 0
since 6α + 4α ≥ 2 · 5α and 2 · 6α > 2 · 5α. If G 6= Hn, by applying step by step this type
of transformations we get Hn, such that χα(Hn) > χα(G).
We have χα(Hn) = k 6α + k 4α for n = 2k and (k − 1)6α + k 4α + 2 · 5α for
n = 2k + 1. We get
χα(S∆(2k, k)) = k(k + 3)
α + (k + 2)α + (k − 2)3α + 4α and
χα(S∆(2k + 1, k)) = (k + 1)(k + 3)
α + (k − 1)3α + 4α.
We shall prove that χα(S∆(2k, k)) ≥ χα(Hn) for n = 2k and k ≥ 3 (equality holds only
for k = 3 and α = 1) and χα(S∆(2k+ 1, k)) > χα(Hn) for n = 2k+ 1 and k ≥ 2. Since
for n = 5 and n = 7 it can be easily verified that there is no unicyclic graph of order n in
Case 2.2, it follows that for n = 2k + 1 we may consider that k ≥ 4. It follows that it is
necessary to show that (3.4) holds for k ≥ 3 (with equality only for k = 3 and α = 1) and
(3.5) is true for k ≥ 4.
k(k + 3)α + (k + 2)α + (k − 2)3α + 4α ≥ k 6α + k 4α (3.4)
(k + 1)(k + 3)α + (k − 1)3α + 4α > (k − 1)6α + k 4α + 2 · 5α (3.5)
For α = 1 (3.4) is equivalent to k2 − 3k ≥ 0 with equality only for k = 3. Suppose that
α > 1 and let
ρ(x) = x(x+ 3)α + (x+ 2)α + (x− 2)3α − (x− 1)4α − x6α.
Since ρ ′(x) is strictly increasing for x ≥ 3, we get
ρ ′(x) ≥ ρ ′(3) = 3α 6α−1 + α 5α−1 + 3α − 4α > 0
for α > 1 [17], which implies ρ(x) ≥ ρ(3) = 5α + 3α − 2 · 4α > 0 for α > 1 by Jensen
inequality. This proves (3.4).
Similarly, let
ϕ(x) = (x+ 1)(x+ 3)α + (x− 1)3α − (x− 1)6α − (x− 1)4α − 2 · 5α.
Since ϕ ′(x) is strictly increasing in x ≥ 4 for α ≥ 1 and
ϕ ′(4) = 7α + 5α7α−1 − 6α − 4α + 3α > 0
for α ≥ 1 [17], it follows that for x ≥ 4 we have
ϕ(x) ≥ ϕ(4) = 5 · 7α + 3 · 3α − 3 · 6α − 3 · 4α − 2 · 5α > 0
I. Tomescu: Maximizing general first Zagreb and sum-connectivity indices for unicyclic graphs . . . 11
for α ≥ 1 [17] and (3.5) is justified.
Consequently, if G 6= H6 we can write
χα(G) ≤ χα(Hn) < χα(S∆(n, bn/2c)) ≤ χα(S∆(n, s))
since by Lemma 2.3 the last term is strictly increasing in s, a contradiction.
References
[1] B. Bollobás and P. Erdős, Graphs of extremal weights, Ars Combin. 50 (1998), 225–233.
[2] K. Ch. Das, K. Xu and I. Gutman, On Zagreb and Harary indices, MATCH Commun. Math.
Comput. Chem. 70 (2013), 301–314, http://match.pmf.kg.ac.rs/electronic_
versions/Match70/n1/match70n1_301-314.pdf.
[3] Z. Du, B. Zhou and N. Trinajstić, Minimum general sum-connectivity index of unicyclic
graphs, J. Math. Chem. 48 (2010), 697–703, doi:10.1007/s10910-010-9702-6.
[4] Z. Du, B. Zhou and N. Trinajstić, On the general sum-connectivity index of trees, Appl. Math.
Lett. 24 (2011), 402–405, doi:10.1016/j.aml.2010.10.038.
[5] R. Garcı́a-Domenech, J. Gálvez, J. V. de Julián-Ortiz and L. Pogliani, Some new trends in
chemical graph theory, Chem. Rev. 108 (2008), 1127–1169, doi:10.1021/cr0780006.
[6] I. Gutman and B. Furtula (eds.), Recent Results in the Theory of Randić Index, volume 6 of
Mathematical Chemistry Monographs, University of Kragujevac, Kragujevac, 2008, http:
//match.pmf.kg.ac.rs/mcm6.htm.
[7] Y. Hu, X. Li and Y. Yuan, Trees with maximum general Randić index, MATCH Commun. Math.
Comput. Chem. 52 (2004), 129–146, http://match.pmf.kg.ac.rs/electronic_
versions/Match52/match52_129-146.pdf.
[8] H. Hua and H. Deng, On unicycle graphs with maximum and minimum zeroth-order general
Randić index, J. Math. Chem. 41 (2007), 173–181, doi:10.1007/s10910-006-9067-z.
[9] L. B. Kier and L. H. Hall, Molecular Connectivity in Chemistry and Drug Research, volume 14
of Medicinal Chemistry, Academic Press, New York, 1976.
[10] L. B. Kier and L. H. Hall, Molecular Connectivity in Structure-Activity Analysis, volume 2 of
Chemometrics Research Studies Press, Wiley, New York, 1986.
[11] X. Li and I. Gutman, Mathematical Aspects of Randić-Type Molecular Structure Descriptors,
volume 1 of Mathematical Chemistry Monographs, University of Kragujevac, Kragujevac,
2006, http://match.pmf.kg.ac.rs/mcm1.htm.
[12] M. Randić, Characterization of molecular branching, J. Am. Chem. Soc. 97 (1975), 6609–6615,
doi:10.1021/ja00856a001.
[13] M. Randić, Novel molecular descriptor for structure-property studies, Chem. Phys. Lett. 211
(1993), 478–483, doi:10.1016/0009-2614(93)87094-j.
[14] R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, volume 11 of Meth-
ods and Principles in Medicinal Chemistry, Wiley-VCH, Weinheim, 2000, doi:10.1002/
9783527613106.
[15] I. Tomescu and M. K. Jamil, Maximum general sum-connectivity index for trees with
given independence number, MATCH Commun. Math. Comput. Chem. 72 (2014), 715–
722, http://match.pmf.kg.ac.rs/electronic_versions/Match72/n3/
match72n3_715-722.pdf.
[16] D. B. West, Introduction to Graph Theory, Prentice-Hall, Upper Saddle River, NJ, 2nd edition,
2001.
12 Art Discrete Appl. Math. 2 (2019) #P1.03
[17] Wolfram Alpha LLC, Wolfram|Alpha, http://www.wolframalpha.com/.
[18] S. Zhang and H. Zhang, Unicyclic graphs with the first three smallest and largest
first general Zagreb index, MATCH Commun. Math. Comput. Chem. 55 (2006), 427–
438, http://match.pmf.kg.ac.rs/electronic_versions/Match55/n2/
match55n2_427-438.pdf.
[19] B. Zhou and N. Trinajstić, On a novel connectivity index, J. Math. Chem. 46 (2009), 1252–
1270, doi:10.1007/s10910-008-9515-z.
[20] B. Zhou and N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47 (2010), 210–
218, doi:10.1007/s10910-009-9542-4.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P1.04
https://doi.org/10.26493/2590-9770.1246.99c
(Also available at http://adam-journal.eu)
Optimal orientations of strong products of paths
Tjaša Paj Erker
University of Maribor, FME, Smetanova 17, 2000 Maribor, Slovenia
Received 9 March 2018, accepted 13 July 2018, published online 8 August 2018
Abstract
Let diammin(G) denote the minimum diameter of a strong orientation of G and let
G  H denote the strong product of graphs G and H . In this paper we prove that
diammin(Pm  Pn) = diam(Pm  Pn) for m,n ≥ 5, m 6= n, and diammin(Pm  Pn) =
diam(Pm  Pn) + 1 for m,n ≥ 5, m = n. We also prove that diammin(G  H) ≤
max {diammin(G),diammin(H)} for any connected bridgeless graphs G and H .
Keywords: Diameter, strong orientation, strong product.
Math. Subj. Class.: 05C12, 05C76
1 Introduction
Let D = (V (D), A(D)) be a directed graph. If (u, v) ∈ A(D), we write u → v. A
uv-path is a directed path u = u1u2 . . . un = v from a vertex u to a vertex v. The length of
the path u = u1u2 . . . un = v is n− 1. If every vertex in D is reachable from every other
vertex in D, we say that directed graph D is strong (there is a directed uv-path in D for
every u, v ∈ V (D)). The distance from u to v is the length of a shortest directed uv-path
in D, denoted by distD(u, v). The greatest distance among all pairs of vertices in D is the
diameter of D, so
diam(D) = max{distD(u, v) | u, v ∈ V (D)}.
Note that the distance of two vertices u, v in undirected graph G, distG(u, v), is the length
of a shortest undirected uv-path in G and the greatest distance between any two vertices in
G is the diameter of G, denoted by diam(G).
Let G be an undirected graph. An orientation of G is a digraph D obtained from G
by assigning to each edge in G a direction. Let D(G) denote the family of all strong
orientations of G. In [9] it is proved that every connected bridgeless graph admits a strong
orientation. We define the minimum diameter of a strong orientation of G as
diammin(G) = min{diam(D) | D ∈ D(G)}.
E-mail address: tjasa.paj@um.si (Tjaša Paj Erker)
cb This work is licensed under http://creativecommons.org/licenses/by/3.0/
2 Art Discrete Appl. Math. 2 (2019) #P1.04
The parameter diammin(G) was studied by many authors, because it is important from
theoretical and practical points of view, as an application in traffic control problems. Ori-
entations of graphs can be viewed as arrangements of one-way streets, if G is thought of
as the system of two-way streets in a city, and we want to make every street in the city
one-way and still get from every point to every other point (see [9, 10]).
For every bridgeless connected graph G of radius r it was shown, see [1], that
diammin(G) ≤ 2r2+2r. There were also some determined values of the minimum diame-
ter of a strong orientation of the Cartesian product of graphs. For Cartesian product of two
paths it was proved that diammin(Pm Pn) = diam(Pm Pn), for m ≥ 3 and n ≥ 6,
see [5]. In [8] it was proved that diammin(Cm Cn) = diam(Cm Cn) for m,n ≥ 6. In
[7] Koh and Tay proved that diammin(T1 T2) = diam(T1 T2) for trees T1 and T2 with
diameters at least 4. They also studied the diameter of orientations of Km Kn, Km Pn,
Pm Cn and Km Cn (see [4, 5, 6]).
In [3], the upper bound for the strong radius and the strong diameter of Cartesian prod-
uct of graphs are determined.
In this article we consider the minimum diameter of strong orientations of strong prod-
ucts of graphs. The strong product of graphs G and H is the graph, denoted by GH , with
the vertex set V (G H) = V (G) × V (H) where two distinct vertices (u, v) and (u′, v′)
are adjacent in GH if and only if uu′ ∈ E(G) and v = v′, or u = u′ and vv′ ∈ E(H),
or uu′ ∈ E(G) and vv′ ∈ E(H). For v ∈ V (H) we define the G-layer Gv:
Gv = {(u, v) | u ∈ V (G)} .
Analogously we define H-layers.
In the next section we prove that diammin(PmPn) = diam(PmPn), for m,n ≥ 5,
m 6= n and that diammin(Pm  Pn) = diam(Pm  Pn) + 1, for m,n ≥ 5, m = n.
2 Orientations of Pm  Pn
In [7] Koh and Tay proved that diammin(Pm Pn) = diam(Pm Pn), for m ≥ 5 and
n ≥ 5. We use some of their notations. So we will define four sections of V (Pm  Pn)
and two basic orientations of Ps Pt, where s, t ≥ 3, similarly as it was introduced in [7].
For m,n ≥ 5 we define
(i) Southwest Section SW =
{
(i, j) | 1 ≤ i ≤
⌈
m
2
⌉
, 1 ≤ j ≤
⌈
n
2
⌉}
;
(ii) Northwest Section NW =
{
(i, j) | 1 ≤ i ≤
⌈
m
2
⌉
,
⌈
n+1
2
⌉
≤ j ≤ n
}
;
(iii) Southeast Section SE =
{
(i, j) |
⌈
m+1
2
⌉
≤ i ≤ m, 1 ≤ j ≤
⌈
n
2
⌉}
;
(iv) Northeast Section NE =
{
(i, j) |
⌈
m+1
2
⌉
≤ i ≤ m,
⌈
n+1
2
⌉
≤ j ≤ n
}
.
We define two basic orientations of Ps  Pt, where s, t ≥ 3: if s ≤ t, we define the
orientation F1 of Ps  Pt as:
(i) For 1 ≤ i ≤ s− 1 and 2 ≤ j ≤ t, (i, j)→ (i+ 1, j − 1);
(ii) For 1 ≤ i ≤ s − 1 and 1 ≤ j ≤ t − 1, (i + 1, j + 1) → (i, j) if j − i ≥ t − s and
(i, j)→ (i+ 1, j + 1) if j − i < t− s;
(iii) For 1 ≤ i ≤ s− 1 and 2 ≤ j ≤ t, (i, j)→ (i, j − 1);
(iv) For 1 ≤ j ≤ t− 1, (s, j)→ (s, j + 1);
T. Paj Erker: Optimal orientations of strong products of paths 3
(v) For 1 ≤ i ≤ s− 1 and 1 ≤ j ≤ t− 1, (i, j)→ (i+ 1, j);
(vi) For 2 ≤ i ≤ s, (i, t)→ (i− 1, t);
and if s > t, we define the orientation F2 of Ps  Pt as:
(i) For 2 ≤ i ≤ s and 1 ≤ j ≤ t− 1, (i, j)→ (i− 1, j + 1);
(ii) For 1 ≤ i ≤ s and 1 ≤ j ≤ t, (i + 1, j + 1) → (i, j) if i − j ≥ s − t and
(i, j)→ (i+ 1, j + 1) if i− j < s− t;
(iii) For 1 ≤ i ≤ s− 1 and 1 ≤ j ≤ t− 1, (i, j)→ (i, j + 1);
(iv) For 2 ≤ j ≤ t, (s, j)→ (s, j − 1);
(v) For 2 ≤ i ≤ s and 1 ≤ j ≤ t− 1, (i, j)→ (i− 1, j);
(vi) For 1 ≤ i ≤ s− 1, (i, t)→ (i+ 1, t).
The orientation F1 of P3  P4 and the orientation F2 of P4  P3 is shown in Figure 1.
F1
(s, t)
F2
(s, t)
Figure 1: Orientations F1 and F2.
Observation 2.1. If s < t, for any (i, j) ∈ V (F1), distF1((i, j), (s, t− 1)) ≤ t− 2.
Proof. Let (i, j) ∈ V (F1). We shall consider four cases.
(i) If j 6= t and j ≥ i+t−s−1, then (i, j)→ (i+1, j)→ · · · → (j−(t−s)+1, j)→
(j− (t− s)+ 2, j+1)→ · · · → (s, t− 1) is a path of length at most s− 1 ≤ t− 2.
(ii) If j 6= t and j < i+ t− s− 1, then (i, j)→ (i+1, j+1)→ · · · → (s, j+ s− i)→
(s, j + s− i+ 1)→ · · · → (s, t− 1) is a path of length at most t− 2.
(iii) If j = t and i 6= s, then (i, t)→ (i+ 1, t− 1)→ (i+ 2, t− 1)→ · · · → (s, t− 1)
is a path of length at most s− 1 ≤ t− 2.
(iv) If j = t and i = s, then (s, t) → (s − 1, t − 1) → (s, t − 1) is a path of length
two.
Observation 2.2. If s < t, for any (i, j) ∈ V (F1), distF1((i, j), (s, t)) ≤ t− 1.
Proof. Since (s, t− 1)→ (s, t), the claim follows by Observation 2.1:
distF1((i, j), (s, t)) = distF1((i, j), (s, t− 1)) + 1 ≤ s− 1 + 1 ≤ t− 1.
Observation 2.3. If s < t, for any (i, j) ∈ V (F1), distF1((s− 1, t), (i, j)) ≤ t− 1.
4 Art Discrete Appl. Math. 2 (2019) #P1.04
Proof. Let (i, j) ∈ V (F1). We shall consider four cases.
(i) If i 6= s and j > i+ t− s, then (s− 1, t)→ (s− 2, t)→ · · · → (i+ (t− j), t)→
(i+ (t− j)− 1, t− 1)→ · · · → (i, j) is a path of length at most s− 2 ≤ t− 2.
(ii) If i 6= s and j ≤ i+ t− s, then (s−1, t)→ (s−1, t−1)→ (s−2, t−2)→ · · · →
(i, i+ t− s)→ (i, i+ t− s− 1)→ · · · → (i, j) is a path of length at most t− 1.
(iii) If i = s and j 6= t, (s − 1, t) → (s − 1, t − 1) → (s − 1, t − 2) → · · · →
(s− 1, j + 1)→ (s, j) is a path of length at most t− 1.
(iv) If i = s and j = t, then (s− 1, t)→ (s, t− 1)→ (s, t) is a path of length two.
Observation 2.4. If s < t, for any (i, j) ∈ V (F1), distF1((s, t), (i, j)) ≤ t− 1.
Proof. Since (s, t)→ (s−1, t) and (s, t)→ (s−1, t−1), the proof is similar as the proof
of Observation 2.3.
Observation 2.5. If s = t, for any (i, j) ∈ V (F1), distF1((i, j), (s, s)) ≤ s.
Proof. Let (i, j) ∈ V (F1). We shall consider three cases.
(i) If j 6= t and j ≥ i−1, then (i, j)→ (i+1, j)→ · · · → (j+1, j)→ (j+2, j+1)→
· · · → (s, s− 1)→ (s, s) is a path of length at most s.
(ii) If j 6= t and j < i − 1, then (i, j) → (i + 1, j + 1) → · · · → (s, j + s − i) →
(s, j + s− i+ 1)→ · · · → (s, s) is a path of length at most s− 1.
(iii) If j = s and i 6= s, then (i, s)→ (i+1, s−1)→ (i+2, s−1)→ · · · → (s, s−1)→
(s, s) is a path of length at most s.
Observation 2.6. If s = t, for any (i, j) ∈ V (F1), distF1((s, s), (i, j)) ≤ s− 1.
Proof. Let (i, j) ∈ V (F1). We shall consider three cases.
(i) If i 6= s and j > i, then (s, s) → (s − 1, s) → · · · → (i + (s − j), s) →
(i+ (s− j)− 1, t− 1)→ . . .→ (i, j) is a path of length at most s− 1.
(ii) If i 6= s and j ≤ i, then (s, s) → (s − 1, s − 1) → · · · → (i, i) → (i, i − 1) →
. . .→ (i, j) is a path of length at most s− 1.
(iii) If i = s and j 6= s − 1, (s, s) → (s − 1, s − 1) → (s − 1, s − 2) → · · · →
(s− 1, j + 1)→ (s, j) is a path of length at most s− 1.
(iv) If i = s and j = s− 1, then (s, s)→ (s− 1, s− 1)→ (s, s− 1) is a path of length
two.
Similarly as above, we can prove next Observations 2.7–2.10.
Observation 2.7. If s > t, for any (i, j) ∈ V (F2), distF2((s, t− 1), (i, j)) ≤ s− 1.
Observation 2.8. If s > t, for any (i, j) ∈ V (F2), distF2((s, t), (i, j)) ≤ s− 1.
Observation 2.9. If s > t, for any (i, j) ∈ V (F2), distF2((i, j), (s− 1, t)) ≤ s− 2.
Observation 2.10. If s > t, for any (i, j) ∈ V (F2), distF2((i, j), (s, t)) ≤ s− 1.
T. Paj Erker: Optimal orientations of strong products of paths 5
In [7], Koh and Tay also introduced a key-vertex v ∈ V (F ) of digraph F . Let F ∈
D(Ps  Pt). We say that a vertex v ∈ V (F ) is a key-vertex of F if
distF (u, v) ≤ max {t, s} and distF (v, u) ≤ max {t, s}
for all u ∈ V (F ). Note that (s, t) is a key-vertex of F1 and of F2.
Analogously as F1 and F2, we define 6 other isomorphic orientations Fi, 3 ≤ i ≤ 8 of
Ps  Pt as shown in Figures 2 and 3.
F1
F4
F5
F8
Figure 2: Orientations F1, F4, F5 and F8.
Obviously vertices denoted by black dots in Figures 2 and 3 are key-vertices of Fi for
i = 1, . . . , 8 (similar arguments as in Observations 2.1–2.6).
Lemma 2.11. Let m,n ≥ 5, m 6= n and m,n ≡ 1 (mod 2). Then
diammin(Pm  Pn) ≤ max {m− 1, n− 1} .
Proof. Let m < n. We define the orientation D of Pm  Pn by F1, F4, F5 and F8:
(a) orient the section NW as F4;
(b) orient the section NE as F8;
(c) orient the section SW as F1;
(d) orient the section SE as F5.
As an illustration, the orientation of P5  P7 is shown in Figure 4. The vertex z =
(m+12 ,
n+1
2 ) is the key-vertex of each Fi, for i = 1, 4, 5, 8. For any u, v ∈ V (D),
distD(u, v) ≤ distD(u, z) + distD(z, v).
6 Art Discrete Appl. Math. 2 (2019) #P1.04
F2
F3 F7
F6
Figure 3: Orientations F2, F3, F6 and F7.
Since distD(u, z) ≤ n−12 and distD(z, v) ≤
n−1
2 (similarly as in Observation 2.2 and
Observation 2.4), we have
distD(u, v) ≤
n− 1
2
+
n− 1
2
= n− 1.
If m > n we define the orientation D of Pm  Pn by F2, F3, F6 and F7. Similarly as
above, we have
distD(u, v) ≤ distD(u, z) + distD(z, v) ≤
m− 1
2
+
m− 1
2
= m− 1
(see Observation 2.10 and Observation 2.8).
Lemma 2.12. Let m,n ≥ 6, m 6= n and m,n ≡ 0 (mod 2). Then
diammin(Pm  Pn) ≤ max {m− 1, n− 1} .
Proof. Let m < n. Denote z1 = (m2 ,
n
2 ), z4 = (
m
2 ,
n
2 + 1), z5 = (
m
2 + 1,
n
2 ) and
z8 = (
m
2 + 1,
n
2 + 1). We define the orientation D of Pm  Pn by F1, F4, F5 and F8 as
follows:
(a) orient the section NW as F4;
(b) orient the section NE as F8;
(c) orient the section SW as F1;
(d) orient the section SE as F5;
(e) Orient z1 → (m2 −1,
n
2+1), (
m
2 +1,
n
2−1)→ z1, z4 → (
m
2 −1,
n
2 ), (
m
2 +1,
n
2+2)→
z4, z5 → (m2 + 2,
n
2 + 1), (
m
2 ,
n
2 − 1) → z5, z8 → (
m
2 + 2,
n
2 ), (
m
2 ,
n
2 + 2) → z8,
and orient all other edges arbitrarily.
T. Paj Erker: Optimal orientations of strong products of paths 7
1 m+12 m
1
n+1
2
n
F1
F4
F5
F8
Figure 4: The orientation D of P5  P7.
F1
z1
F4
z4
F5
z5
F8
z8
1
m
2
m
2 + 1 m
1
n
2
n
2 + 1
n
Figure 5: The orientation D of P6  P8.
8 Art Discrete Appl. Math. 2 (2019) #P1.04
The orientation D is shown in Figure 5. Note that vertices z1, z4, z5 and z8 are key-vertices
of Fi, for i = 1, 4, 5, 8.
Let u, v ∈ V (D). We claim that distD(u, v) ≤ n− 1. There are four cases.
(i) If u and v are in the same section, then we have
distD(u, v) ≤ distD(u, zi) + distD(zi, v) ≤
n
2
− 1 + n
2
− 1 = n− 2
as in Observation 2.2 and Observation 2.4.
(ii) If u ∈ NW and v ∈ SW, then (see Observation 2.2 and Observation 2.3):
distD(u, v) ≤ distD(u, z4) + distD
(
z4, (
m
2 − 1,
n
2 )
)
+ distD
(
(m2 − 1,
n
2 ), v
)
≤ n
2
− 1 + 1 + n
2
− 1 = n− 1.
The argument is similar if u ∈ SW and v ∈ NW, or u ∈ NE and v ∈ SE, or u ∈ SE
and v ∈ NE.
(iii) If u ∈ SW and v ∈ SE, then the claim follows from Observation 2.1 and Observation
2.4, similarly as above. Also, if u ∈ SE and v ∈ SW, or u ∈ NW and v ∈ NE, or
u ∈ NE and v ∈ NW, then the argument is analogous.
(iv) If u ∈ SW and v ∈ NE, then (see Observation 2.1 and Observation 2.3) we have
distD(u, v) ≤ distD
(
u, (m2 ,
n
2 − 1)
)
+ distD
(
(m2 ,
n
2 − 1), z5
)
+
+ distD
(
z5, (
m
2 + 2,
n
2 + 1)
)
+ distD
(
(m2 + 2,
n
2 + 1), v
)
≤ n
2
− 2 + 1 + 1 + n
2
− 1 = n− 1.
The argument is similar for u ∈ NE and v ∈ SW, or u ∈ NW and v ∈ SE, or
u ∈ SE and v ∈ NW.
Analogously if m > n, we have distD(u, v) ≤ m− 1 for any u, v ∈ V (D).
Lemma 2.13. Let m ≥ 5, n ≥ 6, m ≡ 1 (mod 2) and n ≡ 0 (mod 2). Then
diammin(Pm  Pn) ≤ max {m− 1, n− 1} . (2.1)
Proof. Let m < n. Denote z1 = (m+12 ,
n
2 ) and z4 = (
m+1
2 ,
n
2 + 1). We define the
orientation D of Pm  Pn by F1, F4, F5 and F8 as follows:
(a) orient the section NW as F4;
(b) orient the section NE as F8;
(c) orient the section SW as F1;
(d) orient the section SE as F5;
(e) orient z4 → (m+12 − 1,
n
2 ), z1 → z4, z4 → (
m+1
2 + 1,
n
2 ), and orient all other edges
arbitrarily.
The orientation D is shown in Figure 6. Note that vertex z1 is a key-vertex of F1 and F5
and that vertex z4 is a key-vertex of F4 and F8.
T. Paj Erker: Optimal orientations of strong products of paths 9
F1
z1
F4
z4
F8
1
m
2 m
1
n
2
n
2 + 1
n
Figure 6: The orientation D of P5  P8.
Let u, v ∈ V (D). There are three cases.
(i) If u ∈ NW ∪NE and v ∈ NW ∪NE, then we have
distD(u, v) ≤ distD(u, z4) + distD(z4, v) ≤
n
2
− 1 + n
2
− 1 = n− 2
(see Observation 2.2 and Observation 2.4). The case that {u, v} ⊆ SW ∪ SE is
similar.
(ii) If u ∈ SW∪SE and v ∈ NW∪NE, then (see Observation 2.2 and Observation 2.4):
distD(u, v) ≤ distD(u, z1) + distD(z1, z4) + distD(z4, v)
≤ n
2
− 1 + 1 + n
2
− 1 = n− 1.
(iii) If u ∈ NW ∪NE and v ∈ SW, then from Observation 2.2 and Observation 2.3:
distD(u, v) ≤ distD(u, z4) + distD
(
z4,
(
m+1
2 − 1,
n
2
))
+
+ distD
((
m+1
2 − 1,
n
2
)
, v
)
≤ n
2
− 1 + 1 + n
2
− 1 = n− 1.
The case that u ∈ NW ∪NE and v ∈ SE is similar.
Let m > n. Denote z2 = (m+12 ,
n
2 ) and z3 = (
m+1
2 ,
n
2 + 1). We define the orientation
D of Pm  Pn by F2, F3, F6 and F7 as follows:
(a) orient the section NW as F3;
10 Art Discrete Appl. Math. 2 (2019) #P1.04
(b) orient the section NE as F7;
(c) orient the section SW as F2;
(d) orient the section SE as F6;
(e) orient (m+12 −1,
n
2 )→ z3, z3 → z2, (
m+1
2 +1,
n
2 )→ z3 and all other edges oriented
arbitrarily.
The rest of the proof is analogously as above.
Note that if m ≥ 5 and n ≥ 6, m ≡ 0 (mod 2) and n ≡ 1 (mod 2), we also
have (2.1).
Lemma 2.14. Let m ≥ 5, m ≡ 1 (mod 2). Then
diammin(Pm  Pm) ≤ m.
Proof. Denote z = (m+12 ,
m+1
2 ). We define the orientation D of Pm  Pm by F1, F4, F5
and F8 as follows:
(a) orient the section NW as F4;
(b) orient the section NE as F8;
(c) orient the section SW as F1;
(d) orient the section SE as F5.
Note that z is a key-vertex of Fi, for i = 1, 4, 5, 8. For any u, v ∈ D we have
distD(u, v) ≤ distD(u, z) + distD(z, v) ≤
m+ 1
2
+
m− 1
2
= m
as in Observation 2.5 and Observation 2.6.
Lemma 2.15. Let m ≥ 6, m ≡ 0 (mod 2). Then
diammin(Pm  Pm) ≤ m.
Proof. The proof is similarly as the proof of Lemma 2.12 (it follows from Observations 2.1,
2.3, 2.5 and 2.6).
In [2], it is proved that if (u, v) and (u′, v′) are vertices of a strong product GH , then
distGH((u, v), (u
′, v′)) = max{distG(u, u′),distH(v, v′)}.
Since diam(Pm) = m − 1, we get diam(Pm  Pn) = max{m − 1, n − 1}. Since
diam(Pm  Pn) = distPmPm((1, 1), (m,m)) = m − 1 and there is only one path from
(1, 1) to (m,m) in Pm  Pm possessing the length m− 1, it follows that
distD((1, 1), (m,m)) > m− 1 or distD ((m,m), (1, 1)) > m− 1
for any D ∈ D(Pm  Pn). To combine these two observations with Lemmas 2.11–2.15,
we obtain the following theorem:
T. Paj Erker: Optimal orientations of strong products of paths 11
Theorem 2.16. If m,n ≥ 5, then
diammin(Pm  Pn) =
{
diam(Pm  Pn), if m 6= n;
diam(Pm  Pn) + 1, if m = n.
At the end of this section, we give the bounds of diammin(Pn  Pm) for m < 5.
From Figure 7, we see that n − 1 ≤ diammin(Pn  P2) = n for n > 2, n − 1 ≤
diammin(Pn  P3) = n for n > 3 and n− 1 ≤ diammin(Pn  P4) = n+ 1 for n > 4.
· · ·
· · ·
· · ·
Pn
Pn
Pn
P2
P3
P4
Figure 7: Orientations of Pn  P2, Pn  P3 and Pn  P4.
3 Strong orientation of graphs
In this section we shall prove the next theorem.
Theorem 3.1. Let G and H be connected bridgeless graphs. Then
diammin(GH) ≤ max{diammin(G),diammin(H)}.
Proof. Let DG be a strong orientation of G such that diam(DG) = diammin(G) = d1 and
let DH be a strong orientation of H such that diam(DH) = diammin(H) = d2. We define
the orientation DGH of GH as:
(a) Every edge with endvertices in layers Gv , v ∈ V (H) gets the orientation DG.
(b) Every edge with endvertices in layers Hu, u ∈ V (G) gets the orientation DH .
(c) If u→ u′ in G and v → v′ in H , then (u, v)→ (u′, v′), all other edges are oriented
arbitrarily.
12 Art Discrete Appl. Math. 2 (2019) #P1.04
We have to prove that for every pair of vertices (u, v), (u′, v′) in G  H there is a
directed path P from (u, v) to (u′, v′) in DGH , such that the length of P is at most
max {d1, d2}.
If (u, v) and (u′, v) are vertices in the same G-layer or if (u, v) and (u, v′) are vertices
in the same H-layer, then there is a directed path from (u, v) to (u′, v) in DGH of length
at most d1 or a directed path from (u, v) to (u, v′) of length at most d2.
Now let (u, v) and (u′, v′) be arbitrary vertices in DGH . There is a directed path u =
u1u2 . . . um = u
′ in G of length at most d1 and there is a directed path v = v1v2 . . . vn = v′
in H of length at most d2. Without loss of generality we can assume m ≥ n. We have
(u, v)→ (u2, v2)→ (u3, v3)→ · · · → (un, vn)→
(un+1, vn)→ · · · → (um, vn) = (u′, v′)
is a path of length at most d1.
Since diammin(C3) = 2 and diammin(C3  C3) = 2, the bound is tight.
References
[1] V. Chvátal and C. Thomassen, Distances in orientations of graphs, J. Comb. Theory Ser. B 24
(1978), 61–75, doi:10.1016/0095-8956(78)90078-3.
[2] R. Hammack, W. Imrich and S. Klavžar, Handbook of Product Graphs, Discrete Mathematics
and its Applications, CRC Press, Boca Raton, Florida, 2nd edition, 2011, doi:10.1201/b10959.
[3] J. S.-T. Juan, C.-M. Huang and I. Sun, The strong distance problem on the Cartesian product
of graphs, Inform. Process. Lett. 107 (2008), 45–51, doi:10.1016/j.ipl.2008.01.001.
[4] K. M. Koh and E. G. Tay, On optimal orientations of Cartesian products of even cycles and
paths, Networks 30 (1997), 1–7, doi:10.1002/(sici)1097-0037(199708)30:1〈1::aid-net1〉3.0.co;
2-h.
[5] K. M. Koh and E. G. Tay, Optimal orientations of products of paths and cycles, Discrete Appl.
Math. 78 (1997), 163–174, doi:10.1016/s0166-218x(97)00017-6.
[6] K. M. Koh and E. G. Tay, On optimal orientations of Cartesian products of graphs (II): com-
plete graphs and even cycles, Discrete Math. 211 (2000), 75–102, doi:10.1016/s0012-365x(99)
00136-3.
[7] K. M. Koh and E. G. Tay, On optimal orientations of Cartesian products of trees, Graphs
Combin. 17 (2001), 79–97, doi:10.1007/s003730170057.
