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Contents
Perfect matchings, Hamiltonian cycles and edge-colourings in a class of
cubic graphs
Marién Abreu, John Baptist Gauci, Domenico Labbate,
Federico Romaniello, Jean Paul Zerafa . . . . . . . . . . . . . . . . . . . . 349
The search for small association schemes with noncyclotomic eigenvalues
Allen Herman, Roghayeh Maleki . . . . . . . . . . . . . . . . . . . . . . . 367
Comparing Wiener, Szeged and revised Szeged index on cactus graphs
Stefan Hammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
Component (edge) connectivity of pancake graphs
Xiaohui Hua, Lulu Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
Hamilton cycles in primitive graphs of order 2rs
Shaofei Du, Yao Tian, Hao Yu . . . . . . . . . . . . . . . . . . . . . . . . 417
Bootstrap percolation via automated conjecturing
Neal Bushaw, Blake Conka, Vinay Gupta, Aidan Kierans, Hudson Lafayette,
Craig Larson, Kevin McCall, Andriy Mulyar, Christine Sullivan,
Scott Taylor, Evan Wainright, Evan Wilson, Guanyu Wu, Sarah Loeb . . . . 441
On the existence of zero-sum perfect matchings of complete graphs
Teeradej Kittipassorn, Panon Sinsap . . . . . . . . . . . . . . . . . . . . . 455
The A-Möbius function of a finite group
Francesca Dalla Volta, Andrea Lucchini . . . . . . . . . . . . . . . . . . . 467
On adjacency and Laplacian cospectral switching non-isomorphic signed
graphs
Tahir Shamsher, Shariefuddin Pirzada, Mushtaq A. Bhat . . . . . . . . . . 481
The role of the Axiom of Choice in proper and distinguishing colourings
Marcin Stawiski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
Volume 23, Number 3, Fall/Winter 2023, Pages 349–508
xxiii

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P3.01 / 349–366
https://doi.org/10.26493/1855-3974.2672.73b
(Also available at http://amc-journal.eu)
Perfect matchings, Hamiltonian cycles and
edge-colourings in a class of cubic graphs
Marién Abreu
Dipartimento di Matematica, Informatica ed Economia,
Università degli Studi della Basilicata, Italy
John Baptist Gauci *
Department of Mathematics, University of Malta, Malta
Domenico Labbate , Federico Romaniello
Dipartimento di Matematica, Informatica ed Economia,
Università degli Studi della Basilicata, Italy
Jean Paul Zerafa †
Department of Technology and Entrepreneurship Education,
University of Malta, Malta and
Department of Computer Science, Faculty of Mathematics, Physics and Informatics
Comenius University, Mlynská Dolina, 842 48 Bratislava, Slovakia
Received 16 July 2021, accepted 25 August 2022, published online 6 January 2023
Abstract
A graph G has the Perfect-Matching-Hamiltonian property (PMH-property) if for each
one of its perfect matchings, there is another perfect matching of G such that the union of
the two perfect matchings yields a Hamiltonian cycle of G. The study of graphs that have
the PMH-property, initiated in the 1970s by Las Vergnas and Häggkvist, combines three
well-studied properties of graphs, namely matchings, Hamiltonicity and edge-colourings.
In this work, we study these concepts for cubic graphs in an attempt to characterise those
cubic graphs for which every perfect matching corresponds to one of the colours of a proper
3-edge-colouring of the graph. We discuss that this is equivalent to saying that such graphs
are even-2-factorable (E2F), that is, all 2-factors of the graph contain only even cycles.
The case for bipartite cubic graphs is trivial, since if G is bipartite then it is E2F. Thus,
we restrict our attention to non-bipartite cubic graphs. A sufficient, but not necessary,
condition for a cubic graph to be E2F is that it has the PMH-property. The aim of this work
*Corresponding author.
†The author was partially supported by VEGA 1/0743/21, VEGA 1/0727/22, and APVV-19-0308.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
350 Ars Math. Contemp. 23 (2023) #P3.01 / 349–366
is to introduce an infinite family of E2F non-bipartite cubic graphs on two parameters,
which we coin papillon graphs, and determine the values of the respective parameters for
which these graphs have the PMH-property or are just E2F. We also show that no two
papillon graphs with different parameters are isomorphic.
Keywords: Cubic graph, perfect matching, Hamiltonian cycle, 3-edge-colouring.
Math. Subj. Class. (2020): 05C15, 05C45, 05C70
1 Introduction
Let G be a simple connected graph of even order with vertex set V (G) and edge set
E(G). A k-factor of G is a k-regular spanning subgraph of G (not necessarily con-
nected). Two very well-studied concepts in graph theory are perfect matchings and Hamil-
tonian cycles, where the former is the edge set of a 1-factor and the latter is a connected
2-factor of a graph. For t ≥ 3, a cycle of length t (or a t-cycle), denoted by Ct =
(v1, . . . , vt), is a sequence of mutually distinct vertices v1, v2, . . . , vt with corresponding
edge set {v1v2, . . . , vt−1vt, vtv1}. For definitions not explicitly stated here we refer the
reader to [4]. A graph G admitting a perfect matching is said to have the Perfect-Matching-
Hamiltonian property (for short the PMH-property) if for every perfect matching M of
G there exists another perfect matching N of G such that the edges of M ∪ N induce a
Hamiltonian cycle of G. For simplicity, a graph admitting this property is said to be PMH.