[8] J.-C. Konig, D. W. Krumme and E. Lazard, Diameter-preserving orientations of the torus, Net-
works 32 (1998), 1–11, doi:10.1002/(sici)1097-0037(199808)32:1〈1::aid-net1〉3.3.co;2-a.
[9] H. E. Robbins, Questions, discussions, and notes: A theorem on graphs, with an application to
a problem of traffic control, Amer. Math. Monthly 46 (1939), 281–283, doi:10.2307/2303897.
[10] F. S. Roberts and Y. Xu, On the optimal strongly connected orientations of city street graphs I:
Large grids, SIAM J. Discrete Math. 1 (1988), 199–222, doi:10.1137/0401022.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P1.05
https://doi.org/10.26493/2590-9770.1264.94b
(Also available at http://adam-journal.eu)
On the anti-Kekulé problem of cubic graphs∗
Qiuli Li
School of Mathematics and Statistics, Lanzhou University,
Lanzhou, Gansu 730000, China
Wai Chee Shiu †, Pak Kiu Sun
Department of Mathematics, Hong Kong Baptist University,
Kowloon Tong, Hong Kong, China
Dong Ye
Department of Mathematical Sciences, Middle Tennessee State University,
Murfreesboro, TN 37132, United States
Received 6 September 2016, accepted 8 July 2018, published online 12 August 2018
Abstract
An edge set S of a connected graphG is called an anti-Kekulé set ifG−S is connected
and has no perfect matchings, where G − S denotes the subgraph obtained by deleting
all edges in S from G. The anti-Kekulé number of a graph G, denoted by ak(G), is the
cardinality of a smallest anti-Kekulé set of G. It is NP-complete to determine the anti-
Kekulé number of a graph. In this paper, we show that the anti-Kekulé number of a 2-
connected cubic graph is either 3 or 4, and the anti-Kekulé number of a connected cubic
bipartite graph is always equal to 4. Furthermore, a polynomial time algorithm is given to
find all smallest anti-Kekulé sets of a connected cubic graph.
Keywords: Anti-Kekulé set, anti-Kekulé number, cubic graphs.
Math. Subj. Class.: 05C10, 05C70, 05C90
∗This work is supported by NSFC (grant nos. 11401279 and 11371180), the Specialized Research Fund for the
Doctoral Program of Higher Education (No. 20130211120008), the Fundamental Research Funds for the Central
Universities (no. lzujbky-2017-28), and General Research Fund of Hong Kong.
†The corresponding author.
E-mail addresses: qlli@lzu.edu.cn (Qiuli Li), wcshiu@hkbu.edu.hk (Wai Chee Shiu), lionel@hkbu.edu.hk
(Pak Kiu Sun), dong.ye@mtsu.edu (Dong Ye)
cb This work is licensed under http://creativecommons.org/licenses/by/3.0/
2 Art Discrete Appl. Math. 2 (2019) #P1.05
1 Introduction
Let G be a graph. A perfect matching of a graph G is a set of non-adjacent edges that
covers all vertices of G. A perfect matching of a graph is also called a Kekulé structure
in mathematical chemistry and statistical physics. An edge set S of a connected graph G
is called an anti-Kekulé set if G − S is connected and has no perfect matchings, where
G − S denotes the subgraph obtained by deleting all edges in S from G. The anti-Kekulé
number of a graph G, denoted by ak(G), is the cardinality of a smallest anti-Kekulé set
of G. The anti-Kekulé number of a graph is hard to be determined in general. It has been
proved recently that it is an NP-complete problem to determine the anti-Kekulé number of
a connected bipartite graph by Lü, Li and Zhang [17].
In chemistry and physics, graphs are used to represent the skeletons of molecules, and
Kekulé structures (or perfect matchings) are used to model special structures of bonds
between atoms. For example, for a benzenoid hydrocarbons, graphens or fullerenes, a
Kekulé structure of these molecules stands for double bonds between atoms. An anti-
Kekulé set is a set of double bonds whose removal significantly affects the whole molecule
structure by the valence bond (VB) theory (cf. [15]).
A fullerene is a 3-connected plane cubic graph such that every face is either a hexagon
or a pentagon. For example, C60 is a fullerene with 60 vertices such that all pentagons are
disjoint. The anti-Kekulé number of C60 is proved to be 4 by Vukičević [26]. A leapfrog
fullerene is a fullerene obtained by the leapfrog operation (see [16]). Kutnar et al. [16]
obtained a bound for the anti-Kekulé number of leapfrog fullerenes as follows.
Theorem 1.1 ([16]). Let G be a leapfrog fullerene. Then 3 ≤ ak(G) ≤ 4.
The above result was improved by Yang et al. [28] by proving that all fullerenes have
anti-Kekulé number 4. The anti-Kekulé numbers of other interesting graphs, such as ben-
zenoid hydrocarbons [2, 27], fence graphs [22], infinite triangular, rectangular and hexag-
onal grids [25] as well as cata-condensed phenylenes [35], have been investigated.
The result on fullerenes has been generalized to general cubic graphs with high cyclic
edge-connectivity in [30]. A graph G is cyclically k-edge-connected if G cannot be sepa-
rated into two components, each containing a cycle, by deletion of fewer than k edges. The
cyclic edge-connectivity cλ(G) of a graph G is the maximum k such that G is cyclically
k-edge-connected. An edge set S is called an odd cycle edge-transversal of a graph G
if G − S is bipartite. The size of a smallest odd cycle edge-transversal of G is denoted
by τodd(G).
Theorem 1.2 ([30]). Let G be a cyclically 4-edge-connected cubic graph. Then either
ak(G) = 4 or 1 ≤ τodd(G) ≤ 3.
The result above can be used to determine the anti-Kekulé number of fullerenes. Since
a smallest odd cycle-transversal of a fullerene graph contains at least 6 edges and the cyclic
edge-connectivity of a fullerene graph is 5 (see [5, 20]), Theorem 1.2 implies that every
fullerene has anti-Kekulé number 4. However, Theorem 1.2 is not applicable to deter-
mine the anti-Kekulé numbers of some interesting graphs, such as, some boron-nitrogen
fullerenes with low cyclic edge-connectivity, (3, 6)-fullerenes etc.
A (k, 6)-cage (k ≥ 3) is a 3-connected cubic plane graph whose faces are either k-gons
or hexagons. Došlić [5] shows that (k, 6)-cages only exist for k = 3, 4 and 5. A fullerene
is a (5, 6)-cage and the (4, 6)-cages and (3, 6)-cages are usually called (4, 6)-fullerenes (or
Q. Li et al.: On the anti-Kekulé problem of cubic graphs 3
boron-nitrogen fullerens) and (3, 6)-fullerenes, respectively. Many researches have inves-
tigated the properties of these graphs in both mathematics and chemistry, such as hamil-
tonicity [9, 10], resonance [29, 31, 33], the forcing matching number [13, 34], and energy
spectra of (3, 6)-fullerenes [4, 14] which determines their electronic and magnetic proper-
ties [3, 21].
The cyclic edge-connectivity of (k, 6)-cages has been obtained by Došlić in [5]. Let
T be a family of (4, 6)-fullerenes, which consists of a tube with n layers of hexagons (i.e.
each layer is a cyclic chain of three hexagons) capped on both ends by a cap formed by
three quadrangles.
Theorem 1.3 ([5]). Let G be a (k, 6)-cage. Then cλ(G) = 3 if G ∈ T , and cλ(G) = k
otherwise.
In this paper, we consider the anti-Kekulé number of connected cubic graphs including
those with low cyclic edge-connectivity. The following is our first major result.
Theorem 1.4. If G is a 2-connected cubic graph, then 3 ≤ ak(G) ≤ 4.
Since a leapfrog fullerene is 3-connected, Theorem 1.1 is a direct corollary of Theo-
rem 1.4. For bipartite cubic graphs, the result can be strengthened as follows.
Theorem 1.5. If G is a connected cubic bipartite graph, then ak(G) = 4.
Theorems 1.4 and 1.5 can be applied to determine the anti-Kekulé numbers of boron-
nitrogen fullerenes, (3, 6)-fullerenes, toroidal and bipartite Klein-bottle fullerenes (see Sec-
tion 3 for details). Based on Theorems 1.4 and 1.5, a polynomial time algorithm is given
to find all smallest anti-Kekulé sets of a connected cubic graph G in Section 4.
2 Proofs of main results
The well-known theorem of Tutte is essential to our proof of the main results.
Theorem 2.1 (Tutte’s Theorem [23]). A graph G has a perfect matching if and only if
co(G− U) ≤ |U | for any U ⊆ V (G), where co(G− U) is the number of odd components
of G− U .
A bridge is an edge of a connected graph whose deletion disconnects the graph. By
Petersen’s Theorem [19], every cubic graph without bridges has a perfect matching. There-
fore, the anti-Kekulé number of a 2-connected cubic graph G (note that a cubic graph
possesses the same connectivity and edge-connectivity) is at least one, that is, ak(G) ≥ 1.
Indeed, this lower bound can be improved and we present the proof by using Tutte’s The-
orem. For X ⊆ V (G), let ∂(X) denote the set of edges with one end in X and the other
end in V (G)−X . We also denote d(X) = |∂(X)|.
Proof of Theorem 1.4. Let A be an anti-Kekulé set of size ak(G). According to the defini-
tion, G′ := G−A has no perfect matchings. Hence, Theorem 2.1 implies that there exists
S ⊆ V (G′) such that co(G′ − S) > |S|. Choose such an S with the maximum size.
Claim 1. G′ − S has no even components and it has exactly |S|+ 2 odd components.
Suppose by the contrary, G′−S has an even component H . Thus, for any given vertex
v ∈ V (H), H −{v} has at least one odd component. Let S′ = S ∪{v}. Then G′−S′ has
at least co(G′ − S) + 1 odd components. That is,
co(G
′ − S′) ≥ co(G′ − S) + 1 > |S|+ 1 = |S′|,
4 Art Discrete Appl. Math. 2 (2019) #P1.05
contradicting the choice of S. Therefore G′ − S has no even component.
SinceG is a cubic graph, it has an even number of vertices. This implies that co(G′−S)
and |S| are of the same parity, thus co(G′−S) ≥ |S|+2. For any edge e ∈ A, sinceA is an
anti-Kekulé set with the smallest cardinality,G′+e has a perfect matching (note thatG′+e
is the graph with vertex set V (G′) and edge set E(G′)∪{e}). Hence co(G′+e−S) ≤ |S|
by Theorem 2.1. Moreover, adding any edge e to G′ − S will connect at most two odd
components. Therefore, |S| ≥ co(G′ + e − S) ≥ co(G′ − S) − 2 ≥ |S| and thus,
co(G
′ − S)− 2 = |S|.
Claim 2. Let Gi, with 1 ≤ i ≤ |S|+ 2, be the odd components of G′ − S. We have
|S|+2∑
i=1
d(Gi)− 2ak(G) ≤ 3|S|. (2.1)
We count the number of edges between S and the odd components, which is denoted
by N , in two different ways. On one hand, S contributes at most 3|S| to N . On the
other hand, all the odd components send out
∑|S|+2
i=1 d(Gi) − 2ak(G) edges to N . Thus∑|S|+2
i=1 d(Gi)− 2ak(G) = N ≤ 3|S| and the claim holds.
SinceG is 2-edge-connected, d(Gi) ≥ 2 for every i. By a simple computation d(Gi) =
3|V (Gi)| − 2|E(Gi)|, which implies that d(Gi) and |V (Gi)| are of the same parity. Since
everyGi is an odd component, |V (Gi)| is odd and hence d(Gi) is odd, therefore d(Gi) ≥ 3.
Substituting this inequality into Equation (2.1), we have
3(|S|+ 2)− 2ak(G) ≤
|S|+2∑
i=1
d(Gi)− 2ak(G) ≤ 3|S|,
and so ak(G) ≥ 3.
Now we are going to establish an upper bound on ak(G) and it is sufficient to find an
anti-Kekulé set of size 4. Let a ∈ V (G) and let b as well as c be its two distinct neighbors.
Denote the two edges incident with b other than ab by e1 and e2, similarly, denote the two
edges incident with c other than ac by e3 and e4. Hence, removing Ea = {e1, e2, e3, e4}
from G will obtain a subgraph without perfect matchings. Therefore, Ea is an anti-Kekulé
set if G− Ea is connected (there exists some vertex a such that G− Ea is not connected,
see Figure 1). Consider the following two cases according to the different connectivities.
Case 1. G is 3-connected.
We are going to prove that for any vertex a in G, G − Ea is connected. Suppose by
the contrary that G − Ea is not connected. Then the vertices of G are divided into two
parts X and X with a subset E′ ⊆ Ea connecting them. Since G is 3-connected, E′
consists of three or four edges in Ea. If it contains three edges, by symmetry, we assume
E′ = {e1, e2, e3}. Since there are exactly three edges between X and X , the edge ab
lies in the same part. We may assume that both a and b belong to X . Since {ab, e3}
divides G into two parts X ∪ {b} and X \ {b}, {ab, e3} is a 2-edge-cut of G which is a
contradiction (note that a 2-edge-cut is an edge-cut of size 2). If E′ contains four edges,
that is E′ = Ea = {e1, e2, e3, e4}, then by a similar argument as above, we know that
a, b and c lie in the same part, and moreover {ab, ac} forms a 2-edge-cut of G which is a
contradiction.
Q. Li et al.: On the anti-Kekulé problem of cubic graphs 5
c
a
b
Figure 1: Ea are the bold edges.
Note that G−Ea is connected and G−Ea has no perfect matchings. Therefore, Ea is
an anti-Kekulé set of size 4 and we have ak(G) ≤ 4.
Case 2. G has connectivity 2.
Since G is 2-connected but not 3-connected, there exist 2-edge-cuts. Moreover, each
2-edge-cut is an independent set, otherwise the third edge incident to their common end
vertex is a bridge, which contradicts that G is 2-connected. Every 2-edge-cut will split G
into exactly two subgraphs. Among those subgraphs, denote the subgraph with smallest
cardinality by G′ and the corresponding 2-edge-cut by E = {e4, e5}. Also, denote the
end-vertices of e4 and e5 in G′ by v and u, respectively. Moreover, let G′′ be the other
subgraph obtained by deleting E.
Claim 3. uv /∈ E(G).
Assume uv ∈ E(G). Then the edges incident with u or v other than uv, e4 and e5 form
a 2-edge-cut. The deletion of this 2-edge-cut creates a subgraph with cardinality smaller
than G′, contradicting the choice of E.
Claim 4. No 2-edge-cut of G contains an edge e ∈ E(G′).
Suppose the claim is false and there exists e ∈ E(G′) that lies in some 2-edge-cut
E′ = {e, e′}. No matter where e′ lies in, the subgraph induced by V (G′′) ∪ {u} or
V (G′′) ∪ {v} belongs to a component created by the deletion of E′. Thus the cardinality
of the other component is smaller than G′, which contradicts the choice of E and the claim
holds.
Let s be a neighbor of v in G′. Since s is of degree 3, there exists a neighbor t (6= u)
of it in G′. Let e1 and e2 be two incident edges of t other than st, and let e3 be the edge
incident with v other than sv and e2. We claim that {e1, e2, e3, e4} is an anti-Kekulé set. It
is obvious that G− {e1, e2, e3, e4} has no perfect matchings. If G− {e1, e2, e3, e4} is not
connected, then, similar to Case 1, we obtain a 2-edge-cut containing at least one edge in
G′. This is a contradiction and completes the proof.
The condition “2-edge-connected” in Theorem 1.4 is necessary because there exist cu-
bic graphs with bridges and their anti-Kekulé number is less than 3 (see Figure 2). More
precisely, we have the following result.
Theorem 2.2. If G is a connected cubic graph with bridges, then ak(G) ≤ 2.
6 Art Discrete Appl. Math. 2 (2019) #P1.05
Proof. Choose a bridge such that the deletion of it will give a subgraph with the smallest
cardinality, we denote this subgraph by G′ and the corresponding bridge by e. Let the end
vertex of e in G′ be u and let v be a neighbor of u in G′. Moreover, let the two other edges
incident with v other than uv be e1 and e2. Similar to the proof of Case 2 in Theorem 1.4,
we haveG−{e1, e2} is connected. Since any bridge separatesG into two odd components,
any perfect matching M of G should contain e. Also, M contains one edge in {e1, e2} and
thus, G−{e1, e2} has no perfect matchings. As a result, {e1, e2} is an anti-Kekulé set and
so ak(G) ≤ 2.
Figure 2 presents three cubic graphs with anti-Kekulé numbers 0, 1 and 2, and the sets
of bold edges denote their smallest anti-Kekulé sets respectively.
Figure 2: Three cubic graphs with anti-Kekulé numbers 0, 1 and 2 respectively.
If the graphs being considered are bipartite, then a stronger result can be obtained by
using Hall’s Theorem.
Theorem 2.3 (Hall’s Theorem [11]). LetG be a bipartite graph with bipartitionW andB.
ThenG has a perfect matching if and only if |W | = |B| and for anyU ⊆W , |N(U)| ≥ |U |
holds.
Proof of Theorem 1.5. First we show that a connected cubic bipartite graph is essentially
2-connected. By Theorem 2.3, a k-regular bipartite graph contains a perfect matching.
Removing that perfect matching will result in a (k − 1)-regular bipartite graph, and the
same argument can be applied repeatedly. Finally, we deduce that a k-regular bipartite
graph can be decomposed into k disjoint perfect matchings. Since G is a cubic bipartite
graph, it can be decomposed into three disjoint perfect matchings M1, M2 and M3, that is,
E(G) =M1 ∪M2 ∪M3. For any e ∈ E(G), without loss of generality, let e ∈M1. Since
M1 and M2 are disjoint perfect matchings of G, M1 ∪M2 consists of disjoint even cycles
and e lies in one of them. Hence e is not a bridge and G is 2-edge-connected. Furthermore,
a 2-edge-connected cubic graph is 2-connected, thus G is 2-connected.
According to Theorem 1.4, we have 3 ≤ ak(G) ≤ 4. Suppose by the contrary that
ak(G) 6= 4, that is, ak(G) = 3. Let A = {e1, e2, e3} be an anti-Kekulé set. Then G − A
Q. Li et al.: On the anti-Kekulé problem of cubic graphs 7
has no perfect matchings. Assume W and B are the bipartition of G. According to Hall’s
theorem, there exists S ⊆W such that
|NG−A(S)| ≤ |S| − 1. (2.2)
On the other hand, since A is an anti-Kekulé set with the smallest cardinality, we have
|S| ≤ |NG−A+ei(S)| (2.3)
for i = 1, 2 and 3. Adding an edge ei to G−A will increase the neighbors of S by one (at
most). Hence
|NG−A+ei(S)| ≤ |NG−A(S)|+ 1. (2.4)
Combining inequalities (2.2), (2.3) and (2.4), we obtain |S| = |NG−A(S)| + 1. Let S′ =
NG−A(S). The edges going out from S are divided into two parts: either goes into A or
goes into S′. Thus the number of edges between S and S′ is 3|S|−3. Since |S′| = |S|−1,
there is no edge between S′ and W −S. Therefore, A is an edge-cut, which contradicts the
definition of anti-Kekulé set and the proof is complete.
3 Applications
In this section, we apply Theorems 1.4 and 1.5 to obtain the anti-Kekulé numbers of several
families of interesting graphs, such as boron-nitrogen fullerenes and (3, 6)-fullerenes.
Theorem 3.1. If G is a (4, 6)-fullerene, then ak(G) = 4.
Proof. Since G is bipartite, the result follows immediately by Theorem 1.5.
Note that there are two classes of boron-nitrogen fullerenes, one with cyclic edge-
connectivity 3 and the other with cyclic edge-connectivity 4. The anti-Kekulé number of
the latter can be obtained by Theorem 1.2. Now we are going to determine the anti-Kekulé
number of (3, 6)-fullerenes and the following lemma is required. A cyclic 3-edge-cut of a
(3, 6)-fullerene is called trivial if it is formed by the edges incident to a triangle in com-
mon. A 3-edge-cut is called trivial if they are incident to a common vertex. Let Tn (n ≥ 1)
be the graph consisting of n concentric layers of hexagons in which each layer is a cyclic
chain of two hexagons, capped on each end by a cap formed by two adjacent triangles (see
Figure 3).
Lemma 3.2 ([29]).
(i) Every cyclic 3-edge-cut of a (3, 6)-fullerene with connectivity 3 is trivial.
(ii) The connectivity of a (3, 6)-fullerene is 2 if and only if it is isomorphic to Tn for some
n ≥ 1.
Theorem 3.3. If G is a (3, 6)-fullerene, then ak(G) = 3.
Proof. Let G be a (3, 6)-fullerene. Note that a (3, 6)-fullerene has connectivity either 2 or
3. Thus, Theorem 1.4 implies that 3 ≤ ak(G) ≤ 4. To show that ak(G) = 3, it suffices to
give an anti-Kekulé set of size 3.
First, assume that the connectivity of G is 2. By Lemma 3.2, G has two triangles
sharing a common edge. Let S be the edge set of such a triangle (see Figure 3). Then
8 Art Discrete Appl. Math. 2 (2019) #P1.05
Figure 3: A (3, 6)-fullerene T3 and the set of bold edges form an anti-Kekulé set of it.
G − S has two vertices of degree 1 adjacent to a common vertex. Hence G − S has no
perfect matching. Clearly, G− S is connected. So S is an anti-Kekulé set of size 3.
In the following, assume that G is 3-connected. Let S be a 3-edge-cut and let G1 as
well as G2 be the two components of G − S. If S is not trivial, then d(Gi) = 3 and
|V (Gi)| ≥ 2 for i = 1, 2. Since |V (Gi)| and d(Gi) are of the same parity, it follows that
|V (Gi)| ≥ 3. Hence
|E(Gi)| − |V (Gi)| =
3|V (Gi)| − 3
2
− |V (Gi)| =
|V (Gi)| − 3
2
≥ 0.
Therefore both G1 and G2 contain cycles. So S is a cyclic 3-edge-cut. By Lemma 3.2, S
is a trivial cyclic 3-edge-cut. Hence, a 3-edge-cut of G is either a trivial 3-edge-cut or a
trivial cyclic 3-edge-cut.
Let abc be a triangle of G. Let e1 be the edge incident with a but not inside of the
triangle, and e2 be the edge incident with c but not contained in the triangle. The edge set
S = {e1, e2, ac} does not isolate a vertex or a triangle. So S is not an edge-cut. In the
subgraph G− S, both a and c have degree 1 and both of them are adjacent to b. So G− S
has no perfect matching. Therefore, S is an anti-Kekulé set. This completes the proof.
Furthermore, since a toroidal fullerene or a bipartite Klein-bottle fullerene, whose def-
initions can be found in [32], is a cubic bipartite graph, the following result is a direct
consequence of Theorem 1.5. Note that this result can also be deduced from Theorem 1.2.
Corollary 3.4. If G is either a toroidal fullerene or a bipartite Klein-bottle fullerene, then
ak(G) = 4.
4 Finding all the smallest anti-Kekulé sets
The anti-Kekulé problem of graphs can be stated as follows.
Instance: A nonempty graph G = (V,E) having a perfect matching and a positive k.
Question: Is there a subset B ⊆ E with |B| ≤ k such that G′ = (V,E \ B) is connected
and G′ has no Kekulé structure?
Q. Li et al.: On the anti-Kekulé problem of cubic graphs 9
In [17], the authors showed that anti-Kekulé problem on bipartite graphs is NP-complete.
So it is hard to find a smallest anti-Kekulé set of a given graph. However, for cubic graphs,
the problem becomes much easier by Theorems 1.4 and 1.5: all the smallest anti-Kekulé
sets of a cubic graph can be found in polynomial time. The algorithm finding all smallest
anti-Kekulé sets S of a cubic graph G depends on how to find a maximum matching in the
graph G − S. If the maximum matching of G − S has size exactly n/2 where n is the
number of vertices of G, then it is a perfect matching of G− S.
For a given graph G with n vertices, Edmonds [6] found an algorithm to find a max-
imum matching of G in O(n4) steps, which is the blossom algorithm. An efficient im-
plementation of Edmonds’ algorithm takes O(n3) steps to find a maximum matching [7].
For bipartite graphs, Hopcroft and Karp [12] gave an algorithm taking O(n5/2) steps to
find a maximum matching. Later, Micali and Vazirani [18, 24], Gabow and Tarjan [8],
and Blum [1] have given algorithms to find a maximum matching of G in O(
√
nm) steps,
where m is the number of edges of G.
Theorem 4.1 ([1, 8, 18]). LetG be a graph with n vertices andm edges. It takesO(
√
nm)
steps to find a maximum matching of G.
The connectedness of a graph G with n vertices can be determined by the breadth-
first search (BFS) algorithm, which takes O(n) steps. Based on Theorems 1.4 and 1.5, by
applying the BFS algorithm and the maximum matching algorithm to G − S, we can find
all smallest anti-Kekulé sets S of a cubic graph G.
Algorithm (Finding all smallest anti-Kekulé sets)
Input: A cubic graph G with n vertices.
Output: All the smallest anti-kekulé sets of G.
Step 1. Let k = 0. Use the maximum matching algorithm on G. If G has a maximum
matching of size n/2, go to Step 2. Otherwise, ak(G) = 0 and stop. Then ∅ is the
only smallest anti-Kekulé set of G.
Step 2. Set k ← k + 1. Screen all edge subsets S of size k and let Fk := {S | |S| =
k and S ⊂ E(G)}. Go to Step 3.
Step 3. Choose an S from Fk, apply the BFS algorithm to find a spanning tree ofG−S. If
G−S has no spanning tree, go to Step 4. Otherwise, apply the maximum matching
algorithm to G− S. If G− S has a maximum matching of size n/2, go to Step 4.
Otherwise, label S as a smallest anti-Kekulé set and go to Step 4.
Step 4. Set Fk ← Fk \ {S}. If Fk 6= ∅, return to Step 3. Otherwise, go to Step 5.
Step 5. If there is no labeled edge set, go to Step 2. Otherwise, output all labeled sets and
stop.
The screening process in Step 2 takes at most
(
m
k
)
steps. By Theorem 1.4, k ≤ 4.
So the worst case takes
(
m
4
)
steps, which is O(m4) steps. It takes O(n) steps to run BFS
algorithm for G − S and O(
√
nm) steps to find a maximum matching of G − S. So for
a given S, it takes at most O(
√
nm) steps to determine whether it is an anti-Kekulé set or
not. Therefore, the worst case takes O(
√
nm5) steps to find all smallest anti-Kekulé sets
of G. Since G is a cubic graph, m = 3n/2. So we have the following result.
10 Art Discrete Appl. Math. 2 (2019) #P1.05
Theorem 4.2. Let G be a connected cubic graph with n vertices. Then it takes O(n11/2)
steps to find out all the smallest anti-Kekulé sets of G.
References
[1] N. Blum, A new approach to maximum matching in general graphs, in: M. Paterson (ed.),
Automata, Languages and Programming, Springer, Berlin, volume 443 of Lecture Notes in
Computer Science, 1990 pp. 586–597, doi:10.1007/bfb0032060, proceedings of the 17th Inter-
national Colloquium (ICALP 1990) held at Warwick University, England, July 16 – 20, 1990.
[2] J. Cai and H. Zhang, On the anti-Kekulé number of a hexagonal system, MATCH Com-
mun. Math. Comput. Chem. 69 (2013), 733–754, http://match.pmf.kg.ac.rs/
electronic_versions/Match69/n3/match69n3_733-754.pdf.
[3] A. Ceulemans, S. Compernolle, A. Delabie, K. Somers, L. F. Chibotaru, P. W. Fowler,
M. J. Margańska and M. Szopa, Electronic structure of polyhedral carbon cages consisting
of hexagons and triangles, Phys. Rev. B 65 (2002), 115412, doi:10.1103/physrevb.65.115412.
[4] M. DeVos, L. Goddyn, B. Mohar and R. Šámal, Cayley sum graphs and eigenvalues of (3, 6)-
fullerenes, J. Comb. Theory Ser. B 99 (2009), 358–369, doi:10.1016/j.jctb.2008.08.005.
[5] T. Došlić, Cyclical edge-connectivity of fullerene graphs and (k, 6)-cages, J. Math. Chem. 33
(2003), 103–112, doi:10.1023/a:1023299815308.
[6] J. Edmonds, Paths, trees, and flowers, Canad. J. Math. 17 (1965), 449–467, doi:10.4153/
cjm-1965-045-4.
[7] H. N. Gabow, An efficient implementation of Edmonds’ algorithm for maximum matching on
graphs, J. ACM 23 (1976), 221–234, doi:10.1145/321941.321942.
[8] H. N. Gabow and R. E. Tarjan, Faster scaling algorithms for general graph-matching problems,
J. ACM 38 (1991), 815–853, doi:10.1145/115234.115366.
[9] P. R. Goodey, Hamiltonian circuits in polytopes with even sided faces, Israel J. Math. 22 (1975),
52–56, doi:10.1007/bf02757273.
[10] P. R. Goodey, A class of Hamiltonian polytopes, J. Graph Theory 1 (1977), 181–185, doi:
10.1002/jgt.3190010213.
[11] P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935), 26–30, doi:10.1112/
jlms/s1-10.37.26.
[12] J. E. Hopcroft and R. M. Karp, An n5/2 algorithm for maximum matchings in bipartite graphs,
SIAM J. Comput. 2 (1973), 225–231, doi:10.1137/0202019.
[13] X. Jiang and H. Zhang, On forcing matching number of boron-nitrogen fullerene graphs, Dis-
crete Appl. Math. 159 (2011), 1581–1593, doi:10.1016/j.dam.2011.05.006.
[14] P. E. John and H. Sachs, Spectra of toroidal graphs, Discrete Math. 309 (2009), 2663–2681,
doi:10.1016/j.disc.2008.06.034.
[15] D. J. Klein, Valence bond theory for conjugated hydrocarbons, Pure Appl. Chem. 55 (1983),
299–306, doi:10.1351/pac198855020299.
[16] K. Kutnar, J. Sedlar and D. Vukičević, On the anti-Kekulé number of leapfrog fullerenes, J.
Math. Chem. 45 (2009), 431–441, doi:10.1007/s10910-008-9416-1.
[17] H. Lü, X. Li and H. Zhang, Np-completeness of anti-Kekulé and matching preclusion problems,
2017, arXiv:1706.09321 [math.CO].
[18] S. Micali and V. V. Vazirani, An O(
√
|V | · |E|) algorithm for finding maximum matching
in general graphs, in: 21st Annual Symposium on Foundations of Computer Science, IEEE
Computer Society, 1980 pp. 17–27, doi:10.1109/sfcs.1980.12, proceedings of the 21st Annual
Q. Li et al.: On the anti-Kekulé problem of cubic graphs 11
Symposium on Foundations of Computer Science held in Syracuse, New York, USA, October
13 – 15, 1980.
[19] J. Petersen, Die Theorie der Regulären Graphs, Acta Math. 15 (1891), 193–220, doi:10.1007/
bf02392606.
[20] Z. Qi and H. Zhang, A note on the cyclical edge-connectivity of fullerene graphs, J. Math.
Chem. 43 (2008), 134–140, doi:10.1007/s10910-006-9185-7.
[21] M. Szopa, M. Margańska and E. Zipper, Geometry and topology induced electronic properties
of graphene derived quantum systems, Int. J. Theor. Phys. 42 (2003), 1119–1132, doi:10.1023/
a:1025499329135.
[22] S. Tang and H. Deng, On the anti-Kekulé number of three fence graphs, Dig. J. Nanomater.
Bios. 6 (2011), 439–443, http://www.chalcogen.ro/439_Tang.pdf.
[23] W. T. Tutte, The factorization of linear graphs, J. London Math. Soc. 22 (1947), 107–111,
doi:10.1112/jlms/s1-22.2.107.
[24] V. V. Vazirani, A theory of alternating paths and blossoms for proving correctness of the
O(
√
V E) general graph maximum matching algorithm, Combinatorica 14 (1994), 71–109,
doi:10.1007/bf01305952.
[25] D. Veljan and D. Vukičević, The anti-Kekulé number of the infinite triangular, rectangular and
hexagonal grids, Glasnik Matematički 43 (2008), 243–252, doi:10.3336/gm.43.2.02.
[26] D. Vukičević and T. Došlić, Global forcing number of grid graphs, Australas. J. Combin. 38
(2007), 47–62, https://ajc.maths.uq.edu.au/pdf/38/ajc_v38_p047.pdf.
[27] D. Vukičević and N. Trinajstić, On the anti-Kekulé number and anti-forcing number of cata-
condensed benzenoids, J. Math. Chem. 43 (2008), 719–726, doi:10.1007/s10910-006-9223-5.
[28] Q. Yang, D. Ye, H. Zhang and Y. Lin, On the anti-Kekulé number of fullerenes, MATCH
Commun. Math. Comput. Chem. 67 (2012), 281–288, http://match.pmf.kg.ac.rs/
electronic_versions/Match67/n2/match67n2_281-288.pdf.
[29] R. Yang and H. Zhang, Hexagonal resonance of (3, 6)-fullerenes, J. Math. Chem. 50 (2012),
261–273, doi:10.1007/s10910-011-9910-8.
[30] D. Ye, On the anti-Kekulé number and odd cycle transversal of regular graphs, Discrete Appl.
Math. 161 (2013), 2196–2199, doi:10.1016/j.dam.2013.03.014.
[31] D. Ye, Z. Qi and H. Zhang, On k-resonant fullerene graphs, SIAM J. Discrete Math. 23 (2009),
1023–1044, doi:10.1137/080712763.
[32] D. Ye and H. Zhang, 2-extendability of toroidal polyhexes and Klein-bottle polyhexes, Discrete
Appl. Math. 157 (2009), 292–299, doi:10.1016/j.dam.2008.03.009.
[33] H. Zhang and S. Liu, 2-resonance of plane bipartite graphs and its applications to boron-
nitrogen fullerenes, Discrete Appl. Math. 158 (2010), 1559–1569, doi:10.1016/j.dam.2010.05.
012.
[34] H. Zhang, D. Ye and W. C. Shiu, Forcing matching numbers of fullerene graphs, Discrete Appl.
Math. 158 (2010), 573–582, doi:10.1016/j.dam.2009.10.013.
[35] Q. Zhang, H. Bian and E. Vumar, On the anti-Kekulé and anti-forcing number of
cata-condensed phenylenes, MATCH Commun. Math. Comput. Chem. 65 (2011), 799–
806, http://match.pmf.kg.ac.rs/electronic_versions/Match65/n3/
match65n3_799-806.pdf.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P1.06
https://doi.org/10.26493/2590-9770.1266.729
(Also available at http://adam-journal.eu)
Automorphism groups of Walecki tournaments
with zero and odd signatures
Janez Aleš
Heidelberg Colaboratory for Image Processing, Heidelberg University,
INF 205, 69120 Heidelberg, Germany
Received 5 February 2018, accepted 16 June 2018, published online 26 August 2018
Abstract
Walecki tournaments were defined by Alspach in 1966. They form a class of regu-
lar tournaments that posses a natural Hamilton directed cycle decomposition. It has been
conjectured by Kelly in 1964 that every regular tournament possesses such a decomposi-
tion. Therefore Walecki tournaments speak in favor of the conjecture. A second interest
in Walecki tournaments arises from the mapping between cycles of the complementing
circular shift register and isomorphism classes of Walecki tournaments. The problem of
enumerating non-isomorphic Walecki tournaments has not been solved to date. We charac-
terize the arc structure of Walecki tournaments whose corresponding binary sequences have
zero and odd signature. Automorphism groups are determined for zero signature Walecki
tournaments and for odd signature Walecki tournaments with the zero signature Walecki
subtournaments.
Walecki tournaments possess a broad range of subtournaments isomorphic to some
Walecki tournament. Subtournaments of odd signature Walecki tournaments induced by
the outsets of the central vertex are proven to be either regular or almost regular.
Keywords: Tournaments, Hamilton directed cycles, automorphism groups.
Math. Subj. Class.: 05C20, 05C45, 05E18, 20B25
1 Introduction
Walecki tournaments were defined by Alspach in 1966. They form a class of regular tour-
naments that posses a natural Hamilton directed cycle decomposition. It has been conjec-
tured by Kelly in 1964 that every regular tournament possesses such a decomposition. An
approximate version of Kelly’s conjecture has been proven by Kühn et al. [9]. Kelly’s con-
jecture has been verified for regular tournaments on n vertices, whenever n is sufficiently
E-mail address: janez.ales@gmail.com (Janez Aleš)
cb This work is licensed under http://creativecommons.org/licenses/by/3.0/
2 Art Discrete Appl. Math. 2 (2019) #P1.06
large in [8]. Counting Walecki-type Hamiltonian cycle systems up to isomorphism has
been solved by Brugnoli [6]. The problem of enumerating non-isomorphic Walecki tour-
naments has not been solved to date. It was published as an open problem by Alspach [2].
Research of this paper continues the work of Aleš [1].
We first define cycles for the complementing circular shift register on binary sequences
of length n (see Subsection 1.1). Walecki tournaments are defined in Subsection 1.2. Sec-
tion 2 determines the arc structure of Walecki tournaments, while Section 3 focuses on
Walecki tournaments with zero signature. A specific permutation is proven to be an au-
tomorphism for Walecki tournaments. For n odd, transitive subtournaments induced by
outsets and insets of specific vertices are discussed in Subsection 3.1.1 and, for n even,
almost-regular and transitive subtournaments are studied in Subsection 3.1.2. Section 3.2
contains characterization of automorphism groups of Walecki tournaments with zero signa-
ture. Section 4 contains results on odd signature Walecki tournaments and Subsection 4.2
contains characterization of automorphism groups of Walecki tournaments with odd signa-
ture with a zero subsignature.
For theoretical background on tournaments we refer the reader to Beineke and Reid [5],
Moon [12], and for topics on permutation groups to Burnside [7] and Wielandt [13].
Automorphism groups of Walecki tournaments were computed with algorithm NAUTY
(No AUTomorphisms, Yes?). Dr. Brendan McKay has made the graph isomorphism pro-
gram NAUTY available to the academic community and it proved to be an indispensible
tool in this research (see McKay [10, 11]).
1.1 Cycles of the complementing circular shift register
Let En denote the set of all binary sequences e = (e1, e2, . . . , en) with ei = 0 or 1
for all i. When considering particular binary sequences we will use e1e2 · · · en to denote
(e1, e2, . . . , en). We use standard notation ei to denote (ei + 1) modulo 2 and e to denote
(e1, e2, . . . , en).
Let R : En → En, a complementing circular shift register operator, be defined by
(R(e))i = ei+1, if 1 ≤ i ≤ n − 1 and (R(e))n = e1. For an integer k ≥ 2, we have
(Rk(e))i = (R(R
k−1(e)))i. It is clear that Rk(e) ∈ En for all k ≥ 1. If e ∈ En, the
period of e is defined to be the smallest positive integer k such that Rk(e) = e. That is,
Rk(e) = e, and Rj(e) 6= e for 2 ≤ j ≤ k − 1.
A subset {e,R(e), . . . , Rm−1(e)} of En is called an m-cycle of operator R if e has
period m. Define e ∼R f if and only if f = Rk(e) for e, f ∈ En and some integer k. It
is easy to verify that ∼R is an equivalence relation on En. Let [e]R, or [e] if no confusion
will arise, denote the equivalence class containing e ∈ En under the relation ∼R. If not
otherwise stated, we will consider the lexicographically smallest element of [e]R to be the
canonical equivalence class representative. This representation is canonical since no two
distinct binary sequences have the same lexicographic order.
Let f and h be sequences in En1 and En2 , respectively, and let e = fh denote the
sequence of length n1 + n2 in En1+n2 .
For completness reasons we state results obtained by Alspach [4] which have direct im-
plications on the structure of Walecki tournaments with non-trivial automorphism groups.
Lemma 1.1 (Alspach, 1966). Let e ∈ En. If a positive integer k divides n such that n/k
is odd and if
e = ffff . . . ff, for f ∈ Ek,
J. Aleš: Automorphism groups of Walecki tournaments with zero and odd signatures 3
then R2k(e) = e. Moreover, e and f have the same period.
Lemma 1.2 (Alspach, 1966). If e ∈ En has periodm, wherem < 2n, thenm = 2r where
r divides n, n/r is odd, r < n, and e = ffff . . . ff such that f ∈ Er and the period of f
is 2r = m.
1.2 Definition of Walecki tournaments
Let [v(0), v(1), . . . , v(2n)] be a given undirected Hamilton cycle of order 2n+ 1 on the ver-
tex set {v(0), v(1), . . . , v(2n)}. Let τ ∈ S2n+1 be the permutation of {v(0), v(1), . . . , v(2n)}
defined by τ = (v(0))(v(1)v(2)v(4)v(6) · · · v(2n−2)v(2n)v(2n−1)v(2n−3) · · · v(5)v(3)).We de-
fine the v-labeling of the vertices of Walecki tournaments as follows: 2n vertices on the cir-
cumference of a circle are labeled as v(1), v(2), v(4), . . . , v(2n−2), v(2n), v(2n−1), . . . , v(5),
v(3) with the central vertex labeled v(0).
The permutation τ corresponds to the clockwise rotation of the vertices on the circum-
ference of the circle with v(0) as a fixed point in the center. Figure 1 shows the action of
the permutation τ ∈ S2n+1 on vertices of the given Hamilton cycle. A Walecki tourna-
v(0) = t(0)
v(1) = t(1) v(2) = t(2)
v(4) = t(3)
v(3) = t(2n)
v(5) = t(2n−1)
v(2n) = t(n+1)
v(2n−1) = t(n+2) v(2n−2) = t(n)
Figure 1: The diagram shows the action of the permutation τ ∈ S2n+1 on vertices of the
Walecki tournament W (e). Vertex v(0) is fixed by τ .
ment on 2n + 1 vertices is then defined by assigning to each of the n undirected Hamil-
ton cycles H1 = [v(0), v(1), . . . , v(2n)], H2 = [τ(v(0)), τ(v(1)), . . . , τ(v(2n))], . . . ,Hn =
[τn−1(v(0)), τn−1(v(1)), . . . , τn−1(v(2n))] one of the two possible orientations. For ex-
ample, Figure 2 shows the directed Hamilton cycle ⇀H1 = [v(0), v(1), . . . , v(2n)]. Directed
Hamilton cycle ↼H1 has all the arcs of ⇀H1 reversed, that is, ↼H1 = [v(0), v(2n), . . . , v(1)]. It is
easy to see that the union of n Hamilton directed cycles indeed forms a regular tournament.
Let e ∈ En be a binary sequence of length n. Define components of e as follows,
ei = 0, if v(0) → τ i−1(v(1)), and ei = 1, if v(0) ← τ i−1(v(1)), for 1 ≤ i ≤ n, where ar-
rows→ and← denote arcs in a tournament. This establishes a one-to-one correspondence
between all 2n possible orientations of n Hamilton cycles and the elements of En. For
example, Figure 3 shows Walecki tournaments W (0) and W (00) on 3 and 5 vertices, re-
spectively, Figure 4 shows Walecki tournament W (000) on 7 vertices, and Figure 5 shows
Walecki tournament W (0000) on 9 vertices, where W (e) denotes a Walecki tournament
with Hamilton cycles directed according to elements of the sequence e.
4 Art Discrete Appl. Math. 2 (2019) #P1.06
v(0) = t(0)
v(1) = t(1)
v(2) = t(2)
v(4) = t(3)
v(3) = t(2n)
v(5) = t(2n−1)
v(2n) = t(n+1)
v(2n−1) = t(n+2) v(2n−2) = t(n)
v(2n−3) = t(n+3) v(2n−4) = t(n−1)
Figure 2: The directed Hamilton cycle ⇀H1 = [v(0), v(1), . . . , v(2n)].
v(0)
v(1)
v(2)v(3)
v(4)
W (00)
v(0)
v(1)
v(2)
W (0)
Arcs on the directed Hamilton cycle ⇀H1
Arcs on the directed Hamilton cycle ⇀H2
Figure 3: Walecki tournaments W (0) and W (00).
J. Aleš: Automorphism groups of Walecki tournaments with zero and odd signatures 5
v(0)
v(1)
v(2)v(3)
v(4)v(5)
v(6)
W (000)
Arcs on the directed Hamilton cycle ⇀H1
Arcs on the directed Hamilton cycle ⇀H2
Arcs on the directed Hamilton cycle ⇀H3
Figure 4: Walecki tournament W (000).
v(0)
v(1)
v(4)
v(3) v(2)
v(5)
v(6)v(7)
v(8)
W (0000)
Arcs on the directed Hamilton cycle ⇀H1
Arcs on the directed Hamilton cycle ⇀H2
Arcs on the directed Hamilton cycle ⇀H3
Arcs on the directed Hamilton cycle ⇀H4
Figure 5: Walecki tournament W (0000).
6 Art Discrete Appl. Math. 2 (2019) #P1.06
We state a necessary condition for isomorphism of two Walecki tournaments which can
be easily proven using isomorphism τk for 1 ≤ k ≤ 2n.
Proposition 1.3 (Alspach, 1966). Let n be a positive integer and let e ∈ En. If k is an
integer such that 1 ≤ k ≤ 2n, then
W (e) ∼=W (Rk(e)).
Let η denote the permutation τn ∈ S2n+1. That is,
η = (v(0))(v(1) v(2n))(v(2) v(2n−1)) . . . (v(n) v(n+1)).
Notice that τ(v(0)) = η(v(0)) = v(0). The n directed cycles of W (e) are
Hk = [τ
k−1(ηek(v(0))), τk−1(ηek(v(1))), . . . , τk−1(ηek(v(2n)))],
for 1 ≤ k ≤ n.
In order to simplify notation we introduce the t-labeling {t(0), t(1), . . . , t(2n)} of the
vertices of Walecki tournaments, where t(0) = v(0) and t(i) = τ i−1(v(1)), 1 ≤ i ≤ 2n (see
Figure 2). The action of τ on {t(0), t(1), . . . , t(2n)} is given by τ = (t(0))(t(1) t(2) · · · t(2n)).
2 Arc structure of Walecki tournaments
In 1964 Kelly conjectured that every regular tournament admits a decomposition into
Hamilton directed cycles (see Moon [12]). Walecki tournaments are regular and possess
a natural Hamilton directed cycle decomposition (see Alspach [3]). The class of Walecki
tournaments speaks in favor of the above conjecture. Therefore knowledge of their struc-
ture would be of importance. They posses a rich collection of induced subtournaments
ranging from transitive to regular as we shall see later. In some instances outsets of v(0)
induce regular or almost regular subtournaments. In other cases outsets of v(0) induce
subtournaments whose scores differ for at most 2.
We will first state various results which give insight into the arc structure of an arbitrary
Walecki tournament. The reader can find the proofs in [1].
Proposition 2.1. Walecki tournaments are self-complementary and
AntiAut(W (e)) = Aut(W (e))η,
where AntiAut denotes the antiautomorphism group of a tournament.
Proposition 2.2. Let T be a tournament and let V (T ) denote the vertex set of T . For
v ∈ V (T ) and g ∈ Aut(T )v,
g(N+(v)) = N+(v) and g(N−(v)) = N−(v),
where Aut(T )v denotes the subgroup of Aut(T ) which fixes v.
Proposition 2.3. LetW (e) be a Walecki tournament of order 2n+1 and let k be an integer
such that 1 ≤ k ≤ 2n. If t(0)→ t(k) ∈ ⇀Hk, then t(n+k)→ t(0) ∈ ⇀Hk.
Proposition 2.4. Let W (e) be a Walecki tournament of order 2n + 1 and let i and j be
integers such that 0 ≤ i, j ≤ 2n− 1. If t(i)→ t(j), then t(n+i)← t(n+j).
J. Aleš: Automorphism groups of Walecki tournaments with zero and odd signatures 7
The following result is used in many proofs about the structure of Walecki tournaments.
It uses the binary sequence e ∈ En to determine the direction of a particular arc in W (e).
The arcs are grouped according to the Hamilton directed cycle they belong to.
Lemma 2.5. Let e ∈ En and let W (e) be the corresponding Walecki tournament. Let
i and j be integers such that 1 ≤ i < j ≤ 2n. In the case when j − i is even, let
k = i+ 1 + (j − i)/2.
• If 1 ≤ k ≤ n, then t(j+ek)→ t(i)→ t(j+ek) and t(j+n+ek)← t(i+n)← t(j+n+ek).
• If n + 1 ≤ k ≤ 2n, then t(j−n+ek−n) → t(i−n) → t(j−n+ek−n) and t(j+ek−n) ←
t(i)← t(j+ek−n).
In the case when j − i is odd, let ` = i+ 1 + (j − i− 1)/2.
• If 1 ≤ ` ≤ n, then t(j−e`) ← t(i)← t(j−e`) and t(j+n−e`) → t(i+n)→ t(j+n−el).
• If n + 1 ≤ ` ≤ 2n, then t(j−n−e`−n) ← t(i−n) ← t(j−n−e`−n) and t(j−e`−n) →
t(i)→ t(j−el−n).
Proof. Let i and j be as in the conditions of the Proposition 2.4. We first consider the case
when j− i is even. Let k = i+1+(j− i)/2. The structure of the Hamilton directed cycle
⇀Hk implies that if ek = 0, then t(j) → t(i) and t(i) → t(j+1) (see Figure 6). On the other
v(0) = t(0)
t(i)
t(k−2)
t(k−1)
t(k)
t(j)
t(j+1)
t(n+i)
t(n+k−2)
t(n+k−1)
t(n+k)
t(n+j)
t(n+j+1)
Figure 6: The diagram shows the case when j − i is even and ek = 0 from the proof of
Lemma 2.5.
hand, if ek = 1, then t(j+1) → t(i) and t(i) → t(j), implying t(j+ek) → t(i) → t(j+ek).
Proposition 2.4 implies t(j+n+ek)← t(i+n)← t(j+n+ek). To prove the next two statements
substitute above j − n for j and i− n for i. The remaining cases are proven similarly.
3 Zero signature Walecki tournaments
Let [e]R be an equivalence class of binary n-sequences under the relation ∼R defined in
Subsection 1.1. We say that [e]R has zero signature if the lexicographically smallest se-
quence of [e]R is (0, 0, . . . , 0). Tournaments corresponding to the zero signature sequences,
as we shall see, have a surprisingly simple automorphism group. In the case of odd n,
σ = (t(0))(t(1)t(2) · · · t(n)) (t(2n)t(2n−1) · · · t(n+1)) ∈ S2n+1
8 Art Discrete Appl. Math. 2 (2019) #P1.06
is in fact an automorphism of W ((0, 0, . . . , 0)) as proven in Theorem 3.3. We will prove in
Theorem 3.4 and Theorem 3.6 that Walecki tournaments of order 2n+1 with zero signature
possess transitive subtournaments of order n that are induced by the outset of vertex v(1).
When n is odd they also contain circulant subtournaments of order n induced by the outset
of vertex v(0). This furthermore implies that these subtournaments are regular. For example
see Figure 4 which shows Walecki tournament W (000).
The next couple results are needed for characterizing the arc structure of Walecki tour-
naments for the case when n is odd, n ≥ 3, and e = (0, 0, . . . , 0) ∈ En. We omit straight-
forward proofs.
Proposition 3.1. Let e ∈ En and n ≥ 3. Consider the Walecki tournament W (e). If
ei = ei+1 and 1 ≤ i ≤ n−1, then τ is dominance-preserving on⇀Hi and τ−1 is dominance-
preserving on ⇀Hi+1.
Notice that permutation τ is not an automorphism of W (e).
Lemma 3.2. Let e ∈ En and n ≥ 3. Consider the Hamilton directed cycle ⇀Hi, 1 ≤ i ≤ n,
in the Walecki tournament W (e). Let u→ w be any arc of ⇀Hi of the form u = τ i−1(v(2j))
and w = τ i−1(v(2j+1)) or u = τ i−1(v(2j+1)) and w = τ i−1(v(2j+2)), 1 ≤ j ≤ n − 2. If
ρ ∈ S2n+1 is a permutation of {v(0), v(1), . . . , v(2n)} such that ρ = τ on τ i−1(v(2j)), 1 ≤
j ≤ n−2, and ρ = τ−1 on τ i−1(v(2j+1)), 1 ≤ j ≤ n−2. Then ρ is dominance-preserving
on the arc u→ w.
Let n be odd and let the permutation σ ∈ S2n+1 be defined by
σ = (t(0))(t(1) t(2) · · · t(n))(t(2n) t(2n−1) · · · t(n+1)),
(see Figure 7). That is,
σ = (v(0))(v(1) v(2) v(4) · · · v(2n−4) v(2n−2))(v(3) v(5) · · · v(2n−3) v(2n−1) v(2n)).
v(0) = t(0)
v(1) = t(1)
v(2) = t(2)
v(4) = t(3)
v(3) = t(2n)
v(5) = t(2n−1)
v(2n) = t(n+1)
v(2n−1) = t(n+2) v(2n−2) = t(n)
Figure 7: The action of the permutation σ ∈ S2n+1.
Theorem 3.3. Let e = (0, 0, . . . , 0), be the binary n-sequence of all 0s. If n is odd, and
n ≥ 3, then σ is an automorphism of W (e).
J. Aleš: Automorphism groups of Walecki tournaments with zero and odd signatures 9
Proof. We want to show that σ is dominance-preserving on all of W (e). By definition, σ
fixes t(0), cyclically permutes the vertices of the outset of t(0), and cyclically permutes the
vertices of the inset of t(0). Thus, σ is dominance-preserving on the arcs incident with t(0).
Figure 7 shows the action of σ ∈ S2n+1 on vertices of the Walecki tournament W (e), for
e = (0, 0, . . . , 0) ∈ En, n odd and n ≥ 3.
Note that σ restricted to V + = N+(t(0)) − {t(n)} = {t(1), t(2), . . . , t(n−1)} has the
same action as τ. It then follows from Proposition 3.1 that σ is dominance-preserving on
any arc both of whose vertices lie in V + because such an arc is not in ⇀Hn. Similarly, σ
restricted to V − = N−(t(0))− {t(n+1)} = {t(n+2), t(n+3), . . . , t(2n)} has the same action
as τ−1. Again it follows from Proposition 3.1 that σ is dominance-preserving on any arc
both of whose vertices lie in V − because such an arc is not in ⇀H1. (Figure 1 shows the
action of the permutation τ on vertices of the Walecki tournament W (e).)
By Lemma 3.2, σ is dominance-preserving on any arc with one end vertex in V +
and the other end vertex in V − because σ acts like τ on V + and τ−1 on V −. It re-
mains to show that σ is dominance-preserving on any arc incident with t(n) = v(2n−2) or
t(n+1) = v(2n). Since en = 0 we have v(2n−2)→ v(2n) ∈ ⇀Hn. Furthermore, σ(v(2n−2)) =
v(1) = τ0(v(1)) = τn−1(v(2n−1)) and σ(v(2n)) = v(3) = τ−1(v(1)) = τn−1(v(2n)) imply
σ(v(2n−2))→ σ(v(2n)) ∈ ⇀Hn.
We divide the proof for the arcs incident with either v(2n−2) or v(2n) into two cases.
Let u ∈ V − ∪V +. First we consider arcs u→ v(2n−2) and u→ v(2n). We use Lemma 2.5
extensively. We remind the reader that arcs τk−1(v(i)) → τk−1(v(i+1)), 1 ≤ i ≤ 2n − 1,
belong to the Hamilton directed cycle Hk. In the figures accompanying this proof we use
arrows to denote the action of σ and arrows to denote arcs of the
tournament.
Case 1.1. Let u ∈ V −. If k is an integer such that 1 ≤ k ≤ (n − 1)/2, then vertices
t(n+2k+1) and t(n+2k) belong to V −, and arcs t(n+2k+1) = τk−1(v(2(n−k)−1)) → t(n) =
τk−1(v(2(n−k))) and t(n+2k) = τk−1(v(2(n−k)+1))→ t(n+1) = τk−1(v(2(n−k+1))) belong
to the Hamilton directed cycle ⇀Hk (see Figure 8). Now, σ is dominance-preserving on these
v(0) = t(0)
v(1) = t(1)
v(3) = t(2n)
v(2(n−2k)+1)= t(n+2k+1)
v(2(n−2k+1)+1)= t(n+2k)
v(2(n−2k+2)+1)= t(n+2k−1)
v(2n) = t(n+1)
v(2n−2) = t(n)
Figure 8: The diagram shows the action of the permutation σ ∈ S2n+1 on arcs from
Case 1.1 for the proof of Theorem 3.3.
10 Art Discrete Appl. Math. 2 (2019) #P1.06
two arcs since σ(t(n+2k+1)) = t(n+2k) = τk+(n−1)/2(v(n+2k−1)) and σ(t(n)) = t(1) =
τk+(n−1)/2(v(n+2k)) imply
σ(v(2(n−2k)+1))→ σ(v(2n−2)) ∈ ⇀Hk+(n+1)/2.
Also σ(t(n+2k)) = t(n+2k−1) = τk+(n−3)/2(v(n+2k−1)) and σ(t(n+1)) = t(2n) =
τk+(n−3)/2(v(n+2k)) imply
σ(v(2(n−2k+1)+1))→ σ(v(2n)) ∈ ⇀Hk+(n−1)/2.
Case 1.2. Let u ∈ V +. If k is an integer such that (n + 1)/2 ≤ k ≤ n − 1, then vertices
t(2k−n+1) and t(2k−n+2) belong to V +, and arcs t(2k−n+1)→ t(n) and t(2k−n+2)→ t(n+1)
belong to the Hamilton directed cycle ⇀Hk (see Figure 9). Similarly as before we have
σ(v(2(2k−n)))→ σ(v(2n−2)) ∈ ⇀Hk−(n−3)/2
and
σ(v(2(2k−n−1)))→ σ(v(2n)) ∈ ⇀Hk−(n−1)/2.
v(0) = t(0)
v(1) = t(1)
v(3) = t(2n)
v(2(2k−n−1))= t(2k−n)
v(2(2k−n))= t(2k−n+1)
v(2(2k−n+1))= t(2k−n+2)
v(2n) = t(n+1)
v(2n−2) = t(n)
Figure 9: The action of the permutation σ ∈ S2n+1 on arcs from Case 1.2 for the proof of
Theorem 3.3.
Next we consider arcs v(2n−2) → u and v(2n) → u for u ∈ V − ∪ V +. We state all
cases but leave the proofs to the reader.
Case 2.1. Let u ∈ V −. If k is an integer such that 2 ≤ k ≤ (n − 1)/2, then vertices
v(2(n−2k+1)+1) and v(2(n−2k+2)+1) belong to V −. We have to consider vertices v(3) and
v(2n−1) ∈ V − as a special case.
Case 2.2. Let u ∈ V +. If k is an integer such that (n + 3)/2 ≤ k ≤ n − 1, then vertices
v(2(2k−n−1)) and v(2(2k−n+2)) belong to V +. We consider vertex v(1) ∈ V + as a special
case.
Therefore u → w implies σ(u) → σ(w) for every arc u → w in W (e) and so σ ∈
Aut(W (e)).
J. Aleš: Automorphism groups of Walecki tournaments with zero and odd signatures 11
The importance of σ in the theory of automorphism groups of Walecki tournaments was
previously unknown. However, once zero signature sequences were determined as a poten-
tial source of Walecki tournaments with non-trivial automorphism groups, permutation σ
became a natural candidate for a generator.
3.1 Subtournaments of zero signature Walecki tournaments
Next we characterize specific subtournaments of zero signature Walecki tournaments, which
prove to be transitive for n odd and almost regular or transitive for n even.
3.1.1 Transitive subtournaments for n odd
In the following result we prove transitivity of subtournaments of Walecki tournament with
zero signature for n odd. The linear orderings of subsets of vertices that induce transitive
subtournaments are given in the proof.
Theorem 3.4. Let T = W (e) for e = (0, 0, . . . , 0) ∈ En, n odd, and n ≥ 3. For
t(i) ∈ N+(t(0)) and t(j) ∈ N−(t(0)) the tournaments〈
N+(t(i))
〉
, T
〈
N−(t(j))
〉
, T
〈
N−(t(i))− {t(0)}
〉
, and T
〈
N+(t(j))− {t(0)}
〉
are transitive subtournaments of T .
Proof. Proposition 2.1 tells us that W (e) ∼= W (e). Since σ ∈ Aut(T ), it suffices to
prove the theorem for the vertex t(1) ∈ N+(t(0)). Let us consider the outset of vertex t(1).
The arcs v(2i+1) → v(2i+2) lie in ⇀H1 for 0 ≤ i ≤ n − 1 so that t(1) = τ i(v(2i+1)) →
τ i(v(2i+2)) = τ2i+1(v(1)) ∈ ⇀Hi+1. Hence
N+(t(1)) = {t(2i+2) | 0 ≤ i ≤ n− 1}. (3.1)
We prove that the vertices of N+(t(1)) in the order t(2n), t(2), t(2n−2), t(4), . . . , t(2n−2i),
t(2i+2), . . . , t(n+3), t(n−1), t(n+1) determine the score sequence (sj)n−1j=0 , where sj = j for
0 ≤ j ≤ n − 1. That is, s2i = s(t(2n−2i)) = 2i for 0 ≤ i ≤ (n − 3)/2, s2i+1 =
s(t(2i+2)) = 2i + 1 for 0 ≤ i ≤ (n − 3)/2, and sn−1 = s(t(n+1)) = n − 1. We prove
this by showing that all arcs in the subtournament T
〈
N+(t(1))
〉
point from right to left
in the ordering of the vertices given above. Figure 10 shows seven different types of arcs
considered. We divide the proof into several cases and show details for some of them. In
all of them the index i is an integer such that 0 ≤ i ≤ (n− 3)/2.
Case 1.1. Since t(n+1) = τ (n+1)/2+i(v(n−2i−1)) and t(2i+2) = τ (n+1)/2+i(v(n−2i)), the
arcs of type t(n+1)→ t(2i+2) belong to cycles ⇀H(n+3)/2+i.
We omit proofs of t(n+1) → t(2n−2i) ∈ H(n+1)/2−i and t(2i) → t(2n−2i) ∈ ⇀H1. In the
remaining cases the index j is in the range i ≤ j ≤ (n− 3)/2.
Case 1.2. Since t(2j+2) = τ i+j+1(v(2(j−i))) and t(2i+2) = τ i+j+1(v(2(j−i)+1)), it is
t(2j+2)→ t(2i+2) ∈ ⇀Hi+j+2.
We omit proofs of t(2j+2)→ t(2n−2i) ∈ ⇀Hj−i+1, t(2n−2j)→ t(2i+2) ∈ ⇀Hn+i−j+1, and
t(2n−2j)→ t(2n−2i) ∈ ⇀Hn−i−j .
It follows that the scores of vertices in the subtournament T
〈
N+(t(1))
〉
are sj = j for
0 ≤ j ≤ n− 1. Thus, the subtournament T
〈
N+(t(1))
〉
is transitive.
12 Art Discrete Appl. Math. 2 (2019) #P1.06
t(2n−2i) t(2i+2) t(n+1)
t(2n−2i) t(2i+2) t(2n−2j) t(2j+2)
Figure 10: Seven different types of arcs from Case 1.1 and Case 1.2 of the proof of transi-
tivity of the tournament T
〈
N+(v(1))
〉
from Theorem 3.4.
Next we consider the set of vertices N−(t(1)) − {t(0)}. Since N+(t(1)) = {t(2i+2) |
1 ≤ i ≤ n − 1}, it follows that N−(t(1)) − {t(0)} = {t(2i+1) | 1 ≤ i ≤ n − 1}. We will
prove that the labeling of the vertices ofN−(t(1))−{t(0)} in the order t(2n−1), t(3), t(2n−3),
t(5), . . . , t(2n−2i+1), t(2i+1), . . . , t(n+2), t(n) determines the score sequence (sj)n−1j=0 ,where
sj = j for 0 ≤ j ≤ n− 2. That is, s2i−2 = s(t(2n−2i+1)) = 2i− 2, for 1 ≤ i ≤ n−12 , and
s2i−1 = s(t(2i+1)) = 2i − 1, for 1 ≤ i ≤ n−12 . Similarly as in the previous case one can
prove that all arcs in the subtournament T
〈
N−(t(1)) − {t(0)}
〉
point from right to left in
the ordering of the vertices given above. Thus, the subtournament T
〈
N−(t(1)) − {t(0)}
〉
is transitive.
3.1.2 Almost-regular and transitive subtournaments for n even
When n is even σ is not an automorphism because an automorphism group of a tournament
has to have an odd order.
Theorem 3.5. Let T = W (e) for e = (0, 0, . . . , 0) ∈ En, n even, and n ≥ 4. The
subtournaments T
〈
N+(t(0))
〉
and T
〈
N−(t(0))
〉
are almost regular.
Proof. It follows from the definition of Walecki tournaments that N+(t(0)) = {t(i) | 1 ≤
i ≤ n}. Equation (3.1) also holds for n even. Hence, N+(t(1)) = {t(2i) | 1 ≤ i ≤
n}, implying N+(t(0)) ∩ N+(t(1)) = {t(2i) | 1 ≤ i ≤ n/2}. Therefore, |N+(t(0)) ∩
N+(t(1))| = n/2. It is easy to verify that N+(t(2)) = {t(2i+1) | 1 ≤ i ≤ n − 1} ∪
{t(2n)}, which implies N+(t(0))∩N+(t(2)) = {t(2i+1) | 1 ≤ i ≤ n/2− 1}. Furthermore,
|N+(t(0)) ∩N+(t(2))| = n/2− 1.
The scores of the remaining vertices in N+(t(0)) can be obtained similarly, then we
omit the proofs. They alternate between n/2 and n/2− 1 which proves that T
〈
N+(t(0))
〉
is almost regular. Since T is self-complementary, this completes the proof.
Theorem 3.6. Let T = W (e) for e = (0, 0, . . . , 0) ∈ En, n even, and n ≥ 4. For
t+ ∈ N+(t(0)) and t− ∈ N−(t(0)) the tournaments
T
〈
N+(t+)
〉
, T
〈
N−(t+)− {t(0)}
〉
, T
〈
N+(t−)− {t(0)}
〉
, and T
〈
N−(t−)
〉
are transitive subtournaments of T.
J. Aleš: Automorphism groups of Walecki tournaments with zero and odd signatures 13
Proof. Proposition 2.1 tells us that T ∼= T . Hence, it suffices to prove the theorem for
vertices in N+(t(0)). The proof of transitivity of T
〈
N+(t(1))
〉
and T
〈
N−(t(1)) − {t(0)}
〉
is similar to the proof of Theorem 3.4, the difference being that n is even. This changes the
proof in two ways. First, the vertices of N+(t(1)) in the order t(2n), t(2), t(2n−2), t(4), . . . ,
t(2n−2i), t(2i+2), . . . , t(n+2), t(n) determine the score sequence (0, 1, 2, . . . , n− 1).
Furthermore, since n is even we have σ /∈ Aut(T ). Therefore, one has to prove that the
subtournaments T
〈
N+(t+)
〉
and T
〈
N−(t+)−{t(0)}
〉
are transitive for all t+ ∈ N+(t(0)).
The proofs are similar to the initial case and we omit them.
3.2 Automorphism groups of Walecki tournaments with zero signature
The arc structure of Walecki tournaments with zero signature plays a major role in deter-
mining their automorphism groups.
Theorem 3.7. Automorphism groups of Walecki tournaments with zero signature are cyclic:
Aut(W (0)) = Z3, Aut(W (00)) = Z5, Aut(W (e)) = Zn, for n odd, n ≥ 3,
and
Aut(W (e)) = Z1, for n even, n ≥ 4,
where Zn denotes the cyclic group of order n and e = (0, 0, . . . , 0) ∈ En.