This property was first studied in the 1970s by Las Vergnas [15] and Häggkvist [9], and for
more recent results about the PMH-property we suggest the reader to [2, 1, 3, 7, 8]. In [3], a
property stronger than the PMH-property is studied: the Pairing-Hamiltonian property, for
short the PH-property. Before proceeding to the definition of this property, we first define
what a pairing is. For any graph G, KG denotes the complete graph on the same vertex
set V (G) of G. A perfect matching of KG is said to be a pairing of G, and a graph G is
said to have the Pairing-Hamiltonian property if every pairing M of G can be extended
to a Hamiltonian cycle H of KG such that E(H) −M ⊆ E(G). Clearly, a graph having
the PH-property is also PMH, although the converse is not necessarily true. Amongst other
results, the authors of [3] show that the only cubic graphs admitting the PH-property are
the complete graph K4, the complete bipartite graph K3,3, and the cube Q3. However,
this does not mean that these are the only three cubic graphs admitting the PMH-property.
For instance, all cubic 2-factor Hamiltonian graphs (all 2-factors of such a graph form a
Hamiltonian cycle) are PMH (see for example [5, 6, 11, 12, 13]).
If a cubic graph G is PMH, then every perfect matching of G corresponds to one of the
colours of a (proper) 3-edge-colouring of the graph, and we say that every perfect matching
can be extended to a 3-edge-colouring. This is achieved by alternately colouring the edges
of the Hamiltonian cycle containing a predetermined perfect matching using two colours,
and then colouring the edges not belonging to the Hamiltonian cycle using a third colour.
However, there are cubic graphs which are not PMH but have every one of their perfect
matchings that can be extended to a 3-edge-colouring (see for example Figure 1). The
following proposition characterises all cubic graphs for which every one of their perfect
matchings can be extended to a 3-edge-colouring of the graph.
E-mail addresses: marien.abreu@unibas.it (Marién Abreu), john-baptist.gauci@um.edu.mt (John Baptist
Gauci), domenico.labbate@unibas.it (Domenico Labbate), federico.romaniello@unibas.it (Federico
Romaniello), zerafa.jp@gmail.com (Jean Paul Zerafa)
M. Abreu et al.: Perfect matchings, Hamiltonian cycles and edge-colourings . . . 351
b
b
b b
b
b
b b
b
b b
b
Figure 1: The bold dashed edges can be extended to a proper 3-edge-colouring but not to a
Hamiltonian cycle.
Proposition 1.1. Let G be a cubic graph admitting a perfect matching. Every perfect
matching of G can be extended to a 3-edge-colouring of G if and only if all 2-factors of G
contain only even cycles.
Proof. Let F be a 2-factor of G, and let M be the perfect matching E(G)− E(F ). Since
M can be extended to a 3-edge-colouring of G, F can be 2-edge-coloured, and hence F
does not contain any odd cycles. Conversely, let M ′ be a perfect matching of G, and let
F ′ be its complementary 2-factor, that is, E(F ′) = E(G) −M ′. Since F ′ contains only
even cycles, M ′ can be extended to a 3-edge-colouring, by assigning a first colour to all
of its edges and then alternately colouring the edges of the 2-factor F ′ using another two
colours.
We shall call graphs in which all 2-factors consist only of even cycles as even-2-
factorable graphs, denoted by E2F for short. In particular, from Proposition 1.1, if a cubic
graphG has the PMH-property, then it is also E2F. As in the proof of Proposition 1.1, in the
sequel, given a perfect matching M of a cubic graph G, the 2-factor obtained after deleting
the edges of M from G is referred to as the complementary 2-factor of M .
If a cubic graph is bipartite, then trivially, each of its perfect matchings can be ex-
tended to a 3-edge-colouring, since it is E2F. But what about non-bipartite cubic graphs?
In Table 1, we give the number of non-isomorphic non-bipartite 3-connected cubic graphs
(having girth at least 4) such that each one of their perfect matchings can be extended to
a 3-edge-colouring. As is the case of snarks (bridgeless cubic graphs which are not 3-
edge-colourable), these seem to be difficult to find, as one can notice after comparing these
numbers to the total number of non-isomorphic 3-edge-colourable (Class I) non-bipartite
3-connected cubic graphs having girth at least 4, also given in Table 1. The numbers shown
in this table were obtained thanks to a computer check done by Jan Goedgebeur, and the
data is sorted according to the cyclic connectivity of the graphs considered. We remark that
E2F cubic graphs having girth 3 can be obtained by applying a star product between an
E2F cubic graph of smaller order and the complete graph K4—this has been investigated
further by the last two authors in [14]. This is the reason why only graphs having girth
at least 4 are considered in this work. More results on star products (also known in the
literature as 3-cut connections) in cubic graphs can be found in [5, 6, 10, 11, 12, 13].
A complete characterisation of which cubic graphs are PMH is still elusive, so consider-
ing the Class I non-bipartite cubic graphs having the property that each one of their perfect
matchings can be extended to a 3-edge-colouring may look presumptuous. As far as we
352 Ars Math. Contemp. 23 (2023) #P3.01 / 349–366
Cyclic connectivity Total no. of graphs
3 4 5 6 E2F Class I
ratio
E2F : Class I
N
um
be
ro
fv
er
tic
es
8 / 1 / / 1 1 100%
10 / / / / 0 3 0%
12 2 5 2 / 9 17 52.94%
14 2 2 2 / 6 92 6.52%
16 35 56 4 / 95 716 13.27%
18 84 21 9 / 114 7343 1.55%
20 926 655 15 2 1598 93946 1.70%
22 2978 331 17 6 3332 1400203 0.24%
Table 1: The number of non-isomorphic non-bipartite 3-connected cubic graphs with girth
at least 4 which are E2F and Class I.
know this property and the corresponding characterisation problem were never considered
before and tackling the following problem seems a reasonable step to take.
Problem 1.2. Characterise the Class I non-bipartite cubic graphs for which each one of
their perfect matchings can be extended to a 3-edge-colouring, that is, are E2F.
We remark that although the PMH-property is an appealing property in its own right,
Problem 1.2 continues to justify its study in relation to cubic graphs. Observe that in the
family of cubic graphs, whilst snarks are not 3-edge-colourable, even-2-factorable graphs
are quite the opposite being “very much 3-edge-colourable”, since the latter can be 3-edge-
coloured by assigning a colour to one of its perfect matchings, and then alternately colour
the edges of the complementary 2-factor.