Proof. We leave the proof of initial cases as an exercise for the reader. Let T denote the
Walecki tournamentW (e) and letG denote its automorphism group Aut(T ).We use Orbit
Stabilizer Theorem two times to get
|G| = |O(v(0))||Gv(0)| = |O(v(0))||O(v(1))||Gv(0),v(1)|, (3.2)
where O(v(1)) denotes the orbit of vertex v(1) for the subgroup Gv(0) of G.
Case 1. Let us assume that n is odd, n ≥ 3, and e = (0, 0, . . . , 0) ∈ En. We first consider
the cardinality of O(v(0)). T
〈
N+(v(0))
〉
is an almost regular tournament. Therefore, it
is not transitive. On the other hand, T
〈
N+(v(i))
〉
is transitive for v(i) ∈ N+(v(0)) (see
Theorem 3.4). Thus, v(0) cannot be mapped to a vertex from N+(v(0)) by elements of
G. Proposition 2.1 implies T ∼= T with the graph anti-automorphism τn. Therefore, v(0)
cannot be mapped to a vertex from N−(v(0)) by elements of G. We have proven that v(0)
must be fixed under the action of G, and thus
|O(v(0))| = 1. (3.3)
The fact that v(0) cannot be mapped to any vertex in N−(v(0)) can also be proven directly
for n ≥ 5. Let us consider T
〈
N+(v(0)) − v(i)
〉
for v(i) ∈ N+(v(0)). T
〈
N+(v(0))
〉
is
regular of degree (n − 1)/2. Thus, T
〈
N+(v(0)) − v(i)
〉
has (n − 1)/2 vertices of degree
(n−1)/2 and (n−1)/2 vertices of degree (n−3)/2. Therefore, if n ≥ 5, the subtournament
T
〈
N+(v(0)) − {v(i)}
〉
is not transitive. However, T
〈
N+(v(j)) − {v(0)}
〉
is transitive for
v(j) ∈ N−(v(0)). Therefore, v(0) cannot be mapped to v(j) ∈ N−(v(0)) by elements
of G if n ≥ 5. We cannot use the same argument for n = 3 since the subtournament
T
〈
N+(v(0))− v(i)
〉
, for v(i) ∈ N+(v(0)), is a tournament on two vertices and is therefore
transitive.
14 Art Discrete Appl. Math. 2 (2019) #P1.06
Next we determine |O(v(1))|. Since v(0) is a fixed point for any element ρ in G,
ρ(N+(v(0))) = N+(v(0)). Hence, ρ(v(1)) ∈ N+(v(0)) and |O(v(1))| ≤ |N+(v(0))| = n.
We proved that the permutation σ ∈ S2n+1 of V (T ) defined by
σ = (v(1) v(2) v(4) · · · v(2n−4) v(2n−2))(v(3) v(5) · · · v(2n−3) v(2n−1) v(2n)),
is an element in G. Since σ(v(0)) = v(0), σ ∈ Gv(0). Hence,
〈
σ
〉
⊆ Gv(0). The orbit of
v(1) for σ is N+(v(0)) implying
|O(v(1))| = n. (3.4)
Last we prove that Gv(0),v(1) = id. The subtournaments T
〈
N+(v(1))
〉
and T
〈
N−(v(1))−
{v(0)}
〉
are transitive, implying that any automorphism ρ ∈ Gv(0),v(1) fixes all other ver-
tices. Figure 11 shows the partition of the vertices of T with respect to the outsets and
insets of vertices v(0) and v(1). Therefore, Gv(0),v(1) = id, that is,
|Gv(0),v(1)| = 1. (3.5)
Equations (3.2), (3.3), (3.4), and (3.5) imply that |G| = n. Now,
〈
σ
〉
⊆ Gv(0) ⊆ G and
since
〈
σ
〉 ∼= Zn we have G ∼= Zn.
v(0) v(1)
N+(v(0))N−(v(0))
N+(v(1))
N−(v(1))
an arc a set of arcs
Figure 11: The partition of the vertices of the Walecki tournament W (e), for e =
(0, 0, . . . , 0) ∈ En, n odd, and n ≥ 3, with respect to the outsets and insets of vertices
v(0) and v(1). Only the arcs essential for the proof of Theorem 3.7 are drawn.
Case 2. Let us assume n is even, n ≥ 4, and e = (0, 0, . . . , 0) ∈ En. We first consider
the cardinality of O(v(0)). The subtournament T
〈
N+(v(0))
〉
is almost regular (see The-
orem 3.5). Therefore, it is not transitive. On the other hand, T
〈
N+(v(i))
〉
is transitive
for v(i) ∈ N+(v(0)) (see Theorem 3.6). Thus, v(0) cannot be mapped to a vertex from
N+(v(0)) by elements of G.
Proposition 2.1 implies implies T ∼= T via the graph anti-automorphism τn. Therefore,
v(0) cannot be mapped to a vertex from N−(v(0)) by elements of G. We have proven that
J. Aleš: Automorphism groups of Walecki tournaments with zero and odd signatures 15
v(0) must be fixed under the action of G, and thus
|O(v(0))| = 1. (3.6)
Next we determine |O(v(1))|. Since v(0) is a fixed point for any element ρ in G,
ρ(N+(v(0))) = N+(v(0)). Hence, ρ(v(1)) ∈ N+(v(0)). As seen in the proof of Theo-
rem 3.5, |N+(v(0)) ∩N+(v(1))| = n/2 and |N+(v(0)) ∩N−(v(1))| = n/2 − 1. Further-
more, T
〈
N+(v(1))
〉
and T
〈
N−(v(1))
〉
are both transitive which implies that
T
〈
N+(v(0)) ∩ N+(v(1))
〉
and T
〈
N+(v(0)) ∩ N−(v(1))
〉
are transitive. Moreover, v(1)
is dominated by N+(v(0))∩N−(v(1)) implying that T
〈
(N+(v(0))∩N−(v(1)))∪ {v(1)}
〉
is transitive. Let X = N+(v(0)) ∩ N+(v(1)) and Y = (N+(v(0)) ∩ N−(v(1))) ∪ {v(1)}.
Now, vertices of X have score n/2 − 1 in T
〈
N+(v(0))
〉
. Similarly, vertices of Y have
score n/2 in T
〈
N+(v(0))
〉
. Therefore, X and Y have to be fixed setwise. Hence,
|O(v(1))| = 1. (3.7)
Last we prove that Gv(0),v(1) = id. Subtournaments T
〈
N+(v(1))
〉
and T
〈
N−(v(1)) −
{v(0)}
〉
are transitive and thus any automorphism fixing both v(0) and v(1) fixes all other
vertices. Therefore, Gv(0),v(1) = id which implies
|Gv(0),v(1)| = 1. (3.8)
Equations (3.2), (3.6), (3.7), and (3.8) imply that |G| = 1 and G ∼= Z1. This completes the
proof.
4 Odd signature Walecki tournaments
A sequence ff . . . ff ∈ En, for f ∈ Er and n/r > 1 has an odd signature. Notice, that
n/r is odd. If there exists an element in the equivalence class [e]R of odd signature, then
all elements in [e]R have odd signature. We say that such an equivalence class has odd
signature.
Furthermore, a sequence ff . . . ff, for f ∈ Er and n/2r > 1 has an even signa-
ture. Let e be such a sequence. Not all sequences of the equivalence class [e]R have an
even signature. For example, sequences (0, 0, 1, 0) and (0, 1, 0, 1) both belong to the same
equivalence class. However, only the latter has an even signature. We say that an equiva-
lence class [e]R has even signature if there exists a sequence in [e]R with even signature.
To simplify terminology we will refer to an equivalence class with a given signature
as a “sequence” with that signature. We call a sequence periodic if it has either zero, odd,
or even signature. All other sequences are called aperiodic. We will furthermore simplify
terminology by referring to a Walecki tournament whose corresponding binary sequence
has odd signature, for example, as a Walecki tournament with odd signature.
Let e ∈ En, let n be divisible by r, andm = 2r.We introduce a partition of V (W (e))−
{t(0)} into m-sets M1,M2, . . . ,Mn/r, where Mi = {t((i−1)m+j) | 1 ≤ j ≤ m}, for
1 ≤ i ≤ n/r, and |Mi| = m. First we prove the following result that relates the structure of
Walecki tournamentW (e) and Walecki tournamentW (f),where e has either odd signature
e = ff . . . ff ∈ En or even signature e = ff . . . ff ∈ En, and f ∈ Er has either a zero
signature or is aperiodic.
Theorem 4.1. Let n ≥ 1, f ∈ Er, and let r divide n. If e = ff . . . ff ∈ En or e =
ff . . . ff ∈ En, then W (e)〈{t(0)} ∪M1〉 ∼=W (f).
16 Art Discrete Appl. Math. 2 (2019) #P1.06
Proof. Let 1 ≤ k ≤ r. Let t(i) denote a vertex of W (e) and let t(i) denote a vertex of
W (f), likewise for v(i) and v(i). We define a function ψ : {t(0)} ∪M1 → V (W (f)) by
ψ(t(0)) = t(0) and ψ(t(i)) = t(i), for 1 ≤ i ≤ 2r. Clearly, ψ is a bijection. We will
show that the Hamilton directed cycle ⇀Hk inW (f) is a union of ψ-images of directed paths
belonging to Hamilton directed cycles ⇀Hk and ⇀Hr+k in W (e).
Let ⇀Pk denote the directed path [t(0), t(k), . . . , t(2k)] on ⇀Hk and let ⇀Pr+k denote the
directed path [t(0), t(r+k), . . . , t(2k)] on ⇀Hr+k (see Figure 12). A ψ-image of a directed
path ⇀P is a directed path comprised of the ψ-images of vertices of ⇀P. The ψ-image of ⇀Pk
is ⇀P ′k . Similarly, the ψ-image of ⇀Pr+k is ⇀P ′r+k.
⇀
P ′k
↼
P ′r+k
t(0)
t(1)
t(k−1)
t(k)
t(k+1)
t(2k−1)
t(2k)
t(2k+1)
t(r+k−1)
t(r+k)
t(r+k+1)
t(2r)
Figure 12: Hamilton directed cycle from the proof of Theorem 4.1.
The definition of Walecki tournaments implies that ψ is dominance-preserving on paths
⇀Pk and ⇀Pr+k. The signature of e = ff . . . ff implies that if ek = 0, then er+k = 1 and⇀
H ′k =
⇀
P ′k ∪
↼
P ′r+k, where
↼
P ′r+k denotes the path
⇀
P ′r+k with all of its arcs reversed (see
Figure 12). If ek = 1, then er+k = 0 and
↼
H ′k =
↼
P ′r+k∪
⇀
P ′k . Therefore,W (e)〈{t(0)}∪M1〉 ∼=
W (f).
Figure 13 shows Walecki tournament W (000111000)〈{t(0) ∪ M1}〉 ∼= W (000), an
example of a Walecki tournament from Theorem 4.1.
If we consider the case when e ∈ En has period m < 2n, then Lemma 1.2 implies that
m = 2r, n/r is odd, e = ff . . . ff ∈ En, and f ∈ Er. The special form of e implies
various symmetries in the corresponding Walecki tournament.
Lemma 4.2. Let n ≥ 5 and let e ∈ En with period m = 2r < 2n. If k is an integer
such that 1 ≤ k ≤ r, then t(2ri+rfk+k) ∈ N+(t(0)) and t(2ri+rfk+k) ∈ N−(t(0)), for
0 ≤ i ≤ n/r − 1.
Proof. Let k be an integer such that 1 ≤ k ≤ r. Since e = ff . . . ff it follows that
e2ri+k = fk for 0 ≤ i ≤ (n/r − 1)/2. Therefore, if fk = 0, then t(2ri+k) ∈ N+(t(0)) and
if fk = 1, then t(2ri+k) ∈ N−(t(0)) for 0 ≤ i ≤ (n/r − 1)/2.
On the other hand, e2ri+r+k = fk for 0 ≤ i ≤ (n/r − 1)/2− 1. Now, if fk = 0, then
t(2ri+r+k) ∈ N−(t(0)) and if fk = 1, then t(2ri+r+k) ∈ N+(t(0)). If t(0) → t(k) ∈ ⇀Hk,
then t(n+k)→ t(0) ∈ ⇀Hk, which proves the remaining cases.
J. Aleš: Automorphism groups of Walecki tournaments with zero and odd signatures 17
Arcs on Hamilton directed cycle ⇀H1
Arcs on Hamilton directed cycle ⇀H2
Arcs on Hamilton directed cycle ⇀H3
v(0) = t(0)
v(1) = t(1)
v(2) = t(2)
v(4) = t(3)
v(6) = t(4)
v(8) = t(5)
v(10) = t(6)
Figure 13: Walecki tournament W (000) as an induced subtournament of W (000111000).
Lemma 4.3. Let n ≥ 5 and let e ∈ En with period m = 2r < 2n. If k is an integer such
that 1 ≤ k ≤ r, then t(fk+2(2ri+k)−1) ∈ N+(t(1)) and t(fk+2(2ri+k)−1) ∈ N−(t(1)), for
0 ≤ i ≤ (n/r − 1)/2.
Moreover, t(fk+2(2ri+r+k)−1) ∈ N+(t(1)) and t(fk+2(2ri+r+k)−1) ∈ N−(t(1)), for
0 ≤ i ≤ (n/r − 1)/2− 1.
Proof. Let e ∈ En have period m < 2n. Lemma 1.2 implies that m = 2r, n/r is odd,
e = ff . . . ff ∈ En, and f ∈ Er. The special form of e tells us that e2ri+k = fk
for 0 ≤ i ≤ (n/r − 1)/2. The structure of the Hamilton directed cycle ⇀Hk implies that if
fk = 0, then t(2(2ri+k)−1) ∈N−(t(1)) and t(2(2ri+k)) ∈ N+(t(1)). Furthermore, if fk = 1,
then t(2(2ri+k)−1) ∈N+(t(1)) and t(2(2ri+k)) ∈ N−(t(1)). Therefore, t(2(2ri+k)+fk−1) ∈
N+(t(1)) and t(2(2ri+k)+fk−1) ∈ N−(t(1)).
The special form of e also implies e2ri+r+k = fk for 0 ≤ i ≤ (n/r − 1)/2 − 1, then
the remaining cases can be proven similarly as above.
Now, let e ∈ En have odd signature. Theorem 4.1 implies the existence of n/r Walecki
subtournaments T 〈{t(0)} ∪Mi〉, for 1 < i < n/r, of tournament T = W (e). We deter-
mine subtournaments ofW (e) which are isomorphic to some Walecki tournament with odd
signature. This is a generalization of Theorem 4.1 for odd signature Walecki tournaments.
The vertices that induce the subtournament are chosen on the circumference in clockwise
order starting at the vertex t(1).
Theorem 4.4. Let n ≥ 5 and let e = ff . . . ff ∈ En with period m = 2r < 2n, and
f ∈ Er. If ` is an odd integer such that 1 ≤ ` ≤ n/r−2 then, W (e)
〈
{t(0)}∪{M1∪M2∪
· · · ∪M`}
〉 ∼=W (e′), where e′ = ff . . . ff ∈ E`r.
Proof. If ` = 1, the conclusion is just Theorem 4.1. Let ` be an odd integer such that
3 ≤ ` ≤ n/r − 2 and let e′ = ff . . . ff ∈ E`r. Let t(i) denote a vertex of W (e) and t(i)
denote a vertex of W (e′), likewise for v(i) and v(i). We define a function ψ : {t(0)}∪M1 ∪
M2 ∪ · · · ∪M` → V (W (e′)) by ψ(t(0)) = t(0) and ψ(t(i)) = t(i), for 0 ≤ i ≤ 2`r − 1.
18 Art Discrete Appl. Math. 2 (2019) #P1.06
Clearly, ψ is a bijection. We will show that the Hamilton directed cycle ⇀Hk in W (e′) is a
union of ψ-images of directed paths belonging to Hamilton directed cycles⇀Hk and⇀H̀ r+k in
W (e) when `r + k ≤ n, otherwise,⇀H ′k is a union of ψ-images of directed paths belonging
to Hamilton cycles ⇀Hk and ⇀Hn−`r+k.
Case 1. Let us first consider the case when `r + k ≤ n. The ψ-image of ⇀Pk is the path⇀
P ′k as in the proof of the Theorem 4.1, ψ(
⇀Pk) = ⇀P ′k . The ψ-image of ⇀P̀ r+k is ⇀P ′`r+k.
The definition of Walecki tournaments implies that ψ is dominance-preserving on paths ⇀Pk
and ⇀P̀ r+k. By assumption, the difference between `r + k and k ia an odd multiple of r.
Furthermore, odd signature of e implies that if ek = 0 then e`r+k = 1 and
⇀
H ′k =
⇀
P ′k ∪
↼
P ′`r+k
(see Figure 14). If ek = 1 then e`r+k = 0 and
⇀
H ′k =
↼
P ′`r+k ∪
↼
P ′k .
⇀
P ′`r+k
⇀
P ′k
t((0))
t((1))
t((2))
t((k−1))
t((k))
t((k+1))
t((k+2))
t((2k−1))
t((2k))t((2k+1))
t((2k+2))
t((`r+k−1))
t((`r+k))
t((`r+k+1))
t((`r+k+2))
t((2`r))
t((2`r−1))
Figure 14: Hamilton directed cycle constructed from directed paths from Case 1 of the
proof of Theorem 4.4.
Case 2. Let `r + k > n. Note that n − `r < k ≤ `r. Define ⇀Pk as in the previous
case and let ⇀P̀ r+k−n denote the directed path [t(2k), t(2`r), t(2k+1), t(2`r−1), t(2k+1), . . . ,
t(`r+k+2), t(`r+k−1), t(`r+k+1), t(`r+k), t(0)] on ⇀H̀ r+k−n. The ψ-image of ⇀P̀ r+k−n is⇀
P ′`r+k−n. By the definition of Walecki tournaments ψ is dominance-preserving on the path⇀P̀ r+k. Now, n/r is odd and, by assumption, ` is also odd which implies that the difference
`r − n = (`− n/r)r is an even multiple of r. Furthermore, the odd signature of e implies
that if ek = 0 then e`r+k−n = 0 and
⇀
H ′k =
⇀
P ′k ∪
⇀
P ′`r+k−n (see Figure 15). If ek = 1 then
e`r+k−n = 1 and
⇀
H ′k =
↼
P ′`r+k−n ∪
↼
P ′k . This completes the proof.
The importance of τm for the automorphism groups of Walecki tournaments with odd
signature was previously unknown. The subtournaments T
〈
{t(0)}∪Mi
〉
, for 1 ≤ i ≤ n/r,
are isomorphic to W (f). This suggests that the permutation τm, which is a product of
m = 2r disjoint cycles of length n/r, might be an automorphism of W (e).
Proposition 4.5. Let n ≥ 5. If e ∈ En has period m < 2n, then τm ∈ Aut(W (e)).
Proof. We have m = 2r, n/r is odd, e = ff . . . ff ∈ En, f ∈ Er, and e and f have the
same period 2r, and Rm(e) = R2r(e) = e. The statement follows by Theorem 1.3.
J. Aleš: Automorphism groups of Walecki tournaments with zero and odd signatures 19
⇀
P ′k
⇀
P ′n−lr+k
t((0))
t((1))
t((2))
t((k−1))
t((k))
t((k+1))
t((k+2))
t((2k−1))
t((2k))t((2k+1))
t((2k+2))
t((`r+k−1))
t((`r+k))
t((`r+k+1))
t((`r+k+2))
t((2`r))
t((2`r−1))
Figure 15: Hamilton directed cycle constructed from directed paths from Case 2 of the
proof of Theorem 4.4.
4.1 Subtournaments of odd signature Walecki tournaments
A second partition of V (W (e))− {t(0)} is now introduced. The partition is defined by the
orbits O1, O2, . . . , Om, for the permutation τm acting on V (W (e)) − {t(0)}. The orbits
have length n/r and can be expressed as follows:
O` = {t(im+`) | 0 ≤ i ≤ n/r − 1}, (4.1)
for 1 ≤ ` ≤ m and m = 2r.
Let 1 ≤ k ≤ m. One can easily prove that if fk = 0, then Ok ⊆ N+(t(0)) and
Or+k ⊆ N−(t(0)). Moreover, if fk = 1, then Ok ⊆ N−(t(0)) and Or+k ⊆ N+(t(0)).
In order to simplify the notation we introduce the permutation ζ ∈ S2r acting on the set
{1, 2, . . . , 2r} defined by ζ = (1 r + 1)(2 r + 2) · · · (r 2r). That is, ζ(i) = i + r for
1 ≤ i ≤ r, and ζ(i) = i− r for r + 1 ≤ i ≤ 2r. Note that for 1 ≤ k ≤ r,
Oζfk (k) = {t(im+ζfk (k))) | 0 ≤ i ≤ n/r − 1}. (4.2)
It follows from the observations above that Oζfk (k) ⊆ N+(t(0)) and Oζfk (k) ⊆ N
−(t(0)).
Therefore,
N+(t(0)) =
r⋃
k=1
Oζfk (k) (4.3)
and
N−(t(0)) =
r⋃
k=1
O
ζfk (k)
. (4.4)
Orbits and m-sets are orthogonal in the sense that each orbit contains exactly one vertex of
each m-set, and vice-versa.
The arc structure of subtournaments induced by the orbits O`, for the permutation τm,
is determined by the value of e`, for 1 ≤ ` ≤ m.
Theorem 4.6. Let T denote the Walecki tournament W (e) for e ∈ En and n ≥ 5. If e
has period m < 2n, then the orbits O1, O2, . . . , Om for the permutation τm acting on
20 Art Discrete Appl. Math. 2 (2019) #P1.06
V (W (e))−{t(0)} induce regular tournaments T 〈O1〉, T 〈O2〉, . . . , T 〈Om〉. If ` is an inte-
ger such that 1 ≤ ` ≤ m, the subtournaments T 〈O` ∩N+(t(1))〉 and T 〈O` ∩N−(t(1))〉
are transitive and the directions of all their arcs are determined by e`.
Proof. Under the conditions for the sequence e ∈ En, Proposition 4.5 implies that O1, O2,
. . . , Om are orbits for the permutation τm ∈ Aut(T ) which proves that the subtournaments
T 〈O1〉, T 〈O2〉, . . . , T 〈Om〉 are regular. To prove the rest of the theorem we first consider
the subtournaments T 〈O1∩N+(t(1))〉 and T 〈O1∩N−(t(1))〉. We make use of Lemma 4.2
and Lemma 4.3.
Let m = 2r. Lemma 1.1 implies that e = ff . . . ff where f ∈ Er. Vertices of an
arbitrary orbit were determined in (4.1) and (4.2).
We may assume f1 = 0 for if not we may work with W (e) instead. We first consider
the orbit containing vertex t(1). Since f1 = 0 we have Oζf1 (1) = O1 ⊆ N+(t(0)) and
O1 = {t(2ri+1) | 0 ≤ i ≤ n/r− 1}. Let Y + = O1 ∩N+(t(1)) and Y − = O1 ∩N−(t(1)).
Since er+1 = f1 = 1, we have t(2r+1) ∈ Y +. Similarly, e2r+1 = f1 = 0 implies t(4r+1) ∈
Y −. Furthermore, τm ∈ Aut(T ) implies Y + = {t(4ri+2r+1) | 0 ≤ i ≤ (n/r − 3)/2} and
Y − = {t(4ri+1) | 1 ≤ i ≤ (n/r − 1)/2}.
Let us consider Y +. We will prove that considering the vertices of Y + in the or-
der t(2r+1), t(6r+1), . . . , t(2n−4r+1), all arcs point from right to left in the subtournament
T 〈Y +〉. Let 0 ≤ i ≤ (n/r − 3)/2. The equalities
τ4rj+2r(1) = τ2r(j+i+1)(τ2r(j−i)(1)) = τ2r(j+i+1)(4r(j − i))
and
τ4ri+2r(1) = τ2r(j+i+1)(τ2r(i−j)(1)) = τ2r(j+i+1)(4r(j − i) + 1)
imply that the arc t(4rj+2r+1) → t(4ri+2r+1) belongs to the Hamilton directed cycle
⇀H2r(j+i+1)+1. This proves the subtournament T 〈Y +〉 transitive. Moreover, τm(Y +) =
Y − implies that the subtournament T 〈Y −〉 is also transitive.
Next we consider orbits in N+(t(0)) that do not contain vertex t(1). Equation (4.3)
implies that Oζfk (k) ⊆ N+(t(0)), for 2 ≤ k ≤ r. Let us first assume that fk = 0. Using
Equation (4.2) we have Oζfk (k) = Ok = {t(2ri+k) | 0 ≤ i ≤ n/r − 1}. We further divide
the proof depending on the parity of k.
Case 1. Let k be odd. Clearly, k + 1 is even so e(k+1)/2 = f(k+1)/2 and er+(k+1)/2 =
f (k+1)/2 determine the membership of t(k) and t(2r+k), respectively, in N
+(t(1)) and
N−(t(1)).
Case 1.1. If f(k+1)/2 = 0, we have t(k) ∈ N−(t(1)), and f (k+1)/2 = 1 implies t(2r+k) ∈
N+(t(1)). It follows that Y
′
= {t(4ri+k) | 0 ≤ i ≤ (n/r − 1)/2} ⊆ N−(t(1)) and
Y
′′
= {t(4ri+2r+k) | 0 ≤ i ≤ (n/r − 3)/2} ⊆ N+(t(1)). Similarly as above we can prove
that the subtournaments T 〈Y ′〉 and T 〈Y ′′〉 are transitive.
Case 1.2. If f(k+1)/2 = 1, then t(k) ∈ N+(t(1)) and f (k+1)/2 = 0 which implies
that t(2r+k) ∈ N−(t(1)). Since the sequence e has the form ff . . . ff it follows that
Y
′ ⊆ N+(t(1)) and Y ′′ ⊆ N−(t(1)). As in the previous case we can show that the sub-
tournaments T 〈Y ′〉 and T 〈Y ′′〉 are transitive. A change in the value of f(k+1)/2 results in
the reversal of arcs associated with t(1). However, the arcs between vertices of Y
′
depend
on fk only. The same reasoning applies to Y
′′
.
J. Aleš: Automorphism groups of Walecki tournaments with zero and odd signatures 21
Case 2. Let k be even. This implies that ek/2 = fk/2 and er+k/2 = fk/2 determine the
membership of t(k) and t(2r+k), respectively, in N+(t(1)) and N−(t(1)).
Case 2.1. If fk/2 = 0, then t(k) ∈ N+(t(1)), and fk/2 = 1 implies t(2r+k) ∈ N−(t(1)).
Similarly as in the previous case we have Y
′ ⊆ N+(t(1)), Y ′′ ⊆ N−(t(1)), and the sub-
tournaments T 〈Y ′〉 and T 〈Y ′′〉 are transitive.
Case 2.2. Using similar arguments as above, we can prove that if fk/2 = 1, then the
sets Y
′
and Y
′′
determine transitive subtournaments T 〈Y ′〉 and T 〈Y ′′〉, respectively. This
completes the proof for fk = 0.
Assume now fk = 1. This impliesOζfk (k) = Or+k = {t(2ri+r+k) | 0 ≤ i ≤ n/r−1}.
In a way similar to the previous case we define Y
′
= {t(4ri+r+k) | 0 ≤ i ≤ (n/r − 1)/2}
and Y
′′
= {t(4ri+3r+k) | 0 ≤ i ≤ (n/r − 3)/2}, and prove that T 〈Y ′〉 and T 〈Y ′′〉 are
transitive subtournaments. The proof is similar to the case fk = 0, however, the direction
of all arcs considered is reversed since ek = fk = 1. This completes the proof for orbits
Oζfk (k) ⊆ N+(t(0)). The result for orbits Oζfk (k) ⊆ N
−(t(0)) follows since T ∼= T .
Next we consider the subtournaments induced by outsets and insets of vertices in a
Walecki tournament with odd signature e = ff . . . ff ∈ En, whose defining sequence is
f = (0, 0, . . . , 0) ∈ Er. We show that if the vertex is distinct from t(0) then its outset
induces a subtournament that is not regular for n odd. However, the outset of t(0) induces
a regular subtournament. Similarly, for n even the outset of t(0) induces an almost regular
subtournament but the outset of any other vertex induces a subtournament that is not almost
regular. This implies that t(0) must be fixed for any automorphism of a Walecki tournament
with an odd signature e = ff . . . ff where f = (0, 0, . . . , 0).
Theorem 4.7. Let T = W (e) for e = ff . . . ff ∈ En, n ≥ 5, n/r odd, and f =
(0, 0, . . . , 0) ∈ Er. For v ∈ V (W (e))−{t(0)}, the tournaments T
〈
N+(v)
〉
are not regular
and not almost regular subtournaments of T for n odd and n even, respectively.
Proof. Since W (e) ∼= W (e), it suffices to prove the theorem for vertices in N+(t(0)).
Furthermore, since τm ∈ Aut(T ), it is sufficient to prove the theorem for the vertices in
N+(t(0))∩M1. Let M ′ =M1 ∪M3 ∪ · · · ∪Mn/r and M ′′ =M2 ∪M4 ∪ · · · ∪Mn/r−1.
We first assume r > 1 and consider t(1) ∈ N+(t(0))∩M1.We will count the vertices in
N+(t(1))∩N+(t(2n)). First we determine the vertices inN+(t(1)). Since f = (0, 0, . . . , 0),
Theorem 4.1 implies
N+(t(1)) ∩M1 = {t(2i+2) | 0 ≤ i ≤ r − 1}. (4.5)
Using Lemma 2.5 we have
N+(t(1)) ∩M2 = {t(2r+2i+1) | 0 ≤ i ≤ r − 1}. (4.6)
Let X ′ = N+(t(1))∩M ′ and X ′′ = N+(t(1))∩M ′′. The odd signature of the sequence e
and equalities (4.5) and (4.6) imply
X ′ = {t(4rj+2i+2) | 0 ≤ i ≤ r − 1, 0 ≤ j ≤ (n/r − 1)/2} (4.7)
and
X ′′ = {t(4rj+2r+2i+1) | 0 ≤ i ≤ r − 1, 0 ≤ j ≤ (n/r − 3)/2}. (4.8)
22 Art Discrete Appl. Math. 2 (2019) #P1.06
Clearly, N+(t(1)) = X ′ ∪X ′′. Next we determine the vertices in N+(t(2n)). We have
N+(t(2n)) ∩M1 = {t(2i+1) | 1 ≤ i ≤ r − 1} (4.9)
and
N+(t(2n)) ∩M2 = {t(2r+1)} ∪ {t(2r+2i+2) | 0 ≤ i ≤ r − 1}. (4.10)
Let Y ′ denote N+(t(2n)) ∩M ′ and let Y ′′ denote N+(t(2n)) ∩M ′′. The odd signature of
the sequence e and equalities (4.9) and (4.10) imply
Y ′ = {t(4rj+2i+1) | 1 ≤ i ≤ r − 1, 0 ≤ j ≤ (n/r − 3)/2}. (4.11)
Let Y ′′ = Y
′′ ∪ Y
′′
= where
Y
′′
= {t(4rj+2r+1) | 0 ≤ j ≤ (n/r − 3)/2} (4.12)
and
Y
′′
= {t(4rj+2r+2i+2) | 0 ≤ i ≤ r − 1, 0 ≤ j ≤ (n/r − 3)/2}. (4.13)
Clearly, N+(t(2n)) = {t(0)} ∪ Y ′ ∪ Y ′′.
Comparing the indices of the vertices in equalities (4.7), (4.8), (4.11), (4.12), and (4.13)
for vertices in N+(t(1)) and N+(t(2n)), we deduce (X ′ ∪X ′′) ∩ (Y ′ ∪ Y ′′) = Y , which
implies N+(t(1)) ∩ N+(t(2n)) = Y ′′. Hence, the score of vertex t(2n) in T
〈
N+(t(1))
〉
equals |Y ′′| = (n/r − 1)/2. If r > 1, then (n/r − 1)/2 < n/2 − 1 which implies that
T
〈
N+(t(1))
〉
is not regular or almost regular when n is odd or even, respectively. The
proofs for the remaining vertices of N+(t(0)) ∩M1 are similar and we omit them.
The arc structure of T
〈
N+(t(1))
〉
is different in the case when r = 1, that is, when
e = (0, 1, 0, 1, . . . , 0, 1, 0) ∈ En. Notice that n/r odd implies that n has to be odd. In
order to verify non-regularity of T
〈
N+(t(1))
〉
, we consider N+(t(1)) ∩ N+(t(3)). The
signature of e implies
N+(t(1)) = {t(2n)} ∪ {t(4i+2), t(4i+3) | 0 ≤ i ≤ (n− 3)/2}. (4.14)
Since τ2 ∈ Aut(W (e)) we have
N+(t(3)) = {t(2)} ∪ {t(4i+4), t(4i+5) | 0 ≤ i ≤ (n− 3)/2}. (4.15)
Comparing the indices of the vertices in equalities (4.7), (4.8), (4.14), (4.15), for vertices
in N+(t(1)) and N+(t(3)) we deduce that N+(t(1))∩N+(t(3)) = {t(2)}. Hence, the score
of vertex t(3) in T
〈
N+(t(1))
〉
equals 1 implying that T
〈
N+(t(1))
〉
is not regular. This
completes the proof.
4.1.1 Regular subtournaments for n odd
Next we determine the structure of arcs in the subtournaments induced by N+(t(0)) and
N−(t(0)). There are two cases to be considered depending on the parity of n.
Let e = ff . . . ff ∈ En, n odd, be an odd signature sequence with a zero subsignature
f = (0, 0, . . . , 0) ∈ Er, and let T denote W (e). We will consider the subtournaments
T
〈
N+(t(0))
〉
and T
〈
N−(t(0))
〉
.We know that the out-neighbours and in-neighbours of t(0)
are determined by f and f. Therefore, r vertices of each m-set belong to N+(t(0)) and the
J. Aleš: Automorphism groups of Walecki tournaments with zero and odd signatures 23
other r of them to N−(t(0)). On the other hand, the construction of Walecki tournaments
implies that out of two consecutive vertices t(j) and t(j+1) exactly one is an out-neighbour
of the vertex t(i), whenever j − i is even. Therefore one would hope that the score of each
vertex in T
〈
N+(t(0))
〉
is no more than 2n/4 = n/2. This would imply the regularity of
the subtournaments T
〈
N+(t(0))
〉
and T
〈
N−(t(0))
〉
.