1.1 Cycle permutation graphs
Consider two disjoint cycles each of length t, referred to as the first and second t-cycles
and denoted by (x1, . . . , xt) and (y1, . . . , yt), respectively. Let σ be a permutation of the
symmetric group St on the t symbols {1, . . . , t}. The cycle permutation graph correspond-
ing to σ is the cubic graph obtained by considering the first and second t-cycles in which
xi is adjacent to yσ(i), where σ(i) is the image of i under the permutation σ.
b b
b b
b b
bb
b
b
b
b
b
bb
b
Figure 2: Two different drawings of the smallest non-bipartite E2F cubic graph.
M. Abreu et al.: Perfect matchings, Hamiltonian cycles and edge-colourings . . . 353
The smallest non-bipartite cubic graph (from Table 1) which is E2F is in fact a cycle
permutation graph corresponding to σ = (1 2) ∈ S4, where σ(1) = 2, σ(2) = 1, σ(3) =
3, and σ(4) = 4 (see Figure 2). This shows that the edges between the vertices of the first
and second 4-cycles of the cycle permutation graph are x1y2, x2y1, x3y3, x4y4. In what
follows we shall denote permutations in cycle notation and, for simplicity, fixed points shall
be suppressed. With the help of Wolfram Mathematica, in Table 2 we provide the number
of non-isomorphic non-bipartite cycle permutation graphs up to 20 vertices which are PMH
or just E2F. Recall that PMH cubic graphs are also E2F, and so, PMH cycle permutation
graphs should be searched for from amongst the cycle permutation graphs which are E2F.
We also remark that, in the sequel, cycle permutation graphs with total number of vertices
equal to twice an odd number are not considered because, in this case, the first and second
cycles form a 2-factor consisting of two odd cycles, and so they are trivially not E2F.
E2F PMH
N
o.
of
ve
rt
ic
es 8 1 0
12 5 1
16 28 2
20 175 0
Table 2: The number of non-isomorphic non-bipartite cycle permutation graphs with girth
at least 4 which are E2F and PMH.
This work is a first structured attempt at tackling Problem 1.2. We give an infinite
family of non-bipartite cycle permutation graphs which admit the PMH-property or are
just E2F. In Section 2, we generalise the smallest cubic graph which is E2F into a family of
non-bipartite cycle permutation graphs, namely papillon graphs Pr,ℓ (for r, ℓ ∈ N), whose
smallest member P1,1 is, in fact, the graph in Figure 2. We show that papillon graphs are
E2F for all values of r and ℓ (Theorem 2.3) and PMH if and only if both r and ℓ are even
(Theorem 3.8 and Theorem 3.9).
2 Papillon graphs
Let [n] = {1, . . . , n}, for some positive integer n.
Definition 2.1. Let r and ℓ be two positive integers. The papillon graph Pr,ℓ is the graph
on 4r + 4ℓ vertices such that V (Pr,ℓ) = {ui, vi : i ∈ [2r + 2ℓ]}, where:
(i) (u1, u2, . . . , u2r+2ℓ) is a cycle of length 2r + 2ℓ;
(ii) ui is adjacent to vi, for each i ∈ [2r + 2ℓ]; and
(iii) the adjacencies between the vertices vi, for i ∈ [2r + 2ℓ], form a cycle of length
2r + 2ℓ given by the edge set
{v2i−1v2i : i ∈ [r + ℓ]} ∪ {v2i−1v2i+2 : i ∈ [r + ℓ− 1] \ {s}}
∪ {v2v2s+2, v2s−1v2r+2ℓ−1},
where s = min{r, ℓ}.
354 Ars Math. Contemp. 23 (2023) #P3.01 / 349–366
Clearly, the two papillon graphs Pr,ℓ and Pℓ,r are isomorphic, and henceforth, without loss
of generality, we shall tacitly assume that r ≤ ℓ. The papillon graph Pr,ℓ for r ≥ 2 is
depicted in Figure 3. When r and ℓ are equal, say r = ℓ = n, the papillon graph Pr,ℓ is
said to be balanced, and simply denoted by Pn (see, for example, Figure 4). Otherwise,
Pr,ℓ is said to be unbalanced (see, for example, Figure 11). The (2r + 2ℓ)-cycle induced
by the sets of vertices {ui : i ∈ [2r + 2ℓ]} is referred to as the outer-cycle, whilst the
(2r + 2ℓ)-cycle induced by the vertices {vi : i ∈ [2r + 2ℓ]} is referred to as the inner-
cycle. The edges on these two (2r + 2ℓ)-cycles are said to be the outer-edges and inner-
edges accordingly, whilst the edges uivi are referred to as spokes. The edges u1u2r+2ℓ,
v2r−1v2r+2ℓ−1, v2v2r+2, u2ru2r+1 are denoted by a, b, c, d, respectively, and we shall also
denote the set {a, b, c, d} by X . The set X is referred to as the principal 4-edge-cut of Pr,ℓ.
b b
b b
u2r+2ℓ−1 u2r+2ℓ
v2r+2ℓ
b b
b b
u2
v2
b b
b b
v2r+2 v2r+1 b b
b b
v2r v2r−1
u2r−1
v2r+2ℓ−1
a
d
b c
u2r
u1
v1
T1Tr+ℓ
Tr+1 Tr
u2r+1u2r+2
Figure 3: The papillon graph Pr,ℓ, for ℓ ≥ r ≥ 2, and the 4-pole Tj , for j ∈ [r + ℓ].
The graph in Figure 2 is actually the smallest (balanced) papillon graph P1. In general,
since {ui : i ∈ [2r + 2ℓ]} and {vi : i ∈ [2r + 2ℓ]} induce two disjoint (2r + 2ℓ)-cycles in
Pr,ℓ, and since every vertex belonging to the outer-cycle is adjacent to exactly one vertex on
the inner-cycle, there exists an isomorphism π between the papillon graph Pr,ℓ and a cycle
permutation graph corresponding to some σ ∈ S2r+2ℓ satisfying π(xi) = ui and π(yi) =
vσ−1(i), for each i ∈ [2r + 2ℓ]. In fact, the papillon graph Pr,ℓ is the cycle permutation
graph, with (u1, . . . , u2r+2ℓ) as the first cycle, corresponding to the permutation:
• σ1,ℓ := (3 4) . . . (2ℓ+ 1 2ℓ+ 2), with fixed points 1 and 2, when ℓ ≥ 1;
• σ2,2 := (1 2)(3 4)(5 7)(6 8);
• σr,3 := (1 2) . . . (2r − 1 2r)(2r + 1 2r + 5)(2r + 2 2r + 6), with fixed points
2r + 3 and 2r + 4, when r ∈ {2, 3}; and
M. Abreu et al.: Perfect matchings, Hamiltonian cycles and edge-colourings . . . 355
• σr,ℓ := (1 2) . . . (2r−1 2r)(2r+1 2r+2ℓ−1)(2r+2 2r+2ℓ)(2r+3 2r+2ℓ−
3)(2r+4 2r+2ℓ−2) . . . (α β), when ℓ ≥ r ≥ 4, where (α β) = (2r+ℓ 2r+ℓ+2)
if ℓ is even, and (α β) = (2r + ℓ− 1 2r + ℓ+ 3) if ℓ is odd.
We remark that when r > 1, the above permutations has no fixed points when ℓ is even,
but, when ℓ is odd, 2r + ℓ and 2r + ℓ + 1 are fixed points, and thus, in this case, x2r+ℓ
is adjacent to y2r+ℓ, and x2r+ℓ+1 is adjacent to y2r+ℓ+1 in Pr,ℓ. Note that since σr,ℓ is
an involution for all positive integers r and ℓ, the isomorphism π mentioned above can be
rewritten as follows: π(xi) = ui and π(yi) = vσ(i), for each i ∈ [2r + 2ℓ]. The papillon
graph Pr,ℓ admits a natural automorphism ψ which exchanges the two cycles, given by
ψ(ui) = vσr,ℓ(i) and ψ(vi) = uσr,ℓ(i), for each i ∈ [2r + 2ℓ]. In fact, the function ψ is
clearly bijective. Moreover, it maps edges of the outer-cycle to edges of the inner-cycle (and
vice-versa), and maps spokes to spokes, since the edges uivi are mapped to uσr,ℓ(i)vσr,ℓ(i).
b b
b b
u11 u12
v12
b b
b b
u1 u2
v2v1
b b
b b
u10 v10
v9u9
b b
b b
v3 u3
u4v4
b b
b b
v8 v7
u7u8
b b
b b
v6 v5
u5u6
v11
a
d
b c
Figure 4: The balanced papillon graph P3 on 24 vertices.
Before proceeding, we introduce multipoles which generalise the notion of graphs.
This will become useful when describing papillon graphs. A multipole Z consists of a set
of vertices V (Z) and a set of generalised edges such that each generalised edge is either
an edge in the usual sense (that is, it has two endvertices) or a semiedge. A semiedge is a
generalised edge having exactly one endvertex. The set of semiedges of Z is denoted by ∂Z
whilst the set of edges of Z having two endvertices is denoted by E(Z). Two semiedges
are joined if they are both deleted and their endvertices are made adjacent. A k-pole is a
multipole with k semiedges. A perfect matchingM of a k-pole Z is a subset of generalised
edges of Z such that every vertex of Z is incident with exactly one generalised edge of
M . In what follows, we shall construct papillon graphs by joining together semiedges of
a number of multipoles. In this sense, given a perfect matching M of a graph G, and
a multipole Z used as a building block to construct G, we shall say that M contains a
semiedge e of the multipole Z , if M contains the edge in G obtained by joining e to
another semiedge in the process of constructing G.
356 Ars Math. Contemp. 23 (2023) #P3.01 / 349–366
The 4-pole Z with vertex set {z1, z2, z3, z4}, such that E(Z) induces the 4-cycle
(z1, z2, z3, z4) and with exactly one semiedge incident to each of its vertices is referred
to as a C4-pole (see Figure 5). For each i ∈ [4], let the semiedge incident to zi be denoted
by fi. The semiedges f1 and f2 are referred to as the upper left semiedge and the upper
right semiedge of Z , respectively. On the other hand, the semiedges f3 and f4 are referred
to as the lower left semiedge and the lower right semiedge of Z , respectively (see Figure 5).
b b
b b
Z
z1 z2
z3z4
f1 f2
f4f3
b b
b b
Tj
u2j−1 u2j
v2jv2j−1
e
j
1 e
j
2
e
j
4e
j
3
Figure 5: A C4-pole Z and the 4-pole Tj in Pr,ℓ.