Theorem 4.8. Let n be odd and let T denote the Walecki tournament W (e) for e =
ff . . . ff ∈ En, n/r odd, and f = (0, 0, . . . , 0) ∈ Er. If e has period m < 2n, then
the subtournaments T
〈
N+(t(0))
〉
and T
〈
N−(t(0))
〉
are regular.
Proof. The result is clearly true for n ≤ 3. Let T be a tournament as stated in the conditions
of the theorem and n ≥ 5. In order to prove that T
〈
N+(t(0))
〉
is regular, we first determine
the score of the vertex t(1) in T
〈
N+(t(0))
〉
. Let Y denote the set N+(t(0)) ∩ N+(t(1)).
We are interested in the cardinality of Y. Since m < 2n, Lemma 1.2 implies that m = 2r
and n/r is odd. Now, n odd implies that r is also odd. We proceed by proving that
|Y ∩ (Mi ∪ Mi+1)| = r, for 1 ≤ i ≤ n/r − 1. We use Lemma 4.2 and Lemma 4.3
extensively.
Let us consider the cardinality of Y ∩ (M1 ∩M2). We first determine the vertices in
N+(t(1)) ∩M1. If i is an integer such that 1 ≤ i ≤ r − 1, then ei+1 = fi+1. Because
fi+1 = 0, we have
t(2i+1) ∈ N−(t(1)) and t(2i+2) ∈ N+(t(1)). (4.16)
Since T ∼= T , we may assume that e1 = f1 = 0. Thus, t(2) ∈ N+(t(1)). Next we consider
N+(t(0)) ∩ M1. If j is an integer such that 1 ≤ j ≤ r − 1, then ej+1 = fj+1. Since
fj+1 = 0, we have
t(j+1) ∈ N+(t(0)) and t(j+1+r) ∈ N−(t(0)). (4.17)
Being 0 ≤ i ≤ r − 1, we have 2r ≤ 2(r + i) ≤ 4r − 2. The neighbours of the vertex
t(1) in the set M2 are given by
t(2(r+i)+2) ∈ N−(t(1)) and t(2(r+i)+1) ∈ N+(t(1)). (4.18)
If k is an integer such that 0 ≤ k ≤ r − 1, then 2r ≤ 2r + k ≤ 3r − 1. The neighbours of
the vertex t(0) in the set M2 are given by
t(2r+k+1) ∈ N+(t(0)) and t(2r+k+1+r) ∈ N−(t(0)). (4.19)
We use (4.16), (4.17), (4.18), and (4.19) in the following case study. Let i be an integer
such that 1 ≤ i ≤ (r − 3)/2. Since ei+1 = fi+1 = 0, it follows that t(2i+2), t(2r+2i+1) ∈
N+(t(1)). Similarly, ei+(r+1)/2 = fi+(r+1)/2 = 0 implies t(r+2i+1) ∈ N+(t(1)), and
ei+(r+3)/2 = fi+(r+3)/2 = 0 implies t(3r+2i+2) ∈ N+(t(1)). Also e2i+1 = f2i+1 = 0 im-
plies t(2i+1), t(2r+2i+1) ∈ N+(t(0)), and e2i+2 = f2i+2 = 0 implies t(2i+2), t(2r+2i+2) ∈
N+(t(0)). Let 1 ≤ i ≤ (r − 3)/2. Since fi+1, f2i+1, f2i+2, fi+(r+1)/2, and fi+(r+3)/2
are all zero, it follows that exactly two of the vertices t(2i+1), t(2i+2), t(r+2i+1), t(r+2i+2),
t(2r+2i+1), t(2r+2i+2), t(3r+2i+1), and t(3r+2i+2) belong to Y ∩ (M1 ∪M2).
We have yet to consider vertices t(1), t(r), t(r+1), t(2r), t(2r+1), t(3r), t(3r+1), and t(4r).
Their membership in N+(t(1)) ∩ N+(t(0)) is determined by the values of f1, fr and
f(r+1)/2, which are all zero. Notice that er = fr, which implies t(r), t(3r) ∈ N+(t(0)),
24 Art Discrete Appl. Math. 2 (2019) #P1.06
and t(2r), t(4r) ∈ N+(t(1)). Similarly, e(r+1)/2 = f(r+1)/2 implies that if f(r+1)/2 = 0,
then t(r+1), t(3r) ∈ N+(t(1)). Since f1 = 0, we have er+1 = f1 = 1, which implies
t(2r+1) ∈ N+(t(1)) and t(1), t(2r+1) ∈ N+(t(0)). Clearly, t(1) /∈ N+(t(1)). It follows
that exactly two of the vertices t(r), t(r+1), t(2r), t(2r+1), t(3r), t(3r+1), and t(4r) belong to
Y ∩ (M1 ∪M2).
We have considered all vertices in M1 ∪M2 except t(2), t(r+2), t(2r+2), and t(3r+2).
Their membership in N+(t(1)) ∩ N+(t(0)) is determined by the values of f1, f2 and
f(r+3)/2, which are all zero. Now, e1 = f1 = 0 implies t(2) ∈ N+(t(1)), and er+1 = f1 =
1 implies t(2r+2) ∈ N−(t(1)). Notice that e2 = f2 = 0 implies t(2), t(2r+2) ∈ N+(t(0)).
Also, e(r+3)/2 = f(r+3)/2 = 0 implies t(3r+2) ∈ N+(t(1)). Hence, exactly one of the
vertices t(2), t(r+2), t(2r+2), and t(3r+2) belongs to Y ∩ (M1 ∪M2). It follows by above
observations that
|Y ∩ (M1 ∪M2)| = 2(r − 3)/2 + 2 + 1 = r. (4.20)
Similar to the previous case we can show that
|Y ∩ (Mi ∪Mi+1)| = 2(r − 1)/2 + 1 = r,
for 2 ≤ i ≤ n/r − 1, and
|Y ∩ (M1 ∪Mn/r)| ≤ 2(r − 3)/2 + 1 + 1 = r − 1. (4.21)
Let α denote |Y ∩M1|. Since |Y ∩ (M1 ∪M2)| = r, it follows that |Y ∩M2| = r − α.
Since n/r is odd, we have |Y ∩Mn/r−1| = r − α and |Y ∩Mn/r| = α which implies
|Y ∩ (M1 ∪Mn/r)| = 2α ≤ r − 1. Now, r is odd implies |Y ∩ (M1 ∪Mn/r)| ≤ r − 1.
Therefore, 2|Y | ≤ (r − 1) + (n− r) = n− 1 which implies
s(t(1)) = |N+(t(0)) ∩N+(t(1))| ≤ n− 1
2
<
n
2
. (4.22)
Similarly, we can prove that
s(t(i+rfi)) = |N+(t(0)) ∩N+(t(i+rft))| ≤
n− 1
2
,
for 2 ≤ i ≤ r. That is, every vertex in N+(t(0)) ∩M1 has score at most (n− 1)/2. Since
τm is an automorphism of T, the score of every vertex in the tournament T
〈
N+(t(0))
〉
is
at most (n− 1)/2. Now, (
n
2
)
=
∑
v∈N+(t(0)
s(v) ≤ n(n− 1)
2
implies s(v) = (n − 1)/2 for every vertex v ∈ N+(t(0)). Therefore, the subtourna-
ment T
〈
N+(t(0))
〉
is regular. Regularity of the tournament T
〈
N−(t(0))
〉
follows since
T ∼= T .
4.1.2 Almost regular subtournaments for n even
When n is even, T
〈
N+(t(0))
〉
can not be regular. However, one can prove that in the
case of f = (0, 0, . . . , 0) ∈ Er it is almost regular. We follow an analog of the proof of
Theorem 4.8. Oddly enough, the fact that n is even simplifies the proof.
J. Aleš: Automorphism groups of Walecki tournaments with zero and odd signatures 25
Theorem 4.9. Let n be even and let T denote the Walecki tournament W (e) for e =
ff . . . ff ∈ En, n/r odd, and f = (0, 0, . . . , 0) ∈ Er. If e has period m < 2n, then the
subtournaments T
〈
N+(t(0))
〉
and T
〈
N−(t(0))
〉
are almost regular.
Proof. The result is clearly true for n ≤ 4. Let T be a tournament as stated in the conditions
of the theorem. We first prove that T
〈
N+(t(0))
〉
is almost regular. In order to do so we
determine the score of the vertex t(1) in T
〈
N+(t(0))
〉
. Let Y denote the set N+(t(0)) ∪
N+(t(1)). We are interested in the cardinality of Y. Since m < 2n, we have m = 2r
and n/r is odd. Now, n even implies that r is also even. We proceed by considering sets
Y ∩ (Mi ∪Mi+1) for 1 ≤ i ≤ n/r − 1. Similar to the proof of Theorem 4.8 we can prove
|Y ∩ (Mi ∪ Mi+1)| = r, for 1 ≤ i ≤ n/r − 1, and |Y ∩ (Mn/r ∪ M1)| ≤ r. Let α
denote |Y ∩M1|. Since n/r is odd, |Y ∩Mn/r−1| = r − α and |Y ∩Mn/r| = α which
implies |Y ∩ (M1 ∪Mn/r)| = 2α. Hence, 2α ≤ r. Since r is even, we have α ≤ r/2 and
|Y ∩ (M1 ∪ Mn/r)| ≤ r. Therefore, 2|Y | ≤ n which implies s(t(1)) = |N+(t(0)) ∩
N+(t(1))| ≤ n/2. Similarly, we can prove that s(t(i+rfi)) = |N+(t(0))∩N+(t(i+rfi))| ≤
n/2, for 2 ≤ i ≤ r. That is, every vertex in N+(t(0)) ∩M1 has score at most n/2. Since
τm ∈ Aut(T ), the score of every vertex in the subtournament T
〈
N+(t(0))
〉
is at most n/2.
Since T ∼= T with anti-automorphism η = τn ∈ S2n+1, where
η = (v(1)v(2n))(v(2)v(2n−1)) . . . (v(n)v(n+1))
= (t(1)t(n+1))(t(2)t(n+2)) . . . (t(n)t(2n−1)),
(see Figure 16), we have T
〈
N−(t(0))
〉 ∼= T〈N+(t(0))〉. Now, η(t(1)) = t(3) implies
that |N−(t(0)) ∩ N+(t(3))| = s(t(3)) ≤ n/2 in T
〈
N−(t(0))
〉
. Therefore, |N+(t(0)) ∩
N−(t(1))| ≤ n/2 in T
〈
N+(t(0))
〉
. Since
N+(t(0)) = (N+(t(0)) ∩N+(t(1))) ∪ (N+(t(0)) ∩N−(t(1))) ∪ {t(1)},
we have s(t(1)) = |N+(t(0)) ∩ N+(t(1))| ≥ n/2 − 1 in T
〈
N+(t(0))
〉
. Using a similar
argument we can prove that the score of every vertex in T
〈
N+(t(0))
〉
is either n/2 or
n/2− 1.
If k denotes the number of vertices with score n/2 in T
〈
N+(t(0))
〉
, then the number
of vertices of degree n/2− 1 equals n− k. The equation(
n
2
)
=
∑
(v∈N+t(1))
s(v) = k
n
2
+ (n− k)n− 2
2
implies that k = n/2. In other words, the subtournament T 〈N+(t(0))〉 is almost regular.
Furthermore, T 〈N−(t(0))〉 is also almost regular since T ∼= T .
4.2 Automorphism groups of odd signature Walecki tournaments with a zero sub-
signature
Theorem 4.10. Let n be odd, n ≥ 5, and let T denote the Walecki tournament W (e) for
e = ff . . . ff ∈ En, n/r odd, and f = (0, 0, . . . , 0) ∈ Er, then Aut(W (e)) = Zn/r.
Proof. Let us assume that n is odd, n ≥ 5, e = ff . . . ff ∈ En, n/r odd, and f =
(0, 0, . . . , 0) ∈ Er. Let T denote the Walecki tournament W (e) and let G denote its auto-
morphism group Aut(T ). We use Orbit Stabilizer Theorem two times to get
|G| = |O(t(0))| |Gt(0)| = |O(t(0))| |O(t(1))| |Gt(0),t(1)|, (4.23)
26 Art Discrete Appl. Math. 2 (2019) #P1.06
v(0) = t(0)
v(1) = t(1)
v(2) = t(2)v(3) = t(2n)
v(n−1)= t((n+1)/2)
v(n+1)= t((n+3)/2)
v(n)= t(2n+(n−3)/2)
v(n+2)= t(2n+(n−1)/2)
v(2n) = t(n+1)
v(2n−1) = t(n+2) v(2n−2) = t(n)
Figure 16: The diagram shows the action of the permutation η = τn ∈ S2n+1 on vertices
of the Walecki tournament W (e), for e ∈ En, n odd, and n ≥ 1. Vertex v(0) is fixed by the
permutation η which is an involution represented by two-way arrows.
where O(t(1)) denotes the orbit of vertex t(1) for the subgroup Gt(0) of G.
We first consider the cardinality of O(t(0)). T
〈
N+(t(0))
〉
is a regular tournament by
Theorem 4.8. On the other hand T
〈
N+(t(i))
〉
is not regular for t(i) ∈ N+(t(0)) (see
Theorem 4.7). Thus, t(0) cannot be mapped to a vertex from N+(t(0)) by elements of G.
Since T ∼= T with the graph anti-automorphism τn, t(0) cannot be mapped to a vertex from
N−(t(0)) by elements of G. We have proven that t(0) must be fixed under the action of G,
and thus
|O(t(0))| = 1. (4.24)
Next we determine |O(t(1))|. Since t(0) is a fixed point for any element ρ in G,
ρ(N+(t(0))) = N+(t(0)). Hence, ρ(t(1)) ∈ N+(t(0)) and O(t(1)) ⊆ N+(t(0)), imply-
ing
|O(t(1))| ≤ |N+(t(0))| = n. (4.25)
We proved that the permutation τ2r ∈ S2n+1 is an element of G. Since τ2r(t(0)) = t(0),
τ2r ∈ Gt(0). Hence,
〈
τ2r
〉
⊆ Gt(0). The orbit of t(1) for
〈
τ2r
〉
is O1, which implies
|O(t(1))| ≥ |O1| = n/r. (4.26)
If r = 1 then Equations (4.25) and (4.26) imply |O(t(1))| = n. We now assume r > 1. We
will prove that |O(t(1))| = n/r. Suppose the contrary, |O(t(1))| > n/r. Hence, there exists
a vertex t(i) ∈ O(t(1)) − O1. Let us assume that t(i) ∈ Oi where 1 < i ≤ 2r. Since τ2r
is an automorphism, we may assume t(i) ∈ Oi ∩M1. Moreover, f = (0, 0, . . . , 0) implies
t(1), t(i) ∈ N+(t(0)), therefore, 1 < i ≤ r.
It follows that there exist k > 1 such that τk(i−1)t(i) = t(i+k(i−1)+1) where r <
i + k(i − 1) + 1 ≤ 2r. This is a contradiction, for t(i+k(i−1)+1) belongs to N−(t(0))
because ei+k(i−1)+1 = f i+k(i−1)+1−r = 0 = 1, however, it should belong to N
+(t(0))
because t(i) ∈ N+(t(0)). Therefore,
|O(t(1))| = n/r. (4.27)
J. Aleš: Automorphism groups of Walecki tournaments with zero and odd signatures 27
Last we prove that Gv(0),v(1) = id. The subtournaments T
〈
N+(t(0))
〉
are regular for
n odd and almost regular for n even. However, the subtournaments T
〈
N+(t(1))
〉
are not
regular and not almost regular for n odd and n even, respectively, implying that any auto-
morphism ρ ∈ Gt(0),t(1) fixes all other vertices. Therefore, Gv(0),v(1) = id, that is,
|Gt(0),t(1)| = 1. (4.28)
Equations (4.23), (4.24), (4.27), and (4.28) imply that |G| = n/r. Now,
〈
τ2r
〉
⊆ Gt(0) ⊆
G and since
〈
τ2r
〉 ∼= Zn/r we have
G ∼= Zn/r
as required.
References
[1] J. Aleš, Automorphism Groups of Walecki Tournaments, Ph.D. thesis, Simon Fraser Uni-
versity, Burnaby, Canada, ProQuest Dissertations Publishing, 1999, https://search.
proquest.com/docview/304572857.
[2] B. Alspach, Research problems: Problem 99, Discrete Math. 78 (1989), 327, doi:10.1016/
0012-365x(89)90189-1.
[3] B. Alspach, The wonderful Walecki construction, Bull. Inst. Combin. Appl. 52 (2008), 7–20.
[4] B. R. Alspach, A Class of Tournaments, Ph.D. thesis, University of California, Santa
Barbara, ProQuest Dissertations Publishing, 1966, https://search.proquest.com/
docview/302177812.
[5] L. W. Beineke and K. B. Reid, Tournaments, in: L. W. Beineke and R. J. Wilson (eds.), Selected
Topics in Graph Theory, Academic Press, New York, pp. 169–204, 1978.
[6] E. Brugnoli, Enumerating the Walecki-type Hamiltonian cycle systems, J. Combin. Des. 25
(2017), 481–493, doi:10.1002/jcd.21558.
[7] W. Burnside, Theory of Groups of Finite Order, Cambridge University Press, Cambridge, 2nd
edition, 1911.
[8] D. Kühn and D. Osthus, Hamilton decompositions of regular expanders: a proof of Kelly’s
conjecture for large tournaments, Adv. Math. 237 (2013), 62–146, doi:10.1016/j.aim.2013.01.
005.
[9] D. Kühn, D. Osthus and A. Treglown, Hamilton decompositions of regular tournaments, Proc.
Lond. Math. Soc. 101 (2010), 303–335, doi:10.1112/plms/pdp062.
[10] B. D. McKay, Practical graph isomorphism, in: D. S. Meek and H. C. Williams (eds.), Proceed-
ings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, Volume I,
Utilitas Mathematica Publishing, Winnipeg, Manitoba, 1981 pp. 45–87, held at the University
of Manitoba, Winnipeg, Manitoba, October 1 – 4, 1980.
[11] B. D. McKay, Nauty (No AUTomorphisms, Yes?), Version 1.9+, 1995, http://users.
cecs.anu.edu.au/˜bdm/nauty/.
[12] J. W. Moon, Topics on Tournaments, Holt, Rinehart and Winston, New York, 1968.
[13] H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P1.07
https://doi.org/10.26493/2590-9770.1251.b76
(Also available at http://adam-journal.eu)
Parallelism of stable traces
Jernej Rus ∗
Abelium d.o.o., Kajuhova 90, 1000 Ljubljana, Slovenia
Received 18 May 2018, accepted 19 September 2018, published online 8 August 2019
Abstract
A parallel d-stable trace is a closed walk which traverses every edge of a graph exactly
twice in the same direction and for every vertex v, there is no subset X ⊆ N(v) with
1 ≤ |N | ≤ d such that every time the walk enters v from X , it also exits to a vertex in
X . In the past, d-stable traces were investigated as a mathematical model for an innovative
biotechnological procedure – self-assembling of polypeptide structures. Among other, it
was proven that graphs that admit parallel d-stable traces are precisely Eulerian graphs
with minimum degree strictly larger than d. In the present paper we give an alternative,
purely combinatorial proof of this result.
Keywords: Eulerian graph, parallel d-stable trace, nanostructure design, self-assembling, polypep-
tide.
Math. Subj. Class.: 05C45, 05C85, 94C15
1 Introduction
All graphs considered in this paper will be connected, finite, and simple, that is, without
loops and multiple edges. If v is a vertex of a graph G, then its degree will be denoted by
dG(v) or d(v) for short ifG will be clear from the context. The minimum and the maximum
degree of G are denoted with δ(G) and ∆(G), respectively. A directed graph is a graph
where edges have a direction associated with them. In formal terms a directed graph is a
pair G = (V,A), where V is a set of vertices and A is a set of ordered pairs of vertices,
called arcs. A maximal connected subgraph of G is called a component of G, while a
vertex which separates two other vertices of the same component is a cutvertex, and an
edge separating its ends is a bridge. A maximal connected subgraph without a cutvertex is
∗The author is grateful to Sandi Klavžar and anonymous reviewers for several significant remarks and sug-
gestions which were of great help. The authors acknowledge the financial support from the Slovenian Research
Agency H2020 SME2 and the SPIRIT Slovenia - Public Agency for Entrepreneurship, Internationalization, Fore-
ign Investments and Technology - KKIPP.
E-mail address: jernej.rus@gmail.com (Jernej Rus)
cb This work is licensed under http://creativecommons.org/licenses/by/3.0/
2 Art Discrete Appl. Math. 2 (2019) #P1.07
called a block. Thus, every block of a graph G is either a maximal 2-connected subgraph,
or a bridge (with its ends), or an isolated vertex. A subtree T of a graph G is a subgraph of
G that is also a tree (any pair of vertices u, v ∈ V (T ) ⊆ V (G) are connected by exactly
one path in T ). For other general terms and concepts from graph theory not recalled here
we refer to [12].
A circuit is a closed walk allowing repetitions of vertices and edges. An Eulerian circuit
in G is a circuit which traverses every edge of G exactly once. G is called Eulerian if it
admits an Eulerian circuit. A double trace in a graph G is a circuit that traverses every
edge exactly twice. For a set of vertices X ⊆ N(v), we say that a double trace W has
an X-repetition at vertex v (nontrivial X-repetition in [3]), if X is nonempty, X 6= N(v),
and whenever W comes to v from a vertex in X it also continues to a vertex in X . An
X-repetition (at v) is a d-repetition if |X| = d (repetition of order d), see Figure 1. Clearly
if W has an X-repetition at v, then it also has an (N(v) \X)-repetition at v (symmetry of
repetitions). We call a double trace without any repetition of order ≤ d a d-stable trace.
Note, that for every d′ ≤ d, a d-stable trace is also a d′-stable trace.
u
e
(a)
v
(b)
v
(c)
w
(d)
Figure 1: Possible 1-, 2- and 3-repetitions at vertices w, u and w, respectively.
In order to present a mathematical model for the biotechnological procedure from [6]
graphs that admit d-stable traces were characterized in [3] (thus generalizing results of
Sabidussi [11] and Eggleton and Skilton [2] about 1-stable traces and Klavžar and Rus [8]
about 2-stable traces) as follows:
Proposition 1.1 ([3, Proposition 3.4]). A connected graph G admits a d-stable trace if and
only if δ(G) > d.
Let now W be a double trace of a graph G. Then every edge e = uv of G is traversed
exactly twice. If in both cases e is traversed in the same direction (either both times from
u to v or both times from v to u) we say that e is a parallel edge (with respect to W ). If
this is not the case we say that e is an antiparallel edge. The condition that all the edges
of G are of the same type is called a parallelism. A double trace W is a parallel double
trace if every edge of G is parallel and an antiparallel double trace if every edge of G is
antiparallel.
By replacing every edge of a graph with two new edges we can quickly prove that
every graph (resp. every Eulerian graph) admits an antiparallel (resp. parallel) double trace,
observation made by several authors, Klavžar and Rus in [8] among others. While graphs
admitting antiparallel d-stable traces were thoroughly studied in [10], the characterization
of parallel d-stable traces was only mentioned as a consequence in [3]:
Theorem 1.2 ([3, Theorem 5.4]). A graph G admits a parallel d-stable trace if and only if
G is Eulerian and δ(G) > d.
J. Rus: Parallelism of stable traces 3
In the present paper we will give an alternative proof of this result. Instead of graph
embeddings (heavily used in [3]), our approach to the problem will be purely combinato-
rial. Characterizing graphs that admit parallel d-stable traces also represents a new problem
related with forbidden transitions in Eulerian tours of Eulerian graphs (further related prob-
lems can be found, among others in [4, 5]).
We can note right away that parallel double traces do not contain 1-repetitions. Note
also that none of the operations that we will use on double traces (concatenations, contrac-
tions, deletions, inductive constructions, and reordering) will change the orientation of the
edges.
1.1 Biotechological background
In 2013 Gradišar et al. [6] presented a novel self-assembly strategy for polypeptide nanos-
tructure design. Their strategy relied on routing a single polypeptide chain consisting of 12
segments through 6 edges of the tetrahedron in such a way that every edge was traversed
exactly twice. The required mathematical support for the particular case of the tetrahedron
and the general case of a polyhedron was already given in [3, 6, 8], where the authors ex-
plained that polyhedron P that is composed from a single polymer chain can be naturally
represented by a graph G(P ) of the polyhedron. Circuits that traverse every edge of G(P )
precisely twice, called double traces of G(P ), play a key role in modeling the construction
process.
The stability of the constructed polyhedra depends on an additional property whether
in the double trace the neighborhoods of vertices can be split. The reader interested in the
biotehological procedures that motivated our research may also consult the references [7,
9], where the authors also exposed the use of parallel d-stable traces.
2 Graphs admitting parallel 2-stable traces
The first mathematical model for the biotechnological procedure from [6], introduced in [8],
stated that a polyhedral graph P can be realized by interlocking pairs of polypeptide chains
if its corresponding graph G(P ) contains a 2-stable trace. Two important deficiencies of
this model were later found in [3]:
(i) it does not account for vertices of degree ≤ 2, and
(ii) it does not successfully model vertices of degree ≥ 6 (because a polyhedron could
split into two parts in a vertex of degree≥ 6, as can be seen at Figure 1 and therefore
the structure would not be stable).
Since until now, a construction of a polyhedron whose graph would have such properties,
has not yet been attempted, we first study parallel 2-stable traces in this section.
To make the arguments in this section more transparent, we explain how the reader can
graphically imagine 1-repetitions and 2-repetitions in double traces. We say that a double
trace contains a 1-repetition if it has an immediate succession of an edge e by its antiparallel
copy. If v is a vertex of a graph G with a double trace W and u and w are two different
neighbors of v, then we can say that W contains a 2-repetition (through) v if the vertex
sequence u→ v → w appears twice in W in any direction (u→ v → w or w → v → u),
see Figure 1.
We will need the next lemma in the proof of Theorem 2.3.
4 Art Discrete Appl. Math. 2 (2019) #P1.07
Lemma 2.1. Let G be a graph and let T a subtree of G such that every vertex v ∈ V (G) \
V (T ) has at most one neighbor in T . Construct a graph G′ from G by contracting T into
a single vertex t. If G admits a 2-stable trace W then G′ admits a 2-stable trace W ′ that
traverses edges from E(G) ∩ E(G′) in the same direction as W .
Proof. Suppose that the graph G admits a 2-stable trace W . Construct a double trace
W ′ from W as follows. Start in an arbitrary vertex of V (G) \ V (T ) and follow W . Let
a = xy be an arc of W that we are currently traversing on our walk along W . If x, y ∈
V (G) \ V (T ), then we put xy into W ′ so that the order of arcs from W is preserved. If
x ∈ V (T ) and y /∈ V (T ) then we put ty in W ′ instead of a. Similarly, we replace arcs
where x /∈ V (T ) and y ∈ V (T ) with xt. Finally, the occurrences of the arcs from T are
ignored in W ′.
We claim that W ′ is a 2-stable trace of G′. Since every edge is traversed twice in W ,
every edge is traversed twice in W ′. Hence W ′ is a double trace. If W ′ is not a 2-stable
trace, there exists a vertex x ∈ V (G′) such that W ′ has a 1-repetition or a 2-repetition at
x. Denote the neighborhood of vertex t in G′ with N(t). We have to consider three cases.
Case 1: x /∈ N(t).
It is clear from the construction that if W ′ had a 1-repetition or a 2-repetition at x, then
W would have a 1-repetition or a 2-repetition at x, a contradiction.
Case 2: x ∈ N(t).
It is again clear from the construction that ifW ′ had a 1-repetition yxy or a 2-repetition
yxz, where y, z 6= t, then W would have a 1-repetition or a 2-repetition at x, a contradic-
tion.
Assume first that W ′ has a 1-repetition txt. It follows that W should contain hxg,
where h, g ∈ T . Since every vertex in V (G) \V (T ) has at most one neighbor in T , h = g.
Therefore W should contain a 1-repetition hxh, a contradiction.
Assume next that W ′ has a 2-repetition txy for some neighbor y of x. It follows that
W should contain hxy and gxy, where h, g ∈ T . Since every vertex in V (G) \ V (T )
has at most one neighbor in T , h = g. Therefore W should contain a 2-repetition hxy, a
contradiction.
Case 3: x = t.
Assume first that W ′ has a 1-repetition yty for some neighbor y of t. It follows that W
should contain yhAhy, where h is the unique neighbor of y in T and A is a circuit in T .
Since T is a tree, the only possibility that circuit appears in a part of a double trace W that
is completely included in T is with a 1-repetition, a contradiction.
Assume next that W ′ has a 2-repetition ytz for some neighbors y and z of t. It follows
that W should contain yhBgz and yhCgz, where h is a unique neighbor of y in T , g is
a unique neighbor of z in T , while B and C are hg-paths in T . Considering the fact that
in a tree any two vertices are connected with a unique path, we can argue that B = C and
therefore that then W should have a 2-repetition (1-repetition if h = g), a contradiction.
We have thus proved that W ′ is a 2-stable trace in G′. During the construction of W ′
we did not change the direction of any arc from W .
Note that Lemma 2.1 is, by repetition of the procedure described above, also true for
forests (any number of disjoint subtrees).
J. Rus: Parallelism of stable traces 5
The following was proven in [8], where it was also observed that a graph G admits a
parallel double trace if and only if G is Eulerian.
Proposition 2.2 ([8, Proposition 5.4]). A connected graph G admits a parallel 1-stable
trace if and only if G is Eulerian.
Proof. Parallelism of any stable trace of a graph G implies that all the vertices of G are of
even degree and traversing an arbitrary Eulerian circuit of G twice in the same direction
constructs a parallel 1-stable trace.
We next prove Theorem 2.3 about parallel 2-stable traces and then use it in Section 3 to
present an alternative proof of Theorem 1.2.
Theorem 2.3. A graph G admits a parallel 2-stable trace if and only if G is Eulerian and
δ(G) > 2.
Note that for Eulerian graphs the constraint on the minimal degree of a graph from
Theorem 2.3 is equivalent to δ(G) ≥ 4.
Proof. Suppose that a graph G admits a parallel 2-stable trace. By definition, every 2-
stable trace is a 1-stable trace. Thus by Proposition 2.2, G is Eulerian and hence by Propo-
sition 1.1 we infer that δ(G) ≥ 4.
For the converse assume that G fulfills the conditions of the theorem. We proceed by
induction on ∆ = ∆(G).
Let ∆ = 4. Then δ(G) = ∆(G) = 4. By Proposition 2.2, G admits a parallel 1-stable
trace W ′. If W ′ is not already a 2-stable trace, W ′ contains at least one 2-repetition. We
proceed with the second induction on the number k of vertices where W ′ has 2-repetitions.
Let k ≥ 1 and let v be one of the vertices where W ′ has a 2-repetition. If a 1-stable trace
W ′ has a 2-repetition through v, where v is a vertex with dG(v) = 4, then it is not difficult
to see that W ′ has two 2-repetitions through v. Let v1, v2, v3, and v4 be the neighbors
of v. Without loss of generality, we can assume that A = v1 → v → v2 is the first
and B = v3 → v → v4 is the second 2-repetition through v in W ′. That means that
sequences A and B appear twice in W ′. Because W ′ is a parallel 1-stable trace, there are
only two possibilities how occurrences of A and B are arranged in W ′. These possibilities
are AABB (Figure 2, left) and ABAB (Figure 3, left). Note that we left out all the other
vertices in Figures 2 and 3.
In the first case we construct a double trace W from W ′ in G as follows. We start
in an arbitrary vertex of V (G) \ {v} and follow W ′. Let a = xy be an arc of W ′ that
we are currently traversing on our walk along W ′. If x, y ∈ V (G) \ {v, v1, v2, v3, v4},
then we put xy into W so that the order of arcs from W ′ is preserved. Put one occurrence
of v1 → v → v2 and one occurrence of v3 → v → v4 in W as well. Replace the
remaining occurrence of v1 → v → v2 with v1 → v → v4 and the remaining occurrence
of v3 → v → v4 with v3 → v → v2, so that W stays connected, see Figure 2, right.
We construct W analogously in the second case, see Figure 3, right.
We claim that in both cases W is a parallel 1-stable trace of G with at least one vertex
with 2-repetition less than W ′. Note first that any edge e = xy that appears in W (arcs
xy or yx appears in W ) has its unique corresponding edge e′ in W ′. Any edge e = xy in
W , where x 6= v and y 6= v, is traversed twice in the same direction in W because it is
traversed twice in the same direction inW ′. Four remaining edges (vv1, vv2, vv3, and vv4)
6 Art Discrete Appl. Math. 2 (2019) #P1.07
vv4
v2
v3
v1
=⇒ v
v4
v2
v3
v1
Figure 2: Removing 2-repetition through v (case AABB).
vv4 v2
v3
v1
=⇒ v
v4 v2
v3
v1
Figure 3: Removing 2-repetition through v (case ABAB).
J. Rus: Parallelism of stable traces 7
are traversed twice in the same direction by construction. Hence W is a parallel double
trace. It is also clear from the construction that W is a 1-stable trace. Finally we need to
verify that W has at least one vertex with 2-repetition less than W ′. Let x be an arbitrary
vertex of G in which W has a 2-repetition. We have to consider three cases.
Case 1: x /∈ {v, v1, v2, v3, v4}.
It is clear from the construction that if W has a 2-repetition through x, then also W ′
has a 2-repetition through x.
Case 2: x ∈ {v1, v2, v3, v4}.
It is again clear from the construction that if W has a 2-repetition yxz, where y, z 6= v,
then also W ′ has a 2-repetition through x.
Similarly, if W has a 2-repetition vxy for some neighbor y of x, then also W ′ has a
2-repetition through x since the order of arcs adjacent to {v1, v2, v3, v4} did not change
in W .
Case 3: x = v.