For some integer n ≥ 1, let Z1, . . . ,Zn be n copies of the above C4-pole Z . For
each j ∈ [n], let V (Zj) = {zj1, zj2, zj3, zj4}, and let f j1 , f j2 , f j3 , f j4 be the semiedges of Zj
respectively incident to zj1, z
j
2, z
j
3, z
j
4 such that f
j
1 and f
j
2 are the upper left and upper right
semiedges of Zj , whilst f j3 and f j4 are the lower left and lower right semiedges of Zj . A
chain of C4-poles of length n ≥ 2, is the 4-pole obtained by respectively joining f j2 and f j4
(upper and lower right semiedges of Zj) to f j+11 and f j+13 (upper and lower left semiedges
of Zj+1), for every j ∈ [n − 1]. When n = 1, a chain of C4-poles of length 1 is just a
C4-pole. For simplicity, we shall refer to a chain of C4-poles of length n, as a n-chain of
C4-poles, or simply a n-chain. The semiedges f11 and f
1
3 (similarly, f
n
2 and f
n
4 ) are referred
to as the upper left and lower left (respectively, upper right and lower right) semiedges of
the n-chain. A chain of C4-poles of any length has exactly four semiedges. For simplicity,
when we say that e1, e2, e3, e4 are the four semiedges of a chain Z ′ ofC4-poles (possibly of
length 1), we mean that e1 and e2 are respectively the upper left and upper right semiedges
of Z ′, whilst e3 and e4 are respectively the lower left and lower right semiedges of the
same chain Z ′ (see Figure 6). The semiedges e1 and e2 (similarly, e3 and e4) are referred
to collectively as the upper semiedges (respectively, lower semiedges) of Z ′. In a similar
way, the semiedges e1 and e3 (similarly, e2 and e4) are referred to collectively as the left
semiedges (respectively, right semiedges) of Z ′.
b b
b b
e1
e3
b b
b b
b b
b b
e2
e4
Figure 6: A chain of C4-poles of length 3 having semiedges e1, e2, e3, e4.
M. Abreu et al.: Perfect matchings, Hamiltonian cycles and edge-colourings . . . 357
In order to construct the papillon graph Pr,ℓ using C4-poles as building blocks, for each
j ∈ [r + ℓ], we consider the 4-pole Tj arising from the cycle (u2j−1, u2j , v2j , v2j−1) of
Pr,ℓ, whose semiedges are ej1, ej2, ej3, ej4 as in Figure 5. The r-chain and ℓ-chain giving rise
to Pr,ℓ consist of T1, . . . , Tr (referred to as the right r-chain of Pr,ℓ), and Tr+1, . . . , Tr+ℓ
(referred to as the left ℓ-chain of Pr,ℓ), which have semiedges e11, er2, e13, er4, and er+11 , er+ℓ2 ,
er+13 , e
r+ℓ
4 , respectively. The papillon graph Pr,ℓ is then obtained by joining the semiedges
in pairs as follows: e11 to e
r+ℓ
2 , e
r
2 to e
r+1
1 , e
1
3 to e
r+1
3 , and e
r
4 to e
r+ℓ
4 .
2.1 Odd cycles and isomorphisms in the class of papillon graphs
In this section we shall discuss the presence and behaviour of odd cycles in papillon graphs.
Consider the balanced papillon graph Pn and let C be an odd cycle in Pn. Since cycles
intersect C4-poles in 2, 3 or 4 vertices, there must exist some t1 ∈ [2n], such that |V (C) ∩
V (Tt1)| = 3. Without loss of generality, assume that t1 ∈ [n], that is, Tt1 belongs to the
right n-chain of Pn. If t1 ̸∈ {1, n}, we must have exactly one of the following:
• |V (C) ∩ V (Ti)| = 4, for all i ∈ {1, . . . , t1 − 1}; or
• |V (C) ∩ V (Ti)| = 4, for all i ∈ {t1 + 1, . . . , n}.
Without loss of generality, assume that we either have t1 = 1, or |V (C) ∩ V (Ti)| = 4,
for all i ∈ {1, . . . , t1 − 1}. This implies that the number of vertices in C belonging to
∪t1i=1V (Ti) is odd and at least 3. Moreover, the edges a and c must belong to C. We claim
that b ̸∈ E(C). For, suppose that b ∈ E(C). Since X is a 4-edge-cut, d ∈ E(C) as well.
This implies that n > 1 and there exist:
• t2 ∈ {t1 + 1, . . . , n}, such that |V (C) ∩ V (Tt2)| = 3;
• s1 ∈ {n+ 1, . . . , 2n− 1}, such that |V (C) ∩ V (Ts1)| = 3; and
• s2 ∈ {s1 + 1, . . . , 2n}, such that |V (C) ∩ V (Ts2)| = 3.
Let Ω = {1, . . . , t1}∪{t2, . . . , n, n+1, . . . , s1}∪{s2, . . . , 2n}. If Ω\{t1, t2, s1, s2} ≠ ∅,
then for any j ∈ Ω\{t1, t2, s1, s2}, |V (C)∩V (Tj)| = 4. Additionally, for any k ∈ [2n]\Ω,
|V (C) ∩ V (Tk)| = 0. However, this means that C has even length, a contradiction. Thus,
{b, d} ∩ E(C) = ∅. As a result, C intersects none of the C4-poles Tt1+1, . . . , Tn, but
intersects each of the C4-poles Tn+1, . . . , Tn in exactly 2 or 4 vertices. Hence, the length
of C is at least 2n + 3. When n = 1, (u1, u2, v2, v4, u4) is a 5-cycle, and when n > 1,
(u1, u2, v2, v2n+2, u2n+2, u2n+3, u2n+4, . . . , u4n) is an odd cycle of length exactly 2n+3.
Therefore, a shortest odd cycle in Pn has length 2n + 3. By using similar arguments, a
shortest odd cycle in Pr,ℓ has length 2r + 3.
Remark 2.2. The papillon graph Pr,ℓ is not bipartite and has a shortest odd cycle of length
2r + 3.
Consequently, we can show that any two distinct papillon graphs Pr1,ℓ1 and Pr2,ℓ2
are not isomorphic, where by distinct we mean that (r1, ℓ1) ̸= (r2, ℓ2). Suppose not, for
contradiction. Since Pr1,ℓ1 ≃ Pr2,ℓ2 , we must have r1 + ℓ1 = r2 + ℓ2, and so if r1 = r2,
then this implies that ℓ1 = ℓ2, and conversely. Hence, r1 ̸= r2 and ℓ1 ̸= ℓ2. Thus, without
loss of generality, we can assume that r1 < r2. However, this means that a shortest odd
cycle in Pr1,ℓ1 (of length 2r1 + 3), is shorter than a shortest odd cycle in Pr2,ℓ2 (of length
2r2 + 3), a contradiction.
We are now in a position to give our first result.
358 Ars Math. Contemp. 23 (2023) #P3.01 / 349–366
Theorem 2.3. Every papillon graph Pr,ℓ is E2F.
Proof. Let Pr,ℓ be a counterexample to the above statement, and let M be a perfect match-
ing of Pr,ℓ whose complementary 2-factor contains an odd cycle C. As previously dis-
cussed, C must intersect some Tj , for some j ∈ [r+ ℓ], in exactly 3 (consecutive) vertices.
Without loss of generality, assume that these 3 vertices are u2j−1, u2j , v2j . This means that
both the left semiedges (ej1 and e
j
3) of Tj belong to this odd cycle. However, since C is in
the complementary 2-factor of M , the two edges u2j−1v2j−1 and v2j−1v2j (which do not
belong to E(C)) must both belong to M , a contradiction.
3 The PMH-property in papillon graphs
3.1 The balanced case r = ℓ
Let M be a perfect matching of the balanced papillon graph Pn. Since X = {a, b, c, d} is
a 4-edge-cut of Pn, |M ∩ X | ≡ 0 (mod 2), that is, |M ∩ X | is 0, 2 or 4. The following is
a useful lemma which shall be used frequently in the results that follow.
Lemma 3.1. Let M be a perfect matching of the balanced papillon graph Pn and let X be
its principal 4-edge-cut. If |M ∩ X | = k, then |M ∩ ∂Tj | = k, for each j ∈ [2n].
Proof. Let M be a perfect matching of Pn. We first note that the left semiedges of a C4-
pole are contained in a perfect matching if and only if the right semiedges of the C4-pole
are contained in the same perfect matching. The lemma is proved by considering three
cases depending on the possible values of k, that is, 0, 2 or 4. When n = 1, the result
clearly follows since X is made up by joining ∂T1 and ∂T2 accordingly. So assume n ≥ 2.
Case I: k = 0.
Since a and c do not belong to M , the left semiedges of T1 are not contained in M , and so
M cannot contain its right semiedges. Therefore, |M ∩ ∂T1| = 0. Consequently, the left
semiedges of T2 are not contained in M implying again that |M ∩ ∂T2| = 0. By repeating
the same argument up till the nth C4-pole, we have that |M ∩ ∂Tj | = 0, for every j ∈ [n].
By noting that c and d do not belong to M and repeating a similar argument to the 4-poles
in the left n-chain, we can deduce that |M ∩ ∂Tj | = 0 for every j ∈ [2n].
Case II: k = 4.
Since a and c belong toM , the left semiedges of T1 are contained inM , and soM contains
its right semiedges as well. Therefore, |M ∩ ∂T1| = 4. Consequently, the left semiedges
of T2 are contained in M implying again that |M ∩ ∂T2| = 4. As in Case I, by noting
that both c and d belong to M and repeating a similar argument to the 4-poles in the left
n-chain, we can deduce that |M ∩ ∂Tj | = 4 for every j ∈ [2n].
Case III: k = 2.
We first claim that when k = 2, M ∩X must be equal to {a, d} or {b, c}. For, suppose that
M ∩ X = {a, c}, without loss of generality. This means that the right semiedges of T1 are
also contained in M , implying that |M ∩ ∂T1| = 4. This implies that the left semiedges
of T2 are contained in M , which forces |M ∩ ∂Tj | to be equal to 4, for every j ∈ [2n]. In
particular, |M ∩ ∂Tn| = 4, implying that the edges b and d belong to M , a contradiction
since M ∩ X = {a, c}. This proves our claim. Since the natural automorphism ψ of Pn,
M. Abreu et al.: Perfect matchings, Hamiltonian cycles and edge-colourings . . . 359
which exchanges the outer- and inner-cycles, exchanges also {a, d} with {b, c}, without
loss of generality, we may assume thatM∩X = {a, d}. Since c ̸∈M , 1 ≤ |M∩∂T1| < 4.
But, ∂T1 corresponds to a 4-edge-cut in Pn, and so, by using a parity argument, |M ∩ ∂T1|
must be equal to 2, implying that exactly one of the right semiedges of T1 is contained in
M . This means that exactly one left semiedge of T2 is contained in M , and consequently,
by a similar argument now applied to T2, we obtain |M ∩ ∂T2| = 2. By repeating the same
argument and noting that Tn+1 has exactly one left semiedge (corresponding to the edge d)
contained in M , one can deduce that |M ∩ ∂Tj | = 2 for every j ∈ [2n].
The following two results are two consequences of the above lemma and they both
follow directly from the proof of Case III. In a few words, if a perfect matching M of
Pn intersects its principal 4-edge-cut in exactly two of its edges, then these two edges are
either the pair {a, d} or the pair {b, c}, and, for every j ∈ [2n], M contains only one pair
of semiedges of Tj which does not consist of the pair of left semiedges of Tj nor the pair
of right semiedges of Tj .
Corollary 3.2. Let M be a perfect matching of Pn and let X be its principal 4-edge-cut.
If |M ∩ X | = 2, then M ∩ X is equal to {a, d} or {b, c}.
Corollary 3.3. Let M be a perfect matching of Pn and let X be its principal 4-edge-cut
such that |M ∩X | = 2. For each j ∈ [2n], M contains exactly one of the following sets of
semiedges: {ej1, ej2}, {ej3, ej4}, {ej1, ej4}, {ej2, ej3}, that is, of all possible pairs of semiedges
of Tj , {ej1, ej3} and {ej2, ej4} cannot be contained in M .
In the sequel, the process of traversing one path after another shall be called concate-
nation of paths. If two paths P and Q have endvertices x, y and y, z, respectively, we write
PQ to denote the path starting at x and ending at z obtained by traversing P and then Q.