The 1-stable traceW ′ had a 2-repetition (two 2-repetitions to be more accurate) through
v but during the construction of W we manage to remove them both.
We have thus constructed a 1-stable trace W which have at least one vertex with 2-
repetition less thanW ′. Hence, it follows by induction assumption that any 4-regular graph
admits a parallel 2-stable trace.
Assume now that ∆ ≥ 6 and that any graph H with ∆(H) < ∆ that fulfills the
conditions of Theorem 2.3 admits a parallel 2-stable trace. We have to again consider two
cases.
Case 1: ∆ ≡ 2 (mod 4).
Construct the graph G′ from G as follows. For every vertex v of degree ∆ (temporary
denote its neighbors with v1, . . . , v∆) repeat the following procedure. Remove v from G.
Add two new vertices v′ and v′′, connect them by an edge, connect v′ with v1, . . . , v∆
2
, and
connect v′′ with the remaining neighbors of v, see Figure 4.
v
v1
. . .
v∆
(a) G
v′ v′′
v1
. . .
v∆
2
v∆
2 +1
. . .
v∆
(b) G′
Figure 4: Construction from the proof of Theorem 2.3 for the case ∆ ≡ 2 (mod 4).
Note that in G′ all except the newly added vertices are of the same degree as in G,
8 Art Discrete Appl. Math. 2 (2019) #P1.07
while dG′(v′) = ∆2 + 1 and dG′(v
′′) = ∆2 + 1 (the last two statements are true for all new
vertices). It follows that ∆(G′) < ∆. Since ∆ ≥ 6, we then also infer that δ(G′) ≥ 4.
Because ∆ ≡ 2 (mod 4), the degrees dG′(v′) = dG′(v′′) = ∆2 + 1 are even, hence G
is Eulerian and by the induction assumption on ∆, the graph G′ admits a parallel 2-stable
trace. If we use a path containing vertices v′ and v′′ as subtree T , it follows from a repeated
application of Lemma 2.1 that G admits a parallel 2-stable trace.
Case 2: ∆ ≡ 0 (mod 4).
Construct the graph G′ from G as follows. For every vertex v of degree ∆ (temporary
denote its neighbors with v1, . . . , v∆) repeat the following procedure. Remove v from G,
and add three new vertices v′, v′′, and v′′′. Connect v′′ with v′ and v′′′ by an edge, connect
v′ with v1, . . . , v∆
2 −1
, connect v′′ with v∆
2
and v∆
2 +1
, and connect v′′′ with the remaining
neighbors of v, see Figure 5.
v
v1
. . .
v∆
(a) G
v′
v′′
v′′′
v1
. . .
v∆
2 −1
v∆
2 +2
v∆
2
v∆
2 +1
. . .
v∆
(b) G′
Figure 5: Construction from the proof of Theorem 2.3 for the case ∆ ≡ 0 (mod 4).
Analogously as in the first case, note that in G′ all except the newly added vertices are
of the same degree as in G, while dG′(v′) = dG′(v′′′) = ∆2 and dG′(v
′′) = 4 (the last two
statements are true for all new vertices). It follows that ∆(G′) < ∆. Since ∆ ≥ 6, we then
also infer that δ(G′) ≥ 4. Because ∆ ≡ 0 (mod 4), the degrees dG′(v′) = dG′(v′′′) = ∆2
are even, hence G is Eulerian. By the induction assumption on ∆, the graph G′ admits a
parallel 2-stable trace. Similarly as in previous case, if we use a path containing vertices
v′, v′′ and v′′ as subtree T , it follows from repeated application Lemma 2.1 that G admits
a parallel 2-stable trace.
We have thus proved Theorem 2.3.
3 Alternative proof of Theorem 1.2
We now extend the results from previous section to present an alternative proof of Theo-
rem 1.2 (Theorem 5.4 from [3]).
Proof. Assume first that the graph G admits a parallel d-stable trace. From Proposition 1.1
it follows that δ(G) > d for every graph G that admits a d-stable trace. Assume that there
J. Rus: Parallelism of stable traces 9
exists an vertex v of odd degree in G. Since every edge of a parallel double trace is used
twice in the same direction, input and output degree of a parallel double traceW would not
match at v, which is absurd. Therefore it follows that G is Eulerian and δ(G) > d.
For the converse assume that graph G is Eulerian and δ(G) > d. Since G is Eulerian,
δ(G) is an even number. Furthermore, since for parallel 1-stable traces and 2-stable traces
the theorem follows from Proposition 2.2 and Theorem 2.3, respectively, we can assume
that d ≥ 3. Let G′ be a graph obtained from G by replacing every vertex v of degree
dG(v) > 4 with (dG(v) − 2)/2 new vertices, connected into a path Pv and additionally
connecting two endvertices of Pv with three different neighbors of v and each inner vertex
of Pv with two different remaining neighbors, so that each of the vertices from N(v) is
connected to exactly one vertex in Pv . It is not difficult to see that G′ is a 4-regular graph
and therefore by Theorem 2.3 admits a parallel 2-stable trace W ′. Construct a parallel
double trace W in G from W ′ as follows. We start in an arbitrary vertex of G′ and follow
W ′. Let a′ = xy be an arc of W ′ that we are currently traversing on our walk along W ′.
If for every v, dG(v) > 4, x, y /∈ V (Pv), then we put xy into W so that the order of arcs
from W ′ is preserved. If for some v, dG(v) > 4, x ∈ Pv or y ∈ Pv , we replace a′ with
vy or xv, respectively. Finally, occurrences of the arcs with both endvertices contained in
some Pv are ignored in W .
We claim that the parallel double trace W is a parallel d-stable trace of the graph G.
We assume conversely and denote an arbitrary vertex in which W has a repetition of order
≤ d with v. Denote the maximal order of (≤ d)-repetition at v with d′. Since we used
the same construction as in the proof of Theorem 2.3, it follows that W is a parallel 2-
stable trace (and d′ > 2). From the symmetry of repetitions it then also follows that
dG(v) > d
′+2, since otherwise W would have at least one 1-repetition or one 2-repetition
at v (therefore also dG(v) ≥ 8). It is then also not difficult to see that every repetition in a
parallel double trace is of even order. Let X be a subset of N(v) containing vertices from
a maximal repetition at v (note that |X| = d′). There exists a path Pv in G′ that during
the construction replaced v from G. To make the argument more transparent, we imagine
vertices from Pv arranged in a horizontal line with all the neighbors of v except for two,
lying directly above or below vertices of Pv . The remaining two neighbors of vertex v are
aligned at the beginning and at the end of the horizontal line containing vertices from Pv .
Figure 6(b) shows Pv with vertices from N(v) in G′ for dG(v) = 8 (v′, v′′, and v′′′ are the
vertices replacing v in G′). Next, we color vertices from N(v) with two colors—black and
white, so that vertices from X are colored black while vertices from N(v) \X are colored
white. Example of such a coloring can be seen at Figure 6.
Since the subsetN(v)\X is also a repetition, the arguments used hereinafter are true for
black and white vertices and we can, without loss of generality, assume that the neighbor of
N(v), lying farmost to the left in the above mentioned horizontal line is colored white. We
next move along this horizontal line and denote the first black vertex that we meet (below
or above the line) with b. Denote its neighbor in Pv with v′. Since there are at least four
black vertices, v′ is not the farmost right vertex from Pv . Therefore, we can also denote
the right neighbor of v′ from Pv with v′′ and consider two cases. In the first case b is the
only neighbor of v′ (/∈ Pv) colored black (Figure 7(a)), while in the second case also the
second neighbor of v′ (/∈ Pv) is colored black (Figure 7(b)). In both cases we can, without
loss of generality, assume that an edge bv′ is traversed twice in the direction toward v′ in
W ′ (that is, arc bv′ is traversed twice in W , while arc v′b does not appear in W ′). The
fact that W has an X-repetition implies that every time double trace W comes to v from a
10 Art Discrete Appl. Math. 2 (2019) #P1.07
v
(a) v and N(v) in G
v′ v′′ v′′′
(b) Pv and N(v) in G′
Figure 6: Structures of N(v) in G and Pv in G′. Vertices contained in X are colored black.
v′
b
v′′
(a) b is the only black neighbor of v′
w1
w3
w2
v′
b′
b
v′′
(b) Both neighbors of v′ from N(v) are black
Figure 7: Two cases of the structure of Pv (of v′ and b to be more precise). Vertices for
which the color is not determined are colored grey.
J. Rus: Parallelism of stable traces 11
vertex in X it also continues to a vertex in X and, consequently, that every time a double
trace W ′ comes to a vertex in Pv from a (black colored) vertex in X it also leaves to a
(black colored) vertex in X . Note that in between W ′ can traverse other vertices from Pv
and that this applies for all appearances of verb continue from now on until the end of this
section. Analogously is true for (white colored) vertices from N(v) \X . Therefore, in W ′
there exist two subsequences which start with bv′, continue on some other vertices from Pv
and end in two from b different vertices from X . In the first case, when b is the only black
colored neighbor of v′, the subsequence b → v′ → v′′ has to appear twice in W ′, since
otherwise W ′ can not continue (twice) from b to a black colored vertex without previously
traversing white vertex. This contradicts the fact that W ′ is a parallel 2-stable trace, since
bv′v′′ is a 2-repetition at v′. In the second case, we denote the set of white vertices that
appear to the left of b with W = {w1, . . . , wl}. (Note that l is an odd integer.) For an
example, see Figure 7(b), where those vertices are denoted with w1, w2, and w3. Next,
we denote the second black colored neighbor of v′ from N(v) with b′. The subsequence
b→ v′ → b′ can appear at most once inW ′ (otherwiseW ′ would have a 2-repetition at v′).
Assume next that for every w ∈ W , w continues to a vertex inW . Then vertices fromW
form an odd repetition in W ′, which can not appear in a parallel 2-stable trace. Therefore,
at least one vertex w fromW has to continue to a white colored vertex not included inW
(that is,w continues to a white colored vertex to the right of b). If subsequence b→ v′ → b′
does not appear inW ′ it follows that edge v′v′′ (arc v′v′′ and v′′v′) is used more than twice
in W ′: at least once to connect a vertex fromW to a white colored vertex not inW , twice
to connect b to a (black colored) vertex in X different from b′, and twice to connect b′ to a
(black colored) vertex in X different from b, which is absurd. If subsequence b→ v′ → b′
does appear in W ′ it (in addition to multiple appearances of v′v′′) follows that edge v′v′′
is not parallel in W ′. Since all the black colored vertices except b and b′ are to the right of
v′ both b→ v′ → v′′ and v′′ → v′ → b′ have to appear in W ′, which is also absurd.
Since v was an arbitrary vertex in G and d′ was an arbitrary integer, 2 < d′ ≤ d, it
follows that W is a parallel d-stable trace of G and therefore Theorem 1.2 is proved.
4 Concluding remarks
In this section we present two concepts which we assumed could be used for constructing
parallel 2-stable traces. Unfortunately, it has turned out, when proving Theorem 2.3, that
there exist graphs admitting only parallel 2-stable traces which can not be realized using
the here described constructions.
The first construction goes as follows. Let G be an Eulerian graph with n vertices
(denoted with v1, . . . , vn) fulfilling conditions of Theorem 2.3 and let W ′ be an Eulerian
circuit of G. W ′ induces a set of functions Π′ = {π′1, . . . , π′n}, where π′i : N(vi) −→
N(vi), π′i(v) = u if and only if v → vi → u or u → vi → v are sequences in W ′, for
1 ≤ i ≤ n. Note that u 6= v, becauseG is simple andW ′ traverses every edge exactly once.
Suppose that W ′′ is another Eulerian circuit in G such that W ′′ induces a set of functions
Π′′ = {π′′1 , . . . , π′′n} with above described characteristics. In addition demand that edges
are traversed in the same direction as in W ′, and that if π′i(v) = u then π
′′
i (v) 6= u and
π′′i (u) 6= v. Concatenate Eulerian circuitsW ′ andW ′′ into a double traceW in an arbitrary
vertex v. It is obvious from the construction that every edge is traversed twice in the same
direction inW and thatW is without 1-repetitions and 2-repetitions in any vertex other than
v. Hence, if a graph G admits two Eulerian circuits with above described characteristics,
12 Art Discrete Appl. Math. 2 (2019) #P1.07
then G admits parallel 2-stable trace as well.
It turns out that we cannot always construct a parallel 2-stable trace of G by concate-
nating two Eulerian circuits. For instance, the graphG from Figure 8 has a parallel 2-stable
trace:
v1 → v2 → v3 → v1 → v2 → v4 → v1 → v5 → v2 → v3 → v4 → v6 → v5 → v2 →
v4 → v6 → v7 → v9 → v8 → v6 → v7 → v10 → v8 → v11 → v7 → v9 → v10 →
v11 → v7 → v10 → v11 → v9 → v8 → v11 → v9 → v10 → v8 → v6 → v5 → v3 →
v1 → v5 → v3 → v4 → v1,
but because of the cut vertex v6, from any Eulerian circuit W of G we cannot construct
another Eulerian circuit with the described properties.
v3
v2
v1
v5
v4
v6
v8
v7
v11
v10
v9
Figure 8: Graph whose parallel 2-stable traces cannot be constructed by concatenating two
Eulerian circuits.
The main idea of the second construction is to find a parallel 2-stable trace in each
block of a graph G and then concatenate them into a parallel 2-stable trace of the graph G.
Let again G be an Eulerian graph fulfilling the conditions of Theorem 2.3. Denote blocks
of G with B1, . . . , Bk and cutvertices with v1, . . . , vl. Find first a parallel 2-stable trace
Wi in block Bi. Concatenate parallel 2-stable traces into a parallel 2-stable trace of G in
corresponding cutvertices. When concatenating, one has to be careful that no 1-repetitions
and 2-repetitions appear.
Similar as for the first construction, none of the parallel 2-stable traces of the graph G
from Figure 8 can not be constructed by concatenating parallel 2-stable traces in its blocks.
Vertex v6 is a unique cutvertex of the graph G and it is contained in both of its blocks.
Since v6 is of degree 2 in both blocks of the graph G, none of them admit parallel 2-stable
trace. Similar problem occurs if one or more blocks of G are bridges.
Next possible improvement could instead of parallel 2-stable traces in blocks demand
parallel 1-stable traces where 2-repetitions (or 1-repetitions if block is a bridge) would be
allowed at cutvertices but are then later removed during the concatenation into a parallel
2-stable trace of the whole graph.
An attempt to find efficient algorithms for constructing and counting stable traces of
graphs was made in [1]. It would be of interest to characterize graphs that do not have any
of the two above described properties of graphs from Figure 8 and then try to improve the
J. Rus: Parallelism of stable traces 13
algorithms from [1] by using the above described constructions for those special cases of
graphs.
References
[1] N. Bašić, D. Bokal, T. Boothby and J. Rus, An algebraic approach to enumerat-
ing non-equivalent double traces in graphs, MATCH Commun. Math. Comput. Chem.
78 (2017), 581–594, http://match.pmf.kg.ac.rs/electronic_versions/
Match78/n3/match78n3_581-594.pdf.
[2] R. B. Eggleton and D. K. Skilton, Double tracings of graphs, Ars Combin. 17 (1984), 307–323.
[3] G. Fijavž, T. Pisanski and J. Rus, Strong traces model of self-assembly polypeptide structures,
MATCH Commun. Math. Comput. Chem. 71 (2014), 199–212.
[4] H. Fleischner, Eulerian Graphs and Related Topics. Part 1, Volume 1, volume 45 of Annals of
Discrete Mathematics, North-Holland, Amsterdam, 1990.
[5] H. Fleischner, Eulerian Graphs and Related Topics. Part 1, Volume 2, volume 50 of Annals of
Discrete Mathematics, North-Holland, Amsterdam, 1991.
[6] H. Gradišar, S. Božič, T. Doles, D. Vengust, I. Hafner Bratkovič, A. Mertelj, B. Webb, A. Šali,
S. Klavžar and R. Jerala, Design of a single-chain polypeptide tetrahedron assembled from
coiled-coil segments, Nat. Chem. Biol. 9 (2013), 362–366, doi:10.1038/nchembio.1248.
[7] H. Gradišar and R. Jerala, De novo design of orthogonal peptide pairs forming parallel coiled-
coil heterodimers, J. Pept. Sci. 17 (2011), 100–106, doi:10.1002/psc.1331.
[8] S. Klavžar and J. Rus, Stable traces as a model for self-assembly of polypeptide
nanoscale polyhedrons, MATCH Commun. Math. Comput. Chem. 70 (2013), 317–
330, http://match.pmf.kg.ac.rs/electronic_versions/Match70/
n1/match70n1_317-330.pdf.
[9] V. Kočar, S. Božič Abram, T. Doles, N. Bašić, H. Gradišar, T. Pisanski and R. Jerala, TOPO-
FOLD, the designed modular biomolecular folds: polypeptide-based molecular origami nanos-
tructures following the footsteps of DNA, WIREs Nanomed. Nanobiotechnol. 7 (2015), 218–
237, doi:10.1002/wnan.1289.
[10] J. Rus, Antiparallel d-stable traces and a stronger version of Ore problem, J. Math. Biol. 75
(2017), 109–127, doi:10.1007/s00285-016-1077-2.
[11] G. Sabidussi, Tracing graphs without backtracking, in: R. Henn et al. (eds.), Methods of Oper-
ations Research XXV, Part 1, Anton Hain Verlag, Meisenheim, 1977 pp. 314–332, proceedings
of the First Symposium on Operations Research held at the Heidelberg University, September
1 – 3, 1976.
[12] D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ, 1996.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P1.08
https://doi.org/10.26493/2590-9770.1301.165
(Also available at http://adam-journal.eu)
On primitive geometries of rank two∗
Dimitri Leemans †
Université Libre de Bruxelles, C.P.216 - Algèbre et Combinatoire,
Boulevard du Triomphe, 1050 Brussels, Belgium
Bernardo G. Rodrigues
School of Mathematical Sciences, University of KwaZulu-Natal,
Durban 4041, South Africa
Received 9 September 2015, accepted 18 September 2018, published online 14 August 2019
Abstract
In this paper, we describe a new algorithm to classify primitive coset geometries of rank
two for a given group G. This algorithm allows us to classify those geometries for the 12
smallest sporadic simple groups.
Keywords: Primitive coset geometries, sporadic groups, codes, designs.
Math. Subj. Class.: 51E24, 20D45
1 Introduction
Primitive coset geometries have been studied since the 1990’s, first by a team led by Fran-
cis Buekenhout in Brussels [4, 5] and later on by Peter Rowley and Nayil Kilic (see for
instance [8, 9, 10, 11, 12]). These geometries may be used to construct codes, designs,
graphs, etc. Recently, we had the idea of using rank two primitive geometries to con-
struct new binary codes for the McLaughlin group [13]. More precisely, we examined
the binary codes obtained from the row span over F2 of the adjacency matrices of some
strongly regular graphs which occur as subgraphs of the McLaughlin graph, namely those
with parameters (105, 32, 4, 12), (120, 42, 8, 18) and (253, 112, 36, 60). These new codes
were obtained by computing incidence matrices of rank two geometries whose element-
stabilizers are maximal subgroups of the McLaughlin group. In order to be able to generate
these geometries, a new approach was needed. Indeed, the previous algorithms described
by Dehon and Leemans [7] did not allow for a study of a group such as the Mathieu group
∗The authors also thank an anonymous referee for useful comments that improved this paper.
†Corresponding author.
E-mail addresses: dleemans@ulb.ac.be (Dimitri Leemans), rodrigues@ukzn.ac.za (Bernardo G. Rodrigues)
cb This work is licensed under http://creativecommons.org/licenses/by/3.0/
2 Art Discrete Appl. Math. 2 (2019) #P1.08
M24 of order roughly 2.4 · 108. We managed to classify such geometries with our new
algorithm for the 12 smallest sporadic groups, the largest one being the Rudvalis group, of
order roughly 1.4 · 1011. The geometries obtained are available on the following websites:
• https://data.adam-journal.eu/apg/
• http://homepages.ulb.ac.be/˜dleemans/PRI/
Our idea relies on permutation representations of groups. We present in this paper an
algorithm that outperforms the previous algorithms by a factor of 1000 at least. It allows
to classify rank two primitive coset geometries for the five Mathieu groups, the first three
Janko groups, the Higman Sims group, the McLaughlin group, the Held group and the
Rudvalis group. It also allows to confirm the classification of rank two primitive geometries
of M11 given in [5], M23 given in [10] and to correct some infelicities in [12] for M22. We
also obtained a complete classification for M24 while Kilic gives in [11] a partial one. The
results for the Janko groups, the Higman Sims group, the McLaughlin group, the Held
group and the Rudvalis groups are complete and new.
2 Terminology
The basic concepts about geometries constructed from a group and some of its subgroups
are due to Tits [14] (see also [3, Chapter 3]). The following theorem shows how to construct
geometries starting from groups.
Theorem 2.1 (Tits, 1956 [14]). Let n be a positive integer. Let I := {1, . . . , n} be a finite
set and let G be a group together with a family of subgroups (Gi)i∈I . Let X be the set
consisting of all cosets Gig, g ∈ G, i ∈ I . Let t : X → I be defined by t(Gig) = i. Define
an incidence relation ∗ on X ×X by:
Gig1 ∗Gjg2 iff Gig1 ∩Gjg2 is non-empty in G.
Then the 4-tuple Γ := (X, ∗, t, I) is an incidence structure having a chamber. Moreover,
the groupG acts by right multiplication as an automorphism group on Γ. Finally, the group
G is transitive on the flags of rank less than 3.
In the Theorem 2.1 above, a chamber is a set containing n cosets that are pairwise non-
disjoint. Let G be a group and {Gi : i ∈ {1, . . . , n}} be a set of subgroups of G. We
call Γ(G; (Gi)i∈I) the coset geometry associated toG and the subgroups (Gi)i∈I using the
above theorem. By Theorem 2.1, we see that the group G acts on Γ as a type-preserving
automorphism group.
In this paper, we only deal with geometries of rank two, that is coset geometries con-
structed from a group G and two subgroups G1, G2 of G. Let us denote such a geometry
Γ := Γ(G; (G1, G2)). We readily have that Γ is flag-transitive, that is, G acts transitively
on the pairs of non-disjoint cosets of Γ. Moreover, we want Γ to be primitive, meaning
that G1 and G2 must be maximal subgroups of G. Finally, as G1 and G2 are maximal
subgroups of G, we necessarily have 〈G1, G2〉 = G provided G1 6= G2. This implies that
Γ is a firm and connected geometry.
We are thus interested in classifying firm, connected, flag-transitive and primitive coset
geometries of rank two for a given group G. Given two coset geometries Γ1 := Γ(G;
(G1, G2)) and Γ2 := Γ(G; (H1, H2)), we say that Γ1 and Γ2 are isomorphic provided there
D. Leemans and B. G. Rodrigues: On primitive geometries of rank two 3
exists an element g ∈ Aut(G) such that g({G1, G2}) = {H1, H2}. In the next section,
we describe an algorithm to classify, up to isomorphism, all firm, connected, primitive
geometries of rank 2 for a given group G.
Given a geometry Γ := Γ(G; (G1, G2)), its dual Γ∗ is the geometry
Γ∗ := Γ(G; (G2, G1)).
To a coset geometry of rank two, Γ(G; (G1, G2)) we can associate a graph called the
incidence graph G whose vertices are the cosets of G1 and G2, two distinct cosets being
joined by an edge provided their intersection is nonempty. This graph is bipartite and G
acts transitively on the edges of G. Following [2], the gonality g of Γ is half the girth of
G. The point-diameter dp (resp. line-diameter dl) is the largest distance from the coset G1
(resp.G2) to any other coset of Γ. To Γ, we associate the triple [dp, g, dl]. Finally, we recall
that a design Sλ(t, k, v) is a geometry of rank two with v points and lines of size k such
that each set of t points is contained in exactly λ lines.
3 Algorithm
3.1 The old algorithm
The following algorithm was described in [7] (see Section 3, top-down approach) to con-
struct residually weakly primitive geometries. It was the same algorithm that was used by
Dehon in [6] to classify primitive geometries. This algorithm constructs primitive geome-
tries of all possible ranks, not only rank two.
Sketch of the Dehon algorithm. Let G be a group for which we want to compute all firm,
residually connected, flag-transitive and primitive geometries. Construct a set S containing
all maximal subgroups of G and the group W = Aut(G) acting as a permutation group
on S. This group is used to classify geometries up to isomorphism, two geometries (Γ, G)
and (Γ′, G) being isomorphic if there exists an element g in Aut(G) such that g(Γ) = Γ′.
LetC1 be a set of sets containing one representative of each conjugacy class of maximal
subgroups of G. Each element {G1} of C1 (where G1 is thus a maximal subgroup of G)
gives a rank one geometry Γ(G, {G1}).
If C1 is nonempty, let C2 be the empty set and do the following:
For every set {G1} in C1, determine, up to isomorphism, the subgroups G2 in S
such that Γ(G;G1, G2) is firm, residually connected and flag-transitive1. Store these pairs
{G1, G2} in C2.
If C2 is nonempty, let C3 be the empty set and do the following: for every element
{G1, G2} in C2, determine, up to isomorphism, the subgroups G3 in S such that Γ(G;
G1, G2, G3) is firm, residually connected and flag-transitive. Store these triples {G1, G2,
G3} in C3.
And so on, until the set Ci+1 obtained by adding subgroups to elements of Ci is empty.
The sets Cj , j = 1, . . . , i, contain all firm, residually connected, flag-transitive and primi-
tive geometries of rank j of G, up to isomorphism. ♦
In the algorithm above, the bottleneck is the construction of the group W at the very
beginning. That task of computing all maximal subgroups of G becomes quickly tedious
1In the rank two case, every possible G2 6= G1 is good, and the group W is there to make sure we pick only
pairwise non-isomorphic pairs of subgroups (G1, G2).
4 Art Discrete Appl. Math. 2 (2019) #P1.08
and the permutation representation ofG on these subgroups has a degree that is quickly too
large for MAGMA to work efficiently with it.
Given a rank two coset geometry Γ := Γ(G; (G1, G2)) where G1 and G2 are maximal
subgroups ofG, the groupG has a faithful primitive permutation representation on the set Ω
of cosets of G1 (and respectively on the cosets of G2). In that permutation representation,
G2 partitions the cosets into orbits. As G1 is maximal in G and the representation is
faithful, all the conjugates of G1 correspond to stabilizers of one coset of Ω. Hence, by
fixingG2 and taking stabilizers of representatives of the orbits ofG2 on Ω, we construct all
possible rank two geometries Γ(G; (H1, H2)) with H1 conjugate to G1 and H2 conjugate
to G2. One of them is necessarily the starting geometry.
An algorithm follows from the above paragraph: Given a group G and a maximal
subgroup M1 of G, construct the permutation representation of G on the set Ω of right
cosets of M1. This representation is primitive. For any maximal subgroup M2 of G in
that representation, compute the orbits O of M2 on Ω. In each orbit o ∈ O, pick one
representative x. The triple (G; (M2, Gx)) gives a primitive geometry of rank 2. This
geometry is obviously flag-transitive and it will be firm provided |o| 6= 1. This gives an
obvious way to produce primitive geometries of rank 2 for a given group G. Any rank
two primitive geometry constructed from G can be produced in this way. Therefore, we
have a technique to produce all rank two primitive geometries for G. It may happen that
we generate several isomorphic copies of the same geometry. To avoid that, every time
we generate a geometry with the above technique, we check whether it is isomorphic to
one of the geometries obtained so far. We only keep it if it is not isomorphic to any of the
previously obtained geometries.
In order to be able to apply the above algorithm, all that is needed now is that we can
compute permutation representations of G on all its maximal subgroups and that we can
check isomorphism between geometries that may be self-dual. Starting from the Rudvalis
group, these permutation representations can become very large. This is where we decided
to stop.
We implemented the Algorithm 3.1 in MAGMA [1] and determined all rank two prim-
itive geometries for the twelve smallest sporadic simple groups. Our findings are summa-
rized in the next section.
4 Primitive rank two geometries of sporadic groups
In this section, we summarise the classifications of primitive geometries of rank two we
obtained, using the algorithm described in the previous section, for the five Mathieu groups,
the first three Janko groups, the McLaughlin group, the Higman-Sims group, the Held
group and the Rudvalis group. A summary of the results obtained is available in Table 1.
In the last column, the computing time to classify these geometries is given. The computer
used was a workstation with 4 Intel Xeon E5 processors with 6 cores each working at
2.9 GHz and 1 TB of RAM. We give the orbit tables for the groups M11, M12, M22, M23
and J1 in the following sections. We omit the tables of the remaining groups since these
tables become too large.
4.1 The Mathieu group M11
Table 2 gives the orbit lengths for every primitive permutation representation ofM11. Each
column corresponds to a given primitive permutation representation and gives the ways
D. Leemans and B. G. Rodrigues: On primitive geometries of rank two 5
Algorithm 3.1 An algorithm to compute primitive geometries.
Input: G . . . a group
Output: geos . . . a sequence containing pairwise non-isomorphic pairs of maximal sub-
groups of G
Initialize a sequence geos that will be used to store the geometries.
Compute M , a list containing a representative of each conjugacy class of maximal sub-
groups of G.
for i := 1 to #M do
Let M1 be the i-th element of M .
Construct φ : G→ G/M1, the coset action of G on the cosets of M1.
Let φ(M) := [φ(x) : x ∈M ].
for j := 1 to #M do
Let M2 be the j-th element of φ(M).
Compute O, the orbits of K.
for each orbit o of size > 1 in O do
Let x be a representative of o.
Let Gx be the stabilizer of x in φ(G).
if the pair {φ−1(M2), φ−1(Gx)} is not isomorphic to any pair in geos then
add the pair {φ−1(M2), φ−1(Gx)} to geos
end if
end for
end for
end for
return geos
Table 1: The 11 smallest sporadic groups and their rank two primitive geometries.
Group Order Degree Number of geometries Computing time
M11 7 920 11 37 0.28 s
M12 95 040 12 166 13.79 s
M22 443 520 22 81 3.23 s
M23 10 200 960 23 170 54.04 s
M24 244 823 040 24 5074 106704 s ∼ 29 h
J1 175 560 266 669 146.22 s
J2 604 800 100 334 71.27 s
J3 50 232 960 6156 546 5031.72 s
HS 44 352 000 100 339 282.46 s
McL 898 128 000 275 443 2412.12 s
He 4 030 387 200 2058 4074 14.35 days
Ru 145 926 144 000 4060 1511 9.2 days
6 Art Discrete Appl. Math. 2 (2019) #P1.08
the points are split into orbits by the corresponding maximal subgroups. So for instance,
the entry 12 – 18 – 36 corresponding to the line labelled M9 : 2 and the column labelled
66 means that on the 66-points primitive permutation representation of M11 (obtained by
looking at the coset action of M11 on the cosets of a maximal subgroup isomorphic to S5),
a maximal subgroup M9 : 2 has orbits of length 12, 18 and 36.
Counting the number of orbits of size at least two in the upper triangular half, we get 38
possibilities. In fact, the Mathieu group M11 has, up to isomorphism, 37 primitive geome-
tries of rank two as two of them are dual of each other. This confirms the results obtained
in [5]. In Table 4, we give for each of the 37 geometries Γ the designs corresponding to
Γ and to its dual Γ∗. Some geometries do not appear in that table as a rank two geometry
does not necessarily give a design. Entry 28∗ is the well known S1(4, 5, 11), that is the
Steiner system associated to the Mathieu group M11.
Table 2: Orbits of primitive permutation representations of M11.
11 12 55 66 165
M10 1 – 10 12 10 – 45 30 – 36 45 – 120
L2(11) 11 1 – 11 55 11 – 55 55 – 110
M9 : 2 2 – 9 12 1 – 18 – 36 12 – 18 – 36 9 – 12 – 722
S5 5 – 6 2 – 10 10 – 15 – 30 1 – 15 – 20 – 30 10 – 15 – 20 – 602
M8 : S3 3 – 8 4 – 8 3 – 4 – 242 4 – 6 – 8 – 242 1 – 8 – 12 – 244 – 48
4.2 The Mathieu group M12
Table 5 gives the orbit lengths for every primitive permutation representation of M12.
Counting the number of orbits of size at least two in the upper triangular half, we get
268 possibilities. In fact, the Mathieu group M12 has 166 primitive geometries of rank two
as Aut(M12) conjugates several pairs in the 268 possibilities.
4.3 The Mathieu group M22
Table 6 gives the orbit lengths for every primitive permutation representation of M22. Our
programs gave 81 geometries up to isomorphism. This corrects the number that was ob-
tained in [12], namely 86.
4.4 The Mathieu group M23
Table 7 gives the orbit lengths for every primitive permutation representation of M23. Our
programs gave 170 geometries up to isomorphism. This confirms the results obtained by
Kilic in [10].
4.5 The first group of Janko J1
Table 8 gives the orbit lengths for every primitive permutation representation of J1. Our
programs gave 669 geometries up to isomorphism. This is a completely new result.
D. Leemans and B. G. Rodrigues: On primitive geometries of rank two 7
Table 3: The 37 rank two primitive geometries of M11.