Without loss of generality, if x is adjacent to y, that is, P is a path on two vertices, we may
write xyQ instead of PQ.
Lemma 3.4. Let M1 be a perfect matching of Pn such that |M1 ∩ X | = 2.
(i) There exists a perfect matchingM2 of Pn such that |M2∩X | = 2 andM1∩M2 = ∅.
(ii) The complementary 2-factors of M1 and M2 are both Hamiltonian cycles.
Proof. (i) Since |M1 ∩ X | = 2, by Lemma 3.1 we get that |M1 ∩ ∂Tj | = 2 for every
j ∈ [2n]. For each j, let P (j) be the subgraph of Pn which is induced by E(Tj) −M1.
Note that ∪2nj=1V (P (j)) = V (Pn). By Corollary 3.3, each P (j) is a path of length 3.
Letting N be the unique perfect matching of Pn which intersects each E(P (j)) in exactly
two edges, we note that M1 ∩ N = ∅. Let M2 = E(Pn) − (M1 ∪ N). Since M1 and N
are two disjoint perfect matchings, M2 is also a perfect matching of Pn and, in particular,
M2 contains X − (M1 ∩ X ). Thus, |M2 ∩ X | = 2 and M1 ∩M2 = ∅, proving part (i).
(ii) Let M2 be as in part (i), that is, |M2 ∩ X | = 2 and M1 ∩M2 = ∅. When n = 1,
the result clearly follows. So assume n ≥ 2. For distinct i and j in [2n], let Q(i,j) be the
subgraph of Pn which is induced by M2 ∩{xy ∈ E(Pn) : x ∈ V (Ti), y ∈ V (Tj)}, that is,
E(Q(i,j)) is either empty or consists of exactly one edge, that is, Q(i,j) is a path of length
1. When M1 ∩ X = {a, d}, we can form a Hamiltonian cycle of Pn (not containing M1)
by considering the following concatenation of paths:
P (1)Q(1,2) . . . Q(n−1,n)P (n)Q(n,2n)P (2n)Q(2n,2n−1) . . . P (n+1)Q(n+1,1),
360 Ars Math. Contemp. 23 (2023) #P3.01 / 349–366
b b
b b
b b
b b
b b
b b
b b
b b
Figure 7: Perfect matching M1 (bold dashed edges) with |M1 ∩ X | = 2 and its comple-
mentary 2-factor (highlighted edges).
where Q(1,2) and Q(2n,2n−1) are respectively followed by P (2) and P (2n−1), and, Q(n,2n)
andQ(n+1,1) consist of the edges b and c, respectively. On the other hand, whenM1∩X =
{b, c}, we can form a Hamiltonian cycle of Pn (not containing M1) by considering the
following concatenation of paths:
P (1)Q(1,2) . . . Q(n−1,n)P (n)Q(n,n+1)P (n+1)Q(n+1,n+2) . . . P (2n)Q(2n,1),
whereQ(1,2) andQ(n+1,n+2) are respectively followed by P (2) and P (n+2), and,Q(n,n+1)
andQ(2n,1) consist of the edges d and a, respectively. Thus, the complementary 2-factor of
M1 is a Hamiltonian cycle. This is depicted in Figure 7. The proof that the complementary
2-factor of M2 is a Hamiltonian cycle follows analogously.
Proposition 3.5. Let n be a positive odd integer. Then, the balanced papillon graph Pn is
not PMH.
Proof. Consider the following perfect matching of the balanced papillon graph Pn:
M = ∪2ni=1{u2i−1u2i, v2i−1v2i}.
It is clear that when n = 1, the perfect matching M cannot be extended to a Hamiltonian
cycle of the balanced papillon graph P1. So assume that n ≥ 3. We claim that M cannot
be extended to a Hamiltonian cycle of Pn. For, let F be a 2-factor of Pn containing M .
Since u1u2 ∈ M and Pn is cubic, F contains exactly one of the following two edges:
u1u4n or u1v1. In the former case, if u1u4n ∈ E(F ), then, u2nu2n+1 and all the edges of
the outer- and inner-cycle will belong to F (at the same time, the choice of u1u4n forbids
all the spokes of Pn to belong to F ), yielding two disjoint cycles each of length 4n. In the
latter case, if u1v1 ∈ E(F ), then F must also contain all spokes uivi, for 1 < i ≤ 4n. In
fact, the subgraph induced by the set of spokes is exactly the complement of the 2-factor
obtained in the former case. Consequently, F will consist of 2n disjoint 4-cycles.
Consider Pn, with n ≥ 2, and let M be a perfect matching of Pn with M ∩ X = 0,
which by Lemma 3.1 implies that |M ∩ ∂Tj | = 0 for all j ∈ [2n]. Now consider j ∈
[2n] \ {n, 2n} and let T(j,j+1) denote a 2-chain composed of Tj and Tj+1. We say that
T(j,j+1) is symmetric with respect to M if exactly one of the following occurs:
M. Abreu et al.: Perfect matchings, Hamiltonian cycles and edge-colourings . . . 361
(i) {u2j−1v2j−1, u2jv2j , u2j+1v2j+1, u2j+2v2j+2} ⊂M ; or
(ii) {u2j−1u2j , v2j−1v2j , u2j+1u2j+2, v2j+1v2j+2} ⊂M .
If neither (i) nor (ii) occur, T(j,j+1) is said to be asymmetric with respect to M . This is
shown in Figure 8.
b b
b b
b b
b b
b b
b b
symmetric
b b
b b
b b
b b
b b
b b
b b
b b
asymmetric
b b
b b
Figure 8: Symmetric and asymmetric 2-chains with the bold dashed edges belonging toM .