Nr. G1 G2 G1 ∩G2 [dp, g, dl]
1 GL2(3) GL2(3) S3 [5, 3, 5]
2 GL2(3) GL2(3) E4 [4, 2, 4]
3 GL2(3) GL2(3) C2 [4, 2, 4]
4 GL2(3) GL2(3) C2 [5, 2, 5]
5 GL2(3) GL2(3) C2 [3, 2, 4]
6 GL2(3) GL2(3) 1 [3, 2, 3]
7 GL2(3) S5 D12 [5, 3, 5]
8 GL2(3) S5 D8 [4, 2, 4]
9 GL2(3) S5 S3 [4, 2, 4]
10 GL2(3) S5 C2 [3, 2, 3]
11 GL2(3) S5 C2 [3, 2, 3]
12 GL2(3) L2(11) D12 [3, 2, 4]
13 GL2(3) L2(11) S3 [3, 2, 3]
14 GL2(3) M9 : 2 Q8 : 2 [6, 3, 5]
15 GL2(3) M9 : 2 D12 [4, 3, 4]
16 GL2(3) M9 : 2 C2 [3, 2, 3]
17 GL2(3) M9 : 2 C2 [3, 2, 3]
18 GL2(3) M10 Q8 : 2 [3, 2, 4]
19 GL2(3) M10 S3 [3, 2, 3]
20 S5 S5 D8 [5, 2, 5]
21 S5 S5 S3 [3, 2, 3]
22 S5 S5 E4 [3, 2, 3]
23 S5 L2(11) A5 [3, 3, 4]
24 S5 L2(11) 12 [3, 2, 3]
25 S5 M9 : 2 12 [4, 2, 3]
26 S5 M9 : 2 8 [3, 2, 3]
27 S5 M9 : 2 C4 [3, 2, 3]
28 S5 M10 24 [3, 2, 3]
29 S5 M10 20 [3, 2, 3]
30 L2(11) L2(11) A5 [3, 2, 3]
31 L2(11) M9 : 2 D12 [2, 2, 2]
32 L2(11) M10 A5 [2, 2, 2]
33 M9 : 2 M9 : 2 8 [3, 2, 3]
34 M9 : 2 M9 : 2 4 [3, 2, 3]
35 M9 : 2 M10 M9 [3, 3, 4]
36 M9 : 2 M10 16 [3, 2, 3]
37 M10 M10 M9 [3, 2, 3]
8 Art Discrete Appl. Math. 2 (2019) #P1.08
Table 4: Designs Sλ(t, k, v) from the rank two primitive geometries of M11.
Nr. Design
1 S8(1, 8, 165)
1∗ S8(1, 8, 165)
2 S12(1, 12, 165)
2∗ S12(1, 12, 165)
3 S24(1, 24, 165)
3∗ S24(1, 24, 165)
4 S24(1, 24, 165)
4∗ S24(1, 24, 165)
5 S24(1, 24, 165)
5∗ S24(1, 24, 165)
6 S48(1, 48, 165)
6∗ S48(1, 48, 165)
7 S10(1, 4, 66)
7∗ S4(1, 10, 165)
8 S15(1, 6, 66)
8∗ S6(1, 15, 165)
9 S20(1, 8, 66)
9∗ S8(1, 20, 165)
10 S60(1, 24, 66)
10∗ S24(1, 60, 165)
11 S60(1, 24, 66)
11∗ S24(1, 60, 165)
12 S3(3, 4, 12)
12∗ S4(1, 55, 165)
13 S42(3, 8, 12)
13∗ S8(1, 110, 165)
Nr. Design
14 S3(1, 9, 165)
14∗ S9(1, 3, 55)
15 S4(1, 12, 165)
15∗ S12(1, 4, 55)
16 S24(1, 72, 165)
16∗ S72(1, 24, 55)
17 S24(1, 72, 165)
17∗ S72(1, 24, 55)
18 S1(3, 3, 11)
18∗ S3(1, 45, 165)
19 S1(8, 8, 11)
19∗ S8(1, 120, 165)
20 S15(1, 15, 66)
20∗ S15(1, 15, 66)
21 S20(1, 20, 66)
21∗ S20(1, 20, 66)
22 S30(1, 30, 66)
22∗ S30(1, 30, 66)
23 S1(2, 2, 12)
23∗ S2(1, 11, 66)
24 S1(10, 10, 12)
24∗ S10(1, 55, 66)
25 S10(1, 12, 66)
25∗ S12(1, 10, 55)
26 S15(1, 18, 66)
26∗ S18(1, 15, 55)
Nr. Design
27 S30(1, 36, 66)
27∗ S36(1, 30, 55)
28 S5(1, 30, 66)
28∗ S1(4, 5, 11)
29 S6(1, 36, 66)
29∗ S3(4, 6, 11)
30 S1(11, 11, 12)
30∗ S1(11, 11, 12)
33 S18(1, 18, 55)
33∗ S18(1, 18, 55)
34 S36(1, 36, 55)
34∗ S36(1, 36, 55)
35 S1(2, 2, 11)
35∗ S2(1, 10, 55)
36 S1(9, 9, 11)
36∗ S9(1, 45, 55)
37 S1(10, 10, 11)
37∗ S1(10, 10, 11)
D. Leemans and B. G. Rodrigues: On primitive geometries of rank two 9
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10 Art Discrete Appl. Math. 2 (2019) #P1.08
Table 6: Orbits of primitive permutation representations of M22.
22 77 176 176 231
L3(4) 1 – 21 21 – 56 56 – 120 56 – 120 21 – 210
24 : A6 6 – 16 1 – 16 – 60 80 – 96 80 – 96 15 – 96 – 120
A7 7 – 15 35 – 42 1 – 70 – 105 15 – 35 – 126 21 – 1052
A7 7 – 15 35 – 42 15 – 35 – 126 1 – 70 – 105 21 – 1052
24 : S5 2 – 20 5 – 32 – 40 16 – 802 16 – 802 1 – 30 – 40 – 160
23 : L3(2) 8 – 14 7 – 14 – 56 8 – 56 – 112 8 – 56 – 112 7 – 28 – 84 – 112
M10 10 – 12 2 – 30 – 45 20 – 36 – 120 20 – 36 – 120 30 – 36 – 45 – 120
L2(11) 11
2 112 – 55 11 – 55 – 110 11 – 55 – 110 553 – 66
330 616 672
L3(4) 120 – 210 280 – 336 3362
24 : A6 30 – 60 – 240 16 – 240 – 360 962 – 480
A7 15 – 105 – 210 70 – 126 – 420 42 – 210 – 420
A7 15 – 105 – 210 70 – 126 – 420 42 – 210 – 420
24 : S5 10 – 40 – 120 – 160 80 – 96 – 120 – 320 1603 – 192
23 : L3(2) 1 – 7 – 42 – 112 – 168 562 – 168 – 336 1123 – 336
M10 30
2 – 90 – 180 1 – 30 – 45 – 180 – 360 72 – 1202 – 360
L2(11) 55
3 – 165 66 – 1102 – 330 1 – 552 – 66 – 165 – 330
Table 7: Orbits of primitive permutation representations of M23.
23 253 253 506
M22 1 – 22 77 – 176 22 – 231 176 – 330
L3(4) : 2 7 – 16 1 – 112 – 140 21 – 112 – 120 30 – 140 – 336
24 : A7 2 – 21 21 – 112 – 120 1 – 42 – 210 56 – 210 – 240
A8 8 – 15 15 – 70 – 168 28 – 105 – 120 1 – 15 – 210 – 280
M11 11 – 12 22 – 66 – 165 55 – 66 – 132 66 – 110 – 330
24 : (3×A5) : 2 3 – 20 5 – 48 – 80 – 120 3 – 30 – 60 – 160 10 – 16 – 1202 – 240
23 : 11 23 253 253 2532
1288 1771 40320
M22 616 – 672 231 – 1540 40320
L3(4) : 2 112 – 336 – 840 35 – 336 – 560 – 840 40320
24 : A7 280 – 336 – 672 21 – 210 – 420 – 1120 40320
A8 168 – 280 – 840 35 – 56 – 4202 – 840 201602
M11 1 – 165 – 330 – 792 165 – 220 – 330 – 396 – 660 720 – 79205
24 : (3×A5) : 2 120 – 160 – 240 – 288 – 480 1 – 20 – 60 – 90 – 160 – 4803 57607
23 : 11 23 – 2535 2537 1 – 234 – 253159
D. Leemans and B. G. Rodrigues: On primitive geometries of rank two 11
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12 Art Discrete Appl. Math. 2 (2019) #P1.08
5 Conclusions
We have described a new algorithm that allows to compute primitive geometries of rank two
for much larger groups than the Dehon algorithm [7]. It remains to extend this algorithm
to be able to construct geometries of higher ranks as the Dehon algorithm permitted. Since
our first aim was to study codes arising from these rank two geometries, we did not try to
extend our algorithm and we leave it as an interesting topic for future work.
The problems found in [12] for M22 are very likely due to an incorrect determination
of non-isomorphic pairs of subgroups.
References
[1] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J.
Symbolic Comput. 24 (1997), 235–265, doi:10.1006/jsco.1996.0125.
[2] F. Buekenhout, Diagrams for geometries and groups, J. Comb. Theory Ser. A 27 (1979), 121–
151, doi:10.1016/0097-3165(79)90041-4.
[3] F. Buekenhout (ed.), Handbook of Incidence Geometry: Buildings and Foundations, North-
Holland, Amsterdam, 1995.
[4] F. Buekenhout, M. Dehon and P. Cara, Geometries of Small Almost Simple Groups Based on
Maximal Subgroups, Belgian Mathematical Society, Brussel, 1998, published as a supplement
to the Bull. Belg. Math. Soc. Simon Stevin.
[5] F. Buekenhout, M. Dehon and D. Leemans, All geometries of the Mathieu group M11 based
on maximal subgroups, Experiment. Math. 5 (1996), 101–110, http://projecteuclid.
org/euclid.em/1047565641.
[6] M. Dehon, Classifying geometries with CAYLEY, J. Symbolic Comput. 17 (1994), 259–276,
doi:10.1006/jsco.1994.1016.
[7] M. Dehon and D. Leemans, Constructing coset geometries with MAGMA: an application to
the sporadic groups M12 and J1, Atti Sem. Mat. Fis. Univ. Modena 50 (2002), 415–427.
[8] N. Kilic, On rank 3 residually connected geometries for M23, Int. J. Contemp. Math. Sci. 1
(2006), 679–696, doi:10.12988/ijcms.2006.06071.
[9] N. Kilic, Some rank 3 residually connected geometries for M23, Int. J. Contemp. Math. Sci. 1
(2006), 697–718, doi:10.12988/ijcms.2006.06072.
[10] N. Kilic, On rank 2 geometries of the Mathieu group M23, Taiwanese J. Math. 14 (2010),
373–387, doi:10.11650/twjm/1500405795.
[11] N. Kilic, On rank 2 geometries of the Mathieu group M24, An. Ştiinţ. Univ. “Ovid-
ius” Constanţa Ser. Mat. 18 (2010), 131–147, http://www.anstuocmath.ro/
mathematics/pdf21/13.pdf.
[12] N. Kilic and P. Rowley, On rank 2 and rank 3 residually connected geometries for M22, Note
Mat. 22 (2003), 107–154, doi:10.1285/i15900932v22n1p107.
[13] D. Leemans and B. G. Rodrigues, Binary codes of some strongly regular subgraphs of the
McLaughlin graph, Des. Codes Cryptogr. 67 (2013), 93–109, doi:10.1007/s10623-011-9589-7.
[14] J. Tits, Sur les analogues algébriques des groupes semi-simples complexes, in: Colloque
d’algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Établissements Ceuter-
ick, Louvain, Centre Belge de Recherches Mathématiques, pp. 261–289, 1957.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P1.09
https://doi.org/10.26493/2590-9770.1228.eb5
(Also available at http://adam-journal.eu)
Infinite benzenoids
Nino Bašić ∗
FAMNIT, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia
IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia
IMFM, Jadranska 19, 1000 Ljubljana, Slovenia
Received 6 December 2017, accepted 28 January 2019, published online 30 December 2019
Abstract
A family of benzenoids, called convex benzenoids, was introduced in 2012 by Cruz,
Gutman and Rada. In a later paper by the present author et al., several equivalent charac-
terisations of convex benzenoids have been given and their equivalence was proved. Along
the way an infinite benzenoid called the half-plane was used for the purpose of theoretical
reasoning. In this short paper, some properties of infinite benzenoids are discussed. It is
proved that their boundary consists of countably many connected components. Convex in-
finite benzenoids are classified and it is proved that there are only countably many convex
infinite benzenoids, whilst there are uncountably many infinite (non-convex) benzenoids.
We also show that there are countably many infinite benzenoids which have a finite number
of 1s in their boundary-edges code.
Keywords: Infinite benzenoid, hexagonal system, convex benzenoid, boundary-edges code, half-plane,
countable set.
Math. Subj. Class.: 05C10, 92E10, 03E75
1 Introduction
Benzenoids are important compounds in chemistry and have been extensively studied dur-
ing the past few decades. For an exhaustive treatment of the topic see the classical reference
by Cyvin and Gutman [9]. The reader may want to consult references [5, 6] and [7] for ad-
vanced treatments. In the present paper, we study their associated benzenoid graphs from
a purely mathematical viewpoint. Hereinafter, the word benzenoid is used as a synonym
for benzenoid graph. We build upon theoretical treatment of benzenoids from [2]. The
framework of [2] allows for generalisation to structures called infinite benzenoids. In 2012
∗The author has been supported in part by ARRS Slovenia (projects P1-0294 and N1-0032). The author would
like to thank Tomaž Pisanski for fruitful discussions during the preparation of the manuscript.
E-mail address: nino.basic@famnit.upr.si (Nino Bašić)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 2 (2019) #P1.09
a family of convex benzenoids was introduced by Cruz, Gutman and Rada [4]. In a recent
work [3], we presented several alternative characterisations of convex benzenoids. In this
work, we study properties of convex infinite benzenoids. We assume that the reader is fa-
miliar with results of elementary set theory (e.g., the union of countably many countable
sets is countable). For definitions of set-theoretic terms such as countable set and uncount-
able set, consult a standard reference such as [10] or [12]. This paper further develops
several ideas already present in the PhD thesis of the author [1].
2 The hexagonal grid
Let us consider the infinite cubic plane graph H, called the hexagonal grid. (For an intro-
duction to infinite graphs, consult Chapter 8 of [8].) It comprises infinitely many hexagonal
faces each of which is incident with 6 edges and 6 vertices. We say that faces a, b ∈ F (H)
are adjacent if they are different and share an edge. The faces of H will be simply called
hexagons. Two hexagons are neighbours if they have exactly one edge in common. Some-
times, it is convenient to introduce a coordinate system on the hexagonal gridH, as shown
in Figure 1. Let h ∈ H. Then ξ(h) and η(h) will denote the first and second coordinate
(−3, 2) (−2, 2) (−1, 2) (0, 2) (1, 2)
(−2, 1) (−1, 1) (0, 1) (1, 1) (2, 1)
(−2, 0) (−1, 0) (0, 0) (1, 0) (2, 0)
(−1,−1) (0,−1) (1,−1) (2,−1) (3,−1)
(−1,−2) (0,−2) (1,−2) (2,−2) (3,−2)
η = 2
η = 1
η = 0
η = −1
η = −2
ξ
=
3
ξ
=
2
ξ
=
1
ξ
=
0
ξ
=
−1
ζ
=
3
ζ
=
2
ζ
=
1
ζ
=
0
ζ
=
−
1
Figure 1: Coordinate system onH.
of h as shown in Figure 1. We may introduce yet another coordinate ζ(h), but it is not
independent of the previous two since ζ(h) = ξ(h) + η(h). For a detailed description of
the coordinate system onH, see [1] or [3].
N. Bašić: Infinite benzenoids 3
3 Hexagonal systems and benzenoids
An arbitrary subset of faces K ⊆ F (H), together with all vertices and edges that are
incident with at least one member of K, is called a hexagonal system. We will denote
the corresponding hexagonal system simply by K, even though it is formally a subset of
faces. Two hexagonal systems are isomorphic if one can be obtained from the other by an
isometry of the plane that leaves H invariant. Hexagons a ∈ K and b ∈ K belong to the
same connected component of the hexagonal system if there exists a sequence of hexagons
h1 = a, h2, . . . , hn = b such that hi and hi+1 are adjacent for each i = 1, . . . , n − 1 and
{h1, . . . , hn} ⊆ K. When all hexagons of the sequence are pairwise distinct, it is called a
path between a and b in K. The interval between a and b in K, denoted IK(a, b), is the set
of hexagons which are contained on any of the shortest paths between a and b. A hexagonal
system is connected if it has a single connected component. A subset of faces K ⊆ F (H)
gives rise to two hexagonal systems: K and its complement K{.
We define a benzenoid in the following way:
Definition 3.1. A benzenoid is a connected hexagonal system K, such that each connected
component of the complement K{ is infinite.
A benzenoid is finite if it has a finite number of hexagons. Otherwise, it is called an
infinite benzenoid.
Example 3.2. Let k ∈ Z. Infinite hexagonal systems
Ξk = {h ∈ H | ξ(h) = k},
Hk = {h ∈ H | η(h) = k}, and
Zk = {h ∈ H | ζ(h) = k}
will be called lines. They are all benzenoids and they are all isomorphic to each other.
The hexagonal system K1 on Figure 2 is defined as
K1 = {h ∈ H | η(h) ≡ 0 (mod 2)} =
⋃
k∈Z
H2k.
K1 is not a benzenoid, because it is not connected. K1 consists of infinitely many disjoint
K1 K2 K3
Figure 2: Infinite hexagonal systems K1, K2 and K3.
lines and is isomorphic to its complement K{1 .
K2 is obtained from K1 by adding another line (with a different slope), i.e., K2 =
K1 ∪ Z0. The hexagonal system K2 is a benzenoid.
4 Art Discrete Appl. Math. 2 (2019) #P1.09
Hexagonal system K3 is defined as
K3 = {h ∈ H | η(h) ≥ 0}.
We call it a convex half-plane. In the present work, we will simply call it a half-plane, since
we will not consider other types of (non-convex) half-planes. It is an infinite benzenoid that
is isomorphic to its complement K{3 . ♦
The following theorem tells us that there are as many (non-isomorphic) infinite ben-
zenoids out there as there are real numbers.
Theorem 3.3. There exist uncountably many mutually non-isomorphic infinite benzenoids.
Proof. The interval [0, 1) has the cardinality of the continuum. Each number x ∈ [0, 1)
can be written in its binary representation
0.x1x2x3x4 . . . (3.1)
Note that xi ∈ {0, 1} for each i ∈ N and that
x =
∞∑
i=1
xi · 2−i. (3.2)
The sequence {xi}∞i=1 is uniquely determined if we require that for each n ∈ N there exists
an integer m > n such that xm 6= 1. For example, the number 1116 can be written in binary
representation as 0.1011.
Figure 3: Infinite benzenoid P 11
16
.
We will assign an infinite benzenoid to every such sequence {xi}∞i=1. Define
Px = {h ∈ H | η(h) ≤ 0} \ ({(−2, 0), (−1, 0)} ∪ {(2i− 1, 0) | i ∈ N ∧ xi = 1}) .
The infinite benzenoid Px is obtained from the half-plane P = {h ∈ H | η(h) ≤ 0} by
removing hexagons (−2, 0), (−1, 0) and all hexagon (2i− 1, 0) where i ∈ N and xi = 1.
For example, the infinite benzenoid that corresponds to number 1116 is shown in Figure 3.
It is not hard to see that Px  Py if and only if x 6= y. We constructed an injective
mapping from the set [0, 1) to the class of infinite benzenoids. Therefore, the class of all
infinite benzenoids is uncountable.
The boundary of a hexagonal system K consists of all edges and all vertices that are
indicent to (at least) one hexagon of K and (at lest) one hexagon of K{. Those vertices and
edges are called boundary vertices and boundary edges, respectively.
The boundary of a finite benzenoid is isomorphic to a cycle graph. The boundary of
an infinite benzenoid is a disjoint union of one or more infinite paths. The boundary of K3
in Figure 2 is a single infinite path. The boundary of K2 in Figure 2 consists of infinitely
many connected components (that are all infinite paths).
N. Bašić: Infinite benzenoids 5
Proposition 3.4. Let B be an arbitrary benzenoid. The boundary of B consists of countably
many connected components.
Proof. The edges ofH can be labeled with natural numbers in a spiral fashion as shown in
Figure 4. Let us define a mapping from the set of connected components of the boundary
12
3
4 5
6
7
8
9
10
11
12
13
14
15
16
17181920
21
22
23
24
25
. . .
Figure 4: “Spiral” labeling of edges ofH.
of B to natural numbers. Each connected component is mapped to the minimum among
all labels of the edges that belong to the component. This mapping to natural numbers is
injective and the result follows.
3.1 The boundary-edges code
Several combinatorial descriptions of finite benzenoids are readily available. In [11], Han-
sen et al. presented the boundary-edges code (abbreviated as BEC). The boundary of a
finite benzenoid (which is a cycle graph) is described as a cyclic sequence of numbers,
each of which counts the number of edges between two consecutive boundary vertices of
degree 3. All those numbers are from the set {1, 2, 3, 4, 5}. The only exception here is
benzene, which has BEC 6. In Figure 5, there are several finite benzenoids together with
their BECs. The boundary-edges code of B is denoted BEC(B).
The code BEC(B3) = 424242 of B3 from Figure 5 can also be written as (42)3, where
sk stands for ss . . . s︸ ︷︷ ︸
k
.
It is possible to generalise the definition of the BEC to infinite benzenoids. Each con-
nected component of the boundary will be assigned its own BEC. Since a connected com-
ponent of the boundary of an infinite benzenoid is an infinite path, the BEC can be formally
defined as a mapping Z→ {1, 2, 3, 4, 5}, i.e. a doubly infinite sequences.
Example 3.5. The boundary of a half-plane has a single connected component with the
BEC . . . 22222 . . ., which can simply be written as 2∞.
The boundary of benzenoid K2 in Figure 2 has infinitely many connected components.
They all have the same BEC, namely 2∞132∞.
The boundary of benzenoid P 11
16
in Figure 3 has a single connected component with
BEC 2∞31214123231241232∞. ♦
6 Art Discrete Appl. Math. 2 (2019) #P1.09
(a) B1, BEC(B1) = 6 (b) B2, BEC(B2) = 5252
(c) B3, BEC(B3) = 424242 (d) B4, BEC(B4) = 533113513311
Figure 5: Examples of finite benzenoids together with their BECs.
Infinite benzenoids can be classified with respect to the number of connected compo-
nents of the boundary.
Proposition 3.6. Let n ∈ N ∪ {0,∞}. There exists an infinite benzenoid B with n con-
nected components in the boundary of B. Symbol ∞ means that the boundary of B has
(countably) infinitely many connected components.
Proof. The benzenoid H, i.e. the one that contains all hexagons of the infinite hexagonal
grid, is the only one with 0 connected components in the boundary. The boundary of half-
plane P = {h ∈ H | η(h) ≥ 0} has exactly 1 connected component.
Let
Bk = P ∪ Ξ0 ∪ Ξ2 ∪ Ξ4 ∪ · · · ∪ Ξ2k.
It is easy to see that the boundary ofBk has k+2 connected components. In other words, for
every natural number n ≥ 2 there exist an infinite benzenoid, namely Bn−2, with exactly
n connected components in its boundary.
For the case n =∞, take K2 from Figure 2.
4 Convex (infinite) benzenoids
Let us recall the metric definition of a convex hexagonal system from [3]:
Definition 4.1. A hexagonal system K is convex if for any pair of its hexagons a and b the
whole interval IH(a, b) is contained in K.
Theorem 3.3 means that one cannot describe all of the infinite benzenoids algorithmi-
cally. This means that there exist infinite benzenoids for which there does not exist a finite
computer program for constructing them (even if the program runs infinitely long).
Let us consider convex infinite benzenoids. The whole hexagonal gridH and the empty
set ∅ are clearly convex. A half-plane is also convex. Each half-plane has a normal which
is pointing out of the half-plane (see Figure 6). There are exactly 6 possible directions of
N. Bašić: Infinite benzenoids 7
half-planes. We will denote them as follows:
HP+ξ (n) = {h ∈ H | ξ(h) ≥ n}, (4.1)
HP−ξ (n) = {h ∈ H | ξ(h) ≤ n}, (4.2)
HP+η (n) = {h ∈ H | η(h) ≥ n}, (4.3)
HP−η (n) = {h ∈ H | η(h) ≤ n}, (4.4)
HP+ζ (n) = {h ∈ H | ζ(h) ≥ n}, (4.5)
HP−ζ (n) = {h ∈ H | ζ(h) ≤ n}. (4.6)
(a) HP−η (n) (b) HP
−
ζ (n) (c) HP
−
ξ (n) (d) HP
+
η (n)
(e) HP+ζ (n) (f) HP
+
ξ (n)
Figure 6: The normal of a hexagonal half-plane can point in 6 different directions.
Note that the intersection of two half-planes with the same normal is equal to one of
the two. In other words, one is a sub-benzenoid of the other. For example,
HP+ξ (3) ∩HP
+
ξ (5) = HP
+
ξ (5).
Therefore, the interection of any number of half-planes is equal to an intersection of at most
6 of those half-planes. For each direction of the normal that is present in the list, we select
the half-plane that is contained in all other half-planes that have the same direction. Also,
note that all half-planes (4.1) – (4.6) are isomorphic to each other.
Definition 4.2. Let B be a benzenoid (finite or infinite). The smallest convex benzenoid
containing B is called the convex closure of B and is denoted Conv(B).
In [3], the following proposition was proved.
Proposition 4.3. Any intersection of convex (finite or infinite) benzenoids is a convex ben-
zenoid. 
8 Art Discrete Appl. Math. 2 (2019) #P1.09
Note that the empty set ∅ can also be considered as a convex benzenoid. To a chemist,
this means nothing concrete. To a mathematician, it is clear that every two members of an
empty set satisfy the condition in Definition 4.1, i.e., this condition is void.
(a) half-plane, HP (b) anvil, AN (c) wedge, WE
n
(d) strip, ST (n)
n
(e) chomped wedge, CW(n)
n
(f) knife, KN (n)
n
m
(g) sword, SW(n,m)
Figure 7: Families of convex infinite benzenoids.
In addition to the hexagonal grid H, the empty set ∅ and half-plane HP , we introduce
the following families of infinite benzenoids (see Figure 7):
(a) The benzenoid AN in Figure 7(b) is called anvil and is uniquely determined (up to
isomorphism) by BEC(AN ) = 2∞32∞.
(b) The benzenoidWE in Figure 7(c) is called wedge and is uniquely determined (up to
isomorphism) by BEC(WE) = 2∞42∞.
(c) A member of the one-parametric family of benzenoids ST (n), n ≥ 1, in Figure 7(d)
is called a strip (of width n) and cannot be uniquely determined by the BEC. How-
ever, ST (n) ∼= HP+ξ (0) ∩HP
−
ξ (n). Note that ST (1) is called a line.
(d) A member of the one-parametric family of benzenoids CW(n), n ≥ 2, in Fig-
ure 7(e) is called a chomped wedge and is uniquely determined (up to isomorphism)
by BEC(CW(n)) = 2∞32n−232∞.
N. Bašić: Infinite benzenoids 9
(e) A member of the one-parametric family of benzenoidsKN (n), n ≥ 1, in Figure 7(f)
is called a knife (of width n) and is uniquely determined (up to isomorphism) by
BEC(KN (n)) =
{
2∞32n−242∞ if n ≥ 2,
2∞52∞ if n = 1.
The benzenoid KN (1) will also be called a needle.
(f) A member of the two-parametric family of benzenoids SW(n,m), m ≥ n ≥ 2,
in Figure 7(g) is called a sword and is uniquely determined (up to isomorphism) by
BEC(SW(n)) = 2∞32n−232m−232∞.
All benzenoids on the above list can be obtaned as intersections of half-planes and are
therefore convex by Proposition 4.3.
In [3], the following proposition was proved.
Proposition 4.4. A finite benzenoid B is convex if and only if it can be obtained as an
intersection of half-planes, i.e., if there exist integers n+ξ , n
−
ξ , n
+
η , n
−
η , n
+
ζ and n
−
ζ , such
that
B = HP+ξ (n
+
ξ ) ∩HP
−
ξ (n
−
ξ ) ∩HP
+
η (n
+
η ) ∩HP
−
η (n
−
η ) ∩HP
+
ζ (n
+
ζ ) ∩HP
−
ζ (n
−
ζ ). 
Let B be a benzenoid. We define
η+ = max
h∈B
η(h) and η− = min
h∈B
η(h).
If the set {η(h) | h ∈ B} is not bounded from above, we write η+ = ∞. Similarly, if the
set {η(h) | h ∈ B} is not bounded from below, we write η− = −∞. Analogously, we
define
ξ+ = max
h∈B
ξ(h), ξ− = min
h∈B
ξ(h), ζ+ = max
h∈B
ζ(h) and ζ− = min
h∈B
ζ(h).
Lemma 4.5. Let B be a convex benzenoid. Then the following statements hold:
(1) If η+ <∞ and ξ+ <∞ then ζ+ <∞.
(2) If η− > −∞ and ξ− > −∞ then ζ− > −∞.
(3) If ξ+ <∞ and ζ− > −∞ then η− > −∞.
(4) If ξ− > −∞ and ζ+ <∞ then η+ <∞.
(5) If η+ <∞ and ζ− > −∞ then ξ− > −∞.
(6) If η− > −∞ and ζ+ <∞ then ξ+ <∞.
Proof. It is enough to prove statement (1) of the lemma. Other statements will results from
the fact that we may use a rotational symmetry of the hexagonal gridH.
Suppose that η+ <∞ and ξ+ <∞. This implies B ⊆ HP−η (η+) and B ⊆ HP
−
ξ (ξ
+).
Moreover, B ⊆ HP−η (η+) ∩HP
−
ξ (ξ
+). ButHP−η (η+) ∩HP
−
ξ (ξ
+) is a wedge in which
hexagon (ξ+, η+) obtains the maximal value of ζ coordinate. Therefore, ζ+ <∞.
10 Art Discrete Appl. Math. 2 (2019) #P1.09
The next result generalises Proposition 4.4 to infinite benzenoids.
Proposition 4.6. An benzenoid B (finite or infinite) is convex if and only if it can be ob-
tained as an intersection of half-planes.
Proof. The proof technique used here is similar to the one used in [3] to prove Proposi-
tion 4.4. If a benzenoid B is obtained as an intersection of half-planes then it is convex by
Proposition 4.3.
Now let B be a convex benzenoid. There are several cases to consider, depending on
the values of ξ+, ξ−, η+, η−, ζ+ and ζ−.
Suppose that ξ− > −∞, η− > −∞ and ζ+ < ∞. Statements (2), (4) and (6) of
Lemma 4.5 imply that ζ− > −∞, η+ <∞ and ξ+ <∞, respectively. This means that B
is finite and the result follows from Proposition 4.4.
We need to consider several more cases:
(i) ξ− = −∞, η− > −∞ and ζ+ <∞;
(ii) ξ− = −∞, η− = −∞ and ζ+ <∞;
(iii) ξ− = −∞, η− = −∞ and ζ+ =∞.
Note that because of the symmetry, there are only 3 cases to consider and not 23. We
provide the proof of case (i). The proofs of (ii) and (iii) are very similar and use the same
type of arguments.
(i): Suppose that ξ− = −∞, η− > −∞ and ζ+ < ∞. Statements (6) of Lemma 4.5
implies that ξ+ < ∞. Now consider ζ− and η+. If ζ− > −∞ and η+ < ∞ then by
statement (5) of Lemma 4.5 we obtain ξ− > −∞, a contradiction. We have two subcases:
(i.1) ζ− = −∞ and η+ =∞;
(i.2) ζ− = −∞ and η+ <∞.
Again, by the symmetry argument, we do not need to consider the case ζ− > −∞ and
η+ =∞. We will now consider case (i.1). The case (i.2) is analogous.
(i.1): From η− > −∞ it follows that there exists a hexagon hη− ∈ B such that
η(hη−) = η
−. From ζ+ < ∞ it follows that there exists a hexagon hζ+ ∈ B such
that ζ(hζ+) = ζ+. And from ξ+ < ∞ it follows that there exists a hexagon hξ+ ∈ B
such that ξ(hξ+) = ξ+. See Figure 8 for an illustration. Let h1, h2 ∈ H such that
η(h1) = η
−, ξ(h1) = ξ+, ζ(h2) = ζ+ and ξ(h2) = ξ+. Because h1 ∈ IH(hη−, hξ+) and
h2 ∈ IH(hξ+ , hζ+), it follows that h1, h2 ∈ B.
Let n be an arbitrary large integer. From ζ− = −∞ it follow that there exists a hexagon
h′ ∈ B such that ζ(h′) = −n. Let h′′ ∈ H be the hexagon with ζ(h′′) = −n and
η(h′′) = η−. Since h′′ ∈ IH(h′, h1) it follow that h′′ ∈ B. This means that B contains
the needle {h ∈ H | η(h) = η− ∧ ξ(h) ≤ ξ+}. Similarly, η+ = ∞ implies that B also
contains the needle {h ∈ H | ζ(h) = ζ+ ∧ ξ(h) ≤ ξ+}. The line segment between h1 and
h2 is also contained in B.
This means that all boundary hexagons of HP+η (η−) ∩ HP
−
ξ (ξ
+) ∩ HP−ζ (ζ+) are
contained in B. But B ⊆ HP+η (η−) ∩HP
−
ξ (ξ
+) ∩HP−ζ (ζ+) and therefore
B = HP+η (η−) ∩HP
−
ξ (ξ
+) ∩HP−ζ (ζ
+).
The remaining cases can be proved using the same approach.
N. Bašić: Infinite benzenoids 11
hζ+
hξ+
hη− h1
h2
h′
h′′
Figure 8: The case (i.1).
We are now able to prove the following theorem.
Theorem 4.7. An infinite benzenoid is convex if and only if it is isomorphic to one of the
following:
(a) the hexagonal gridH,
(b) the half-planeHP ,
(c) the anvil AN ,
(d) the wedgeWE ,
(e) a strip ST (n) for some n ≥ 1,
(f) a chomped wedge CW(n) for some n ≥ 2,
(g) a knife KN (n) for some n ≥ 1,
(h) a sword SW(n,m) for some m ≥ n ≥ 2.
Moreover, all benzenoids from the above list are pairwise non-isomorphic.
Proof. We already know that all infinite benzenoids listed in the statement of the theorem
are convex. It is not hard to see that they are also pairwise non-isomorphic.
Suppose that B is a convex infinite benzenoid. From Proposition 4.6 it follows that
it can be obtained as an intersection of half-planes. We can analyse (case by case) all
possibilities for the intersection of (any subset) of the 6 half-planes. In this analysis, we
either obtain a finite convex benzenoid (they were classified in [3]), which contradicts the
assumption that B is infinite, or one of the above.