Remark 3.6. Let n ≥ 2. Consider a perfect matching M1 of Pn such that M1 does
not intersect the principal 4-edge-cut X of Pn, that is, M1 ∩ X = ∅, and consider a 2-
chain of Pn, say T(j,j+1) with j ∈ [2n] \ {n, 2n}, having semiedges e1, e2, e3, e4, where
e1 = e
j
1, e2 = e
j+1
2 , e3 = e
j
3 and e4 = e
j+1
4 . Assume there exists a perfect matching M2
of Pn such that |M2 ∩ X | = 2 and M1 ∩M2 = ∅ (see Figure 9). If T(j,j+1) is symmetric
with respect to M1, then we have exactly one of the following instances:
M2 ∩ ∂T(j,j+1) = {e1, e2} (upper); or M2 ∩ ∂T(j,j+1) = {e3, e4} (lower).
Otherwise, if T(j,j+1) is asymmetric with respect to M1, then exactly one of the following
must occur:
M2 ∩ ∂T(j,j+1) = {e1, e4} (upper left, lower right); or
M2 ∩ ∂T(j,j+1) = {e2, e3} (upper right, lower left).
Notwithstanding whether T(j,j+1) is symmetric or asymmetric with respect to M1,
(M1 ∪M2) ∩ E(T(j,j+1)) induces a path (see Figure 9) which contains all the vertices of
V (T(j,j+1)), and whose endvertices are the endvertices of the semiedges inM2∩∂T(j,j+1).
Remark 3.7. Let n ≥ 2. Consider a perfect matching M1 of Pn such that M1 does not
intersect the principal 4-edge-cut X of Pn, that is, M1 ∩ X = ∅, and consider a 2-chain
of Pn, say T(j,j+1) with j ∈ [2n] \ {n, 2n}. Let M2 be the perfect matching of Pn such
that |M2 ∩ X | = 4. Clearly M1 ∩M2 = ∅. Notwithstanding whether T(j,j+1) is sym-
metric or asymmetric with respect to M1, we have that (M1 ∪M2) ∩ E(T(j,j+1)) induces
two disjoint paths of equal length (see Figure 10) whose union contains all the vertices of Tj
362 Ars Math. Contemp. 23 (2023) #P3.01 / 349–366
b b
b b
b b
b b
b b
b b
symmetric
b b
b b
b b
b b
b b
b b
b b
b b
asymmetric
b b
b b
e1
e3 e4
e2
e1
e3
e2
e4
e1
e3
e2
e4
e1
e3
e2
e4
Figure 9: 2-chains when M1 ∩X = ∅ and |M2 ∩X | = 2 (bold dashed edges belong to M1
and highlighted edges to M2).
and Tj+1. Let Q be one of these paths. We first note that Q contains exactly one vertex
from {uj , vj+1} and exactly one vertex from {uj+3, vj+2}. If T(j,j+1) is symmetric with
respect to M1, then Q contains uj if and only if Q contains uj+3. Otherwise, if T(j,j+1) is
asymmetric with respect to M1, then Q contains uj if and only if Q contains vj+2.
b b
b b
b b
b b
b b
b b
symmetric
b b
b b
b b
b b
b b
b b
b b
b b
asymmetric
b b
b b
Figure 10: 2-chains when M1 ∩ X = ∅ and |M2 ∩ X | = 4 (bold dashed edges belong to
M1 and highlighted edges to M2).
Theorem 3.8. Let n be a positive even integer. Then, the balanced papillon graph Pn is
PMH.
Proof. Let M1 be a perfect matching of Pn. We need to show that there exists a perfect
matching M2 of Pn such that M1 ∪M2 induces a Hamiltonian cycle of Pn. Three cases,
depending on the intersection of M1 with the principal 4-edge-cut X of Pn, are consid-
ered. If |M1 ∩ X | = 2, then, by Lemma 3.4, there exists a perfect matching N of Pn such
M. Abreu et al.: Perfect matchings, Hamiltonian cycles and edge-colourings . . . 363
that |N ∩ X | = 2 and M1 ∩ N = ∅. Moreover, the complementary 2-factor of N is a
Hamiltonian cycle. Since M1 is contained in the mentioned 2-factor, the result follows.
When |M1 ∩ X | = 4, we can define M2 to be the following perfect matching:
M2 = {u1v1, u2v2}
⋃
∪2nj=2{u2j−1u2j , v2j−1v2j}.
In fact, M1 ∪ M2 induces the following Hamiltonian cycle: (u1, v1, v4, . . . , v2n, v2n−1,
v4n−1, v4n, v4n−3, . . . , v2n+1, v2n+2, v2, u2, u3, u4, . . . , u4n), where v4 and v4n−3 are re-
spectively followed by v3 and v4n−2.
What remains to be considered is the case when |M1∩X | = 0. Clearly, |M2∩X | cannot
be zero, because, if so, choosing M2 to be disjoint from M1, M1 ∪M2 induces 2n disjoint
4-cycles. Therefore, |M2∩X | must be equal to 2 or 4. Let R = {T(1,2), . . . , T(n−1,n)} and
L = {T(n+1,n+2), . . . , T(2n−1,2n)} be the sets of 2-chains within the left and right n-chains
of Pn—namely the right and left n-chains each split into n2 2-chains. We consider two
cases depending on the parity of the number of 2-chains in L and R which are asymmetric
with respect to M1. Let the function Φ: R ∪ L → {−1,+1} be defined on the 2-chains
T ∈ R ∪ L such that:
Φ(T ) =
{
+1 if T is symmetric with respect to M1,
−1 otherwise.
Case 1: L and R each have an even number (possibly zero) of asymmetric 2-chains with
respect to M1.
We claim that there exists a perfect matching such that its union withM1 gives a Hamil-
tonian cycle of Pn. Since the number of asymmetric 2-chains in R is even,
∏
T ∈R Φ(T ) =
+1, and consequently, by appropriately concatenating paths as in Remark 3.6, there exists
a path R with endvertices u1 and u2n whose vertex set is ∪2ni=1{ui, vi} such that it contains
all the edges in M1 ∩ (∪ni