Let FO be the class of all such infinite benzenoids B with the property that the total
number of 1s in BECs of all connected components of the boundary of B is finite.
12 Art Discrete Appl. Math. 2 (2019) #P1.09
Theorem 4.8. There are countably many benzenoids in the class FO.
Proof. Let On, n ≥ 0, be the finite benzenoid defined by BEC(On) = (32n)6.
Let B be any benzenoid of the class FO (see Figure 9 for an example). Because there
Figure 9: A benzenoid B ∈ FO.
are finitely many 1s in BECs of B, there exists some O′ = On for n large enough, such
that O′ contains all edges that correspond to 1s in BECs of B as internal edges (see Fig-
ure 10(a)). Then B \ O′ is a hexagonal system that comprises finitely many finite ben-
zenoids and finitely many infinite benzenoids. (In the example in Figure 10(a), there is 1
finite benzenoid and 2 infinite benzenoids.)
O′
(a)
O′′
(b)
Figure 10: A benzenoid B ∈ FO with O′ and O′′.
We can choose a (possibly larger) number m ≥ n, such that O′′ = Om contains O′
and all finite connected components of B \ O′ (see Figure 10(b)). If we remove all edges
of O′′ from the boundary of B, we obtain an even number of semi-infinite paths. No edge
on those semi-infinite paths is an edge that corresponds to a 1s in BECs of B, because such
N. Bašić: Infinite benzenoids 13
edges are all contained in O′. The subsequence of a BEC that corresponds to such a semi-
infinite path contains infinitely many 2s, but only a finite number of 3s, 4s and 5s. Those
numbers cause bends in the boundary and too many bends would result in a collision of the
boundary (see Figure 11), which is impossible. To encode the benzenoid B, we need the
collision
Figure 11: Collision of the boundary of B.
following information:
(i) the number m, which determines the O′′ (we have countably many choices),
(ii) the subset of hexagons of O′′ that determine B ∩ O′′ (there are finitely many such
subsets),
(iii) vetices on the boundary ofO′′ that are starting vertices of semi-infinite paths (finitely
many choices),
(iv) positions and types of bends on each semi-infinite path (there are finitely many bends
and each type and position can be encoded by a natural number).
From the above encoding of B it follows that there are countably many different benzenoids
in the class FO.
Corollary 4.9. There exist countably many mutually non-isomorphic convex infinite ben-
zenoids.
Proof. By Theorem 4.7, a convex infinite benzenoid is isomorphic to one of those listed in
the statement of Theorem 4.7. None of them has a 1 in the boundary-edges code of any
connected component of the boundary. Therefore, the class of convex infinite benzenoids
is a subclass of FO and the results follows by Theorem 4.8.
References
[1] N. Bašić, Algebraic Approach to Several Families of Chemical Graphs, Ph.D. thesis, University
of Ljubljana, 2016.
[2] N. Bašić, P. W. Fowler and T. Pisanski, Coronoids, patches and generalised altans, J. Math.
Chem. 54 (2016), 977–1009, doi:10.1007/s10910-016-0599-6.
14 Art Discrete Appl. Math. 2 (2019) #P1.09
[3] N. Bašić, P. W. Fowler and T. Pisanski, Stratified enumeration of convex benzenoids, MATCH
Commun. Math. Comput. Chem. 80 (2018), 153–172, http://match.pmf.kg.ac.rs/
electronic_versions/Match80/n1/match80n1_153-172.pdf.
[4] R. Cruz, I. Gutman and J. Rada, Convex hexagonal systems and their topological indices,
MATCH Commun. Math. Comput. Chem. 68 (2012), 97–108, http://www.pmf.kg.ac.
rs/match/electronic_versions/match68/n1/match68n1_97-108.pdf.
[5] S. J. Cyvin, J. Brunvoll and B. N. Cyvin, Theory of Coronoid Hydrocarbons, volume 54 of
Lecture Notes in Chemistry, Springer-Verlag, Berlin, 1991, doi:10.1007/978-3-642-51110-3.
[6] S. J. Cyvin, J. Brunvoll, B. N. Cyvin, R. S. Chen and F. J. Zhang, Theory of Coronoid Hy-
drocarbons II, volume 62 of Lecture Notes in Chemistry, Springer-Verlag, Berlin, 1994, doi:
10.1007/978-3-642-50157-9.
[7] S. J. Cyvin and I. Gutman, Kekulé Structures in Benzenoid Hydrocarbons, volume 46 of Lecture
Notes in Chemistry, Springer-Verlag, Berlin, 1988, doi:10.1007/978-3-662-00892-8.
[8] R. Diestel, Graph Theory, volume 173 of Graduate Texts in Mathematics, Springer, Berlin, 5th
edition, 2017, doi:10.1007/978-3-662-53622-3.
[9] I. Gutman and S. J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons, Springer-
Verlag, 1989, doi:10.1007/978-3-642-87143-6.
[10] P. R. Halmos, Naive Set Theory, Undergraduate Texts in Mathematics, Springer-Verlag, New
York, 1st edition, 1974, doi:10.1007/978-1-4757-1645-0.
[11] P. Hansen, C. Lebatteux and M. Zheng, The boundary-edges code for polyhexes, J. Mol. Struct.
(THEOCHEM) 363 (1996), 237–247, doi:10.1016/0166-1280(95)04139-7.
[12] T. Jech, Set Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, doi:
10.1007/3-540-44761-x.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P1.10
https://doi.org/10.26493/2590-9770.1330.993
(Also available at http://adam-journal.eu)
Polynomials of degree 4 over finite fields
representing quadratic residues∗
Shaofei Du †
Capital Normal University, School of Mathematical Sciences,
Bejing 100048, People’s Republic of China
Klavdija Kutnar ‡
University of Primorska, UP FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia
University of Primorska, UP IAM, Muzejski trg 2, 6000 Koper, Slovenia
Dragan Marušič §
University of Primorska, UP FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia
University of Primorska, UP IAM, Muzejski trg 2, 6000 Koper, Slovenia
IMFM, Jadranska 19, 1000 Ljubljana, Slovenia
Received 29 September 2019, accepted 13 October 2019, published online 30 December 2019
Abstract
It is proved that in a finite field F of prime order p, where p is not one of finitely many
exceptions, for every polynomial f(x) ∈ F [x] of degree 4 that has a nonzero constant
term and is not of the form αg(x)2 there exists a primitive root β ∈ F such that f(β)
is a quadratic residue in F . This refines a result of Madden and Vélez from 1982 about
polynomials that represent quadratic residues at primitive roots.
Keywords: Finite field, polynomial, quadratic residues.
Math. Subj. Class.: 12E99
∗The authors wish to thank Ademir Hujdurović and Kai Yuan for helpful conversations about the material of
this paper.
†This work is supported in part by the National Natural Science Foundation of China (11671276).
‡Corresponding author. This work is supported in part by the Slovenian Research Agency (research program
P1-0285 and research projects N1-0038, N1-0062, J1-6720, J1-6743, J1-7051, J1-9110, J1-1695, and J1-1715).
§This work is supported in part by the Slovenian Research Agency (I0-0035, research program P1-0285 and
research projects N1-0038, N1-0062, J1-6720, J1-9108, and J1-1695), and in part by H2020 Teaming InnoRenew
CoE (grant no. 739574).
E-mail addresses: dushf@mail.cnu.edu.cn (Shaofei Du), klavdija.kutnar@upr.si (Klavdija Kutnar),
dragan.marusic@upr.si (Dragan Marušič)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 2 (2019) #P1.10
1 Introduction
The motivation for this paper is twofold: first refining the result of Madden and Vélez
about polynomials that represent quadratic residues at primitive roots [9], and in doing so
obtaining a tool with which hamiltonicity of certain families of vertex-transitive graphs of
order a product of two primes is proved via a structural analysis of their quotients with
respect to an automorphism of prime order. Such a connection between algebraic graph
theory and finite fields is not surprising, see, for example, [6, 14] for a similar application
of finite fields.
In 1969 Lovász [8] asked for a construction of a finite connected vertex-transitive graph
without a Hamilton path, that is, a path containing all vertices of the graph. This problem
has spurred quite a bit of interest in the mathematical community, resulting in a number of
papers affirming the existence of Hamilton paths and in some cases even Hamilton cycles
(see the survey paper [7]). The main obstacle to making a substantial progress with regards
to this problem is a lack of structural results for such graphs. Consequently, tools and
methods from other areas of mathematics applicable in this context are more than welcome.
Such is, for example, the case with the so-called polycirculant conjecture which states that
every 2-closed group contains a fixed-point-free automorphism of prime order (see, for
example, [3, 4, 10, 12, 13]). Fixed-point-free automorphism of prime order have been of
great practical use in constructions of Hamilton cycles in vertex-transitive graphs via the
so-called lifting cycle technique [1, 11]. And it is precisely here that the results of this
paper are of crucial importance as they allow a successful application of this technique for
a complete solution of Lovász problem for connected vertex-transitive graphs of order a
product of two primes (see [5]).
More precisely, the goal of this paper is to obtain a novel result on polynomials of
degree 4 over finite fields of prime order with regards to a polynomial representation of
quadratic residues at primitive roots, thus refining results from [9] (see Theorem 1.1). (The
set of nonzero quadratic residues modulo r, that is, nonzero elements of a finite field F of
order r that are congruent to a perfect square modulo r, will be called squares.)
Theorem 1.1. Let F be a finite field of prime order p, where p is an odd prime not given
in Tables 1 and 2. Then for every polynomial f(x) ∈ F [x] of degree 4 that has a nonzero
constant term and is not of the form αg(x)2 there exists a primitive root β ∈ F such that
f(β) is a square in F .
2 Polynomials of degree 4 over finite fields representing quadratic
residues
In early eighties, motivated by a question posed by Alspach, Heinrich and Rosenfeld [2]
in the context of decompositions of complete symmetric digraphs, Madden and Vélez [9]
investigated polynomials that represent quadratic residues at primitive roots. They proved
that, with finally many exceptions, for any finite field F of odd characteristic, for every
polynomial f(x) ∈ F [x] of degree r ≥ 1 not of the form αg(x)2 or αxg(x)2, there
exists a primitive root β such that f(β) is a nonzero square in F . It is the purpose of this
paper to refine their result for polynomials of degree 4. This will then be used in [5] in
the constructions of Hamilton cycles for some of the basic orbital graphs arising from the
action of PSL(2, p) on cosets of Dp−1. This refinement, stated in Theorem 1.1, will be
proved following a series of lemmas.
S. Du et al.: Polynomials of degree 4 over finite fields representing quadratic residues 3
The following result, proved in [9], is a basis of our argument and will be used through-
out this section.
Proposition 2.1 ([9, Corollary 1]). Let F be a finite field with pn elements. If s and t are
integers such that
(i) s and t are coprime,
(ii) a prime q divides pn − 1 if and only if q divides st, and
(iii) 2φ(t)/t > 1 + (rs− 2)pn/2/(pn − 1) + (rs+ 2)/(pn − 1),
then, given any polynomial f(x) ∈ F [x] of degree r, square-free and with nonzero constant
term, there exists a primitive root γ ∈ F such that f(γ) is a nonzero square in F .
Throughout this section let p be an odd prime and let q1 = 2, q2, . . . , qm be the increas-
ing sequence of prime divisors of p− 1 = qi11 q
i2
2 · · · qimm . As in [9] we define the following
functions with respect to this sequence:
d(n,m) = 2
(
1− 1
qn
)(
1− 1
qn+1
)
· · ·
(
1− 1
qm
)
, (2.1)
cr(n,m) = 2r
√
q1q2 · · · qn−1
qnqn+1 · · · qm
, (2.2)
and k(m) as the unique integer such that d(k(m) − 1,m) ≤ 1 < d(k(m),m). Hence
k(m) ≥ 2. Analogously the functions d and cr can be defined for any positive integers
r ≥ 1, n < m and an arbitrary sequence {q1, . . . , qm} of primes. The following lemma is
a generalization of [9, Lemma 3].
Lemma 2.2. Let {2 = q1, q2, . . . , qm} be a finite sequence of primes satisfying m ≥
2k(m) + 2, and let r = 4. Then
d(k(m) + 1,m)− cr(k(m) + 1,m) > 1. (2.3)
Proof. Since 2 ≤ k(m) ≤ m2 − 1, we have m ≥ 6. Since
d(k(m) + 1,m) =
(
1 +
1
qk(m) − 1
)
d(k(m),m) > 1 +
1
qk(m) − 1
,
(2.3) holds if
1 +
1
qk(m) − 1
− 2r
(
q1q2 · · · qk(m)
qk(m)+1qk(m)+2 · · · qm
) 1
2
> 1,
which may be rewritten in the following form
q2q3 · · · qk(m)(qk(m) − 1)2 <
1
128
qk(m)+1 · · · qm−1qm, (2.4)
in view of the fact that r = 4 and q1 = 2.
We divide the proof into two cases, depending on whether m ≥ 7 or m = 6.
4 Art Discrete Appl. Math. 2 (2019) #P1.10
Case 1. m ≥ 7.
Let Ω be the increasing sequence of all prime numbers and let
Jq = {q1 = 2, q2, q3, . . . , ql = q, ql+1, . . . , qm}
be a subsequence of Ω. Then we shall in fact prove a more general result:
q2q3 · · · ql(ql − 1)2 <
1
128
ql+1 · · · qm−1qm,
where m ≥ 7 and l ≤ m2 − 1 is any integer. To show this for the sequence Jq we define
a subsequence Iq = {w1 = 2, w2, w3, . . . , wl = q, wl+1, . . . , wm} of Ω not missing any
prime in Ω from the interval [w2, wm]. Then the lemma will be proven in case we show
that the following holds:
w2w3 · · ·wl(wl − 1)2 <
1
128
wl+1 · · ·wm−1wm, (2.5)
where m ≥ 7 and l ≤ m2 − 1 is any integer. If wm ≥ 128, then (2.5) is clearly true. So we
only need to consider primes that are smaller than or equal to 127. If
(m− l)− (l − 1 + 2) = m− 2l − 1 ≥ 2, (2.6)
then (2.5) holds provided wm−1wm > 128 holds. Note that this is true if wm ≥ 13, which
is the case since m ≥ 7. Next, note that for either m being even and l < m2 − 2 or m being
odd, (2.6) holds. So we may assume that m is even and that l = m/2− 1 ≥ 2.
Now we prove that (2.5) holds under this assumption for any even integer m ≥ 8 by
induction. Suppose first that m = 8. Then l = 3 and (2.5) rewrites as
w2w3(w3 − 1)2 <
1
128
w4w5w6w7w8. (2.7)
A computer search shows that (2.7) holds for all primes w8 ≤ 127. Suppose now that (2.5)
is true for an even integer m ≥ 8. Then we have
w2w3w4 · · ·wlwl+1(wl+1 − 1)2 = w2(w3 · · ·wlwl+1(wl+1 − 1)2)
< w2(wl+2wl+3 · · ·wmwm+1)
< (wl+2wl+3 · · ·wmwm+1)wm+2.
Therefore (2.5) is true for all even integers m ≥ 8 and then for all integers m ≥ 7. Hence
(2.4) holds, and so does (2.3).
Case 2. m = 6.
Now k(m) = 2. Inserting l = 2 and m = 6 in (2.5), we have
w2(w2 − 1)2 <
1
128
w3w4w5w6. (2.8)
A computer search for all the primes less than 131 shows that (2.8) does not hold only for
wk(m) = w2 ∈ {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 71}.
S. Du et al.: Polynomials of degree 4 over finite fields representing quadratic residues 5
For these exceptional cases, we go back to work on (2.3) directly. Let l = k(m) = 2 in
Jq . Let d(n,m)′ and c4(n,m)′ be the corresponding values for Iq as defined by functions d
and cr in (2.1) and (2.2). Then one can easily see that d(3, 6)′ ≤ d(3, 6) and that c4(3, 6)′ ≥
c4(3, 6), which implies d(3, 6)− c4(3, 6) ≥ d(3, 6)′ − c4(3, 6)′. Therefore, (2.3) holds for
Jq if it holds for Iq . So it suffices to check (2.3) for Iq . In fact, an additional computer
search for the set of primes less than 131 shows that for w1 = 2 and w2 being each of these
exceptional cases, (2.3) holds for Iq . This completes the proof of Lemma 2.2.
The following result proved in [9] will be needed in the next lemma.
Proposition 2.3 ([9, Lemma 5]). Let {2 = q1, q2, . . . , qm} be a finite sequence of primes
satisfying m ≤ 2k(m) + 1. Then m ≤ 9 and qk(m)−1 ≤ 5. In fact the sequence must
satisfy one of the following:
(i) k(m) = 4, qk(m)−1 = 5 and m = 9,
(ii) k(m) = 3, qk(m)−1 = 5 and m ≤ 7,
(iii) k(m) = 3, qk(m)−1 = 3 and m ≤ 7, or
(iv) k(m) = 2, qk(m)−1 = 2 and m ≤ 5.
Lemma 2.4. Let {2 = q1, q2, . . . , qm} be a finite sequence of primes satisfying m ≤
2k(m) + 1, and let p − 1 = qi11 q
i2
2 · · · qimm with qm ≥ 131. Then there exist s and t such
that
(i) s and t are coprime,
(ii) a prime q divides p− 1 if and only if q divides st, and
(iii) 2φ(t)/t > 1 + (4s− 2)√p/(p− 1) + (4s+ 2)/(p− 1).
Proof. Since m ≤ 2k(m) + 1 the four cases (i) – (iv) of Proposition 2.3 need to be con-
sidered. In each case, as in [9, Lemma 7], we will prescribe a choice for s (which then
determines t uniquely) and use the conditions in each of these four cases to find the lower
bound α for the expression (2φ(t)t−1 − 1), that is, (2φ(t)t−1 − 1) ≥ α. We will then be
able to use the assumption qm ≥ 131 to show that
α >
(4s− 2)√p+ 4s+ 2
p− 1
. (2.9)
Suppose first that Proposition 2.3(i) holds, that is, k(m) = 4, qk(m)−1 = 5 and m = 9.
Then q9 ≥ 131. Also, one can easily see that such a sequence of primes must begin with
q1 = 2, q2 = 3 and q3 = 5. Let s = 2 · 3 · 5 and t = q4q5 · · · q9. Then
2
φ(t)
t
− 1 ≥ 2
(
1− 1
7
)(
1− 1
11
)(
1− 1
13
)(
1− 1
17
)(
1− 1
19
)(
1− 1
131
)
− 1
≥ 0.27287.
Thus p satisfies (2.9) with α = 0.27287 and s = 30 if and only if p > 187899. Suppose
now that there is a prime p ≤ 187899 that satisfies the conditions of the case under analysis.
We know that 2 ·3 ·5 · q9 divides p−1 with q9 ≥ 131. However this requires q4q5q6q7q8 <
6 Art Discrete Appl. Math. 2 (2019) #P1.10
187899/(2 · 3 · 5 · 131) ≤ 48 which is clearly not possible, since q4q5q6q7q8 ≥ 7 · 11 · 13 ·
17 · 19 = 323323.
We now consider the other three cases of Proposition 2.3, that is, suppose that Propo-
sition 2.3(ii), (iii) or (iv) holds. In all three cases k(m) ≤ 3. By assumption q1 = 2, and
we now consider the various possibilities for q2. First, assume that q2 = 3 (note that this is
possible in the last two cases) and therefore m ≤ 7. We set s = 2 · 3 and t = q3q4q5q6q7.
Thus
2
φ(t)
t
− 1 ≥ 2
(
1− 1
5
)(
1− 1
7
)(
1− 1
11
)(
1− 1
13
)(
1− 1
131
)
− 1 ≥ 0.14206.
Now p satisfies (2.9) with α = 0.14206 and s = 6 if and only if p ≥ 24351. If p < 24351
we see that q3q4 · · · qm−1 < 24351/(2·3·131) < 31. Since qi ≥ 5 for i ∈ {3, 4, . . . ,m−1}
one can see that either m = 3 or m = 4. In other words, either t = q3 or t = q3q4, and
thus we can improve the value for α with
2
φ(t)
t
− 1 ≥ 2
(
1− 1
5
)(
1− 1
131
)
− 1 ≥ 0.58778.
In this case p satisfies (2.9) with α = 0.58778 if and only if p > 1490. If p ≤ 1490
observe that the assumption that 6qm divides p − 1 with qm ≥ 131 implies that q3 < 2, a
contradiction.
We now use the same approach for the case q2 = 5. We choose s = 2 · 5 and t =
q3q4 · · · qm. Here we have
2
φ(t)
t
− 1 ≥ 2
(
1− 1
7
)(
1− 1
11
)(
1− 1
13
)(
1− 1
17
)(
1− 1
131
)
− 1 ≥ 0.34361.
Hence p satisfies (2.9) with α = 0.34361 if and only if p > 12475. If, however, p ≤ 12475
then since 10qm divides p − 1 we have that q3 < 10, and so either m = 3 or m = 4 and
q3 = 7. In both cases we can improve the value for α since t = q2q3 or t = q3q4. In
particular,
2
φ(t)
t
− 1 ≥ 2
(
1− 1
7
)(
1− 1
131
)
− 1 ≥ 0.70119956.
In this case p satisfies (2.9) with α = 0.70119956 if and only if p > 3057. If p ≤ 3057
observe that the assumption that 10qm divides p− 1 with qm ≥ 131 implies that q3 < 3, a
contradiction.
Finally we consider the case q2 ≥ 7. Then, by Proposition 2.3, we have k(m) = 2 and
m ≤ 5. Here we choose s = 2 and use the same technique as above to complete the proof.
In particular, we have
2
φ(t)
t
− 1 ≥ 2
(
1− 1
7
)(
1− 1
11
)(
1− 1
13
)(
1− 1
131
)
− 1 ≥ 0.42758.
In this case p satisfies (2.9) with α = 0.42758 if and only if p > 243. If p ≤ 243
observe that the assumption that 2qm divides p − 1 with qm ≥ 131 implies that q3 < 2, a
contradiction.
In summary we have seen that given any finite sequence of primes with qm ≥ 131 we
can choose n in such a way that when s = q1q2 · · · qn and t = qn+1qn+2 · · · qm we have
2φ(t)
t
> 1 +
(4s− 2)
√
st+ 1
st
+
4s+ 2
st
, (2.10)
S. Du et al.: Polynomials of degree 4 over finite fields representing quadratic residues 7
completing the proof of Lemma 2.4.
In order to proceed with the proof of Theorem 1.1 we now need to identify all those se-
quences {2 = q1, q2, . . . , qm} with qm < 131 for which one cannot choose s = q1q2 · · · qn
and t = qn+1qn+2 · · · qm so as to satisfy (2.10). Since Lemma 2.2 holds for each qm we can
assume that for each of these sequences Proposition 2.3 applies. A computer search of these
finitely many sequences yields the exceptional sequences which are listed in Tables 1 and 2.
For each of these exceptional sequences we fix s = q1q2 · · · qn and t = qn+1qn+2 · · · qm,
and we then search for a constant k such that x > k implies the inequality
2φ(t)
t
> 1 +
2(2s− 1)
√
x
x− 1
+
4s+ 2
x− 1
. (2.11)
For each of these sequences Tables 1 and 2 give the smallest bound k obtained in this
way. The third column of these tables indicates for which choice of t the given bound k is
obtained:
Type 1 means that the bound k was obtained with t = qm−1qm,
Type 2 means that the bound was obtained with t = qm, and
Type 3 means that the bound was obtained with t = 1.
A computer search then identifies those primes that are smaller than or equal to the bound
k, as summarized in the proposition below.
Proposition 2.5. Let {2 = q1, q2, . . . , qm} be a finite sequence of primes satisfying m ≤
2k(m) + 1, and let p − 1 = qi11 q
i2
2 · · · qimm with qm < 131. If p is not listed in Tables 1
and 2 then there exist s and t such that
(i) s and t are coprime,
(ii) a prime q divides p− 1 if and only if q divides st, and
(iii) 2φ(t)/t > 1 + (4s− 2)√p/(p− 1) + (4s+ 2)/(p− 1).
We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1. It follows by Proposition 2.1 that a polynomial f(x) represents a
nonzero square at some primitive root in F if there exist s and t satisfying the following
three conditions:
(i) s and t are coprime,
(ii) a prime q divides p− 1 if and only if q divides st, and
(iii) 2φ(t)/t > 1 + (4s− 2)√p/(p− 1) + (4s+ 2)/(p− 1).
Our goal is therefore to show that such s and t exist for all odd primes p that are not listed
in Tables 1 and 2.
Let {q1 = 2, q2, . . . , qm} be an increasing sequence of prime divisors of p − 1. If
m ≤ 2k(m) + 1 then Lemma 2.4 applies for qm ≥ 131, and Proposition 2.5 applies for
qm < 131.
8 Art Discrete Appl. Math. 2 (2019) #P1.10
Table 1: The list of sequences not satisfying (2.10), part I.
p ≡ 1 (mod 4) ≤ k
Sequence T k Type p ≤ k with T with T , (p+ 1)/2 prime
2 55 3 3, 5, 17 5
2, 3, 5, 11 2458 1 331, 661, 991, 1321 661, 1321
2, 3, 5, 43 1622 1 1291 no
2, 3, 7, 17 1372 1 no no
2, 3, 5, 7, 13 7040 t = 455 2731 no
2, 3, 43 460 1 no no
2, 3, 31 496 1 373 no
2, 3, 61 435 1 367 no
2, 3, 5, 7, 23 5145 t = 805 4831 no
2, 3, 23 547 1 139, 277 277
2, 3, 67 430 1 no no
2, 3, 7, 13 1517 1 547, 1093 1093
2, 3, 17 632 1 103, 307, 409, 613 613
2, 3, 5, 13 2238 1 1171, 1951 no
2, 3, 11 788 2 67, 199, 397, 727 397
2, 7 99 2 29 no
2, 3, 13 739 2 79, 157, 313 157, 313
2, 3, 7 1023 2 43, 127, 337, 379, 673, 757
673, 757, 883, 1009
2, 23 65 2 47 no
2, 3, 5, 37 1656 1 no no
2, 5 133 2 11, 41, 101 no
2, 3, 5, 41 1632 1 1231 no
2, 3, 59 437 1 no no
2, 3, 53 444 1 no no
2, 3, 7, 19 1327 1 no no
2, 3, 5, 29 1727 1 no no
2, 17 69 2 no no
2, 11 78 2 23 no
2, 3, 5, 19 1921 1 571 no
2, 3, 41 464 1 no no
Suppose now that m ≥ 2k(m) + 2. Then, by Lemma 2.2, we have
d(k(m) + 1,m) > 1 + c4(k(m) + 1,m).
If we let s = q1q2 · · · qk(m) and t = qk(m)+1 · · · qm we have 2φ(t)/t = d(k(m) + 1,m),
S. Du et al.: Polynomials of degree 4 over finite fields representing quadratic residues 9
Table 2: The list of sequences not satisfying (2.10), part II.
p ≡ 1 (mod 4) ≤ k
Sequence T k Type p ≤ k with T with T , (p+ 1)/2 prime
2, 3, 5, 7, 11 8160 t = 385 2311, 4621 4621
2, 3, 5 1432 2 31, 61, 151, 181, 61, 541, 1201
241, 271, 541, 601,
751, 811, 1201
2, 3, 5, 47 1604 1 no no
2, 3, 5, 31 1705 1 no no
2, 3, 7, 23 1265 1 967 no
2, 5, 17 180 1 no no
2, 3, 11, 13 1130 1 859 no
2, 13 74 2 53 no
2, 5, 11 218 1 no no
2, 5, 13 200 1 131 no
2, 3, 37 475 1 223 no
2, 3, 5, 7 3649 1 211, 421, 631, 1051, 421
1471, 2521, 3361
2, 3, 5, 7, 19 5580 t = 665 no no
2, 3 384 2 7, 13, 19, 37, 73, 13, 37, 73, 193
97, 109, 163, 193
2, 5, 7 315 1 71, 281 no
2, 3, 5, 23 1819 1 691, 1381 1381
2, 3, 47 453 1 283 no
2, 3, 5, 7, 17 5905 t = 595 3571 no
2, 3, 29 506 1 349 no
2, 3, 7, 11 1646 1 463 no
2, 3, 5, 17 1995 1 1021, 1531 no
2, 29 63 2 59 no
2, 3, 19 596 1 229, 457 457
2, 19 68 2 no no
and
c4(k(m) + 1,m) = 8 ·
√
q1q2 · · · qk(m)
qk(m)+1qk(m)+2 · · · qm
=
8s
√
q1q2 · · · qm
≥ 8s√
p− 1
Since s is even and 4(p− 1) ≥ 4s ≥ 3 we may apply [9, Lemma 6] to see that
(4s− 2)√p
p− 1
≤ 4s√
p− 1
.
10 Art Discrete Appl. Math. 2 (2019) #P1.10
It follows that
2φ(t)
t
= d(k(m) + 1,m) ≥ 1 + c4(k(m) + 1,m) ≥ 1 +
8s√
p− 1
≥ 1 +
(4s− 2)√p
p− 1
+
4s+ 2
p− 1
.
(Note that the last inequality holds since p ≥ 7.)
For the sake of completeness we would like to add the following proposition (obtained
with a computer search) which deals with exceptional primes p not covered by Theorem 1.1
which are congruent to 1 modulo 4 and for which (p + 1)/2 is also a prime (primes given
in the last column of Tables 1 and 2). As is the case with Theorem 1.1 this proposition too
is used in the construction of Hamilton cycles in vertex-transitive graphs of order a product
of two primes in [5].
Proposition 2.6. Let F be a finite field of odd prime order p, and let k ∈ F . If
p ∈ {5, 13, 37, 61, 73, 157, 193, 277, 313, 397, 421, 457, 541,
613, 661, 673, 757, 1093, 1201, 1321, 1381, 4621}
then there exists a primitive root β of F such that f(β) = β4 + kβ2 + 1 is a square in F
except when
(p, k) ∈ {(5, 4), (13, 1), (13, 4), (13, 5), (13, 6), (13, 7), (13, 10),
(37, 3), (37, 28), (37, 29), (61, 18), (61, 37), (61, 40)}.
Amongst these exceptions only for (p, k) ∈ {(13, 1), (37, 28), (61, 18)} there exists ξ ∈
S∗ ∩ (S∗ + 1) such that k = 2(1 − 2ξ). In particular, ξ = 10 for (p, k) = (13, 1),
ξ = 12 for (p, k) = (37, 28), and ξ = 57 for (p, k) = (61, 18). Moreover, amongst these
exceptions only for (p, k) ∈ {(13, 1), (37, 28), (61, 18)} there exists ξ̄ ∈ S∗ ∩ (S∗ + 1)
such that k = −2(1 − 2ξ̄). In particular, ξ̄ = 4 for (p, k) = (13, 1), ξ̄ = 26 for (p, k) =
(37, 28), and ξ̄ = 5 for (p, k) = (61, 18).
References
[1] B. Alspach, Lifting Hamilton cycles of quotient graphs, Discrete Math. 78 (1989), 25–36, doi:
10.1016/0012-365x(89)90157-x.
[2] B. Alspach, K. Heinrich and M. Rosenfeld, Edge partitions of the complete symmetric directed
graph and related designs, Israel J. Math. 40 (1981), 118–128, doi:10.1007/bf02761904.
[3] P. J. Cameron, M. Giudici, G. A. Jones, W. M. Kantor, M. H. Klin, D. Marušič and L. A.
Nowitz, Transitive permutation groups without semiregular subgroups, J. London Math. Soc.
66 (2002), 325–333, doi:10.1112/s0024610702003484.
[4] P. J. Cameron (ed.), Research problems from the Fifteenth British Combinatorial Conference,
Discrete Math. 167/168 (1997), 605–615, doi:10.1016/s0012-365x(96)00212-9.
[5] S. F. Du, K. Kutnar and D. Marušič, Hamilton cycles in vertex-transitive graphs of order a
product of two primes, arXiv:1808.08553 [math.CO].
[6] Y.-Q. Feng, D.-W. Yang and J.-X. Zhou, Arc-transitive cyclic and dihedral covers of pentavalent
symmetric graphs of order twice a prime, Ars Math. Contemp. 15 (2018), 499–522, doi:10.
26493/1855-3974.1409.e54.
S. Du et al.: Polynomials of degree 4 over finite fields representing quadratic residues 11
[7] K. Kutnar and D. Marušič, Hamilton cycles and paths in vertex-transitive graphs—current di-
rections, Discrete Math. 309 (2009), 5491–5500, doi:10.1016/j.disc.2009.02.017.
[8] L. Lovász, The factorization of graphs, in: R. Guy, H. Hanam, N. Sauer and J. Schonheim
(eds.), Combinatorial Structures and Their Applications, Gordon and Breach, New York, 1970
pp. 243–246, proceedings of the Calgary International Conference on Combinatorial Structures
and their Applications held at the University of Calgary, Calgary, Alberta, Canada, June 1969.
[9] D. J. Madden and W. Y. Vélez, Polynomials that represent quadratic residues at primitive
roots, Pacific J. Math. 98 (1982), 123–137, http://projecteuclid.org/euclid.
pjm/1102734391.
[10] D. Marušič, On vertex symmetric digraphs, Discrete Math. 36 (1981), 69–81, doi:10.1016/
0012-365x(81)90174-6.
[11] D. Marušič, Hamiltonian circuits in Cayley graphs, Discrete Math. 46 (1983), 49–54, doi:
10.1016/0012-365x(83)90269-8.
[12] D. Marušič, Semiregular automorphisms in vertex-transitive graphs with a solvable group of
automorphisms, Ars Math. Contemp. 13 (2017), 461–468, doi:10.26493/1855-3974.1486.a33.
[13] G. Verret, Arc-transitive graphs of valency 8 have a semiregular automorphism, Ars Math.
Contemp. 8 (2015), 29–34, doi:10.26493/1855-3974.492.37d.
[14] Y. Wang and Y.-Q. Feng, Half-arc-transitive graphs of prime-cube order of small valencies, Ars
Math. Contemp. 13 (2017), 343–353, doi:10.26493/1855-3974.964.594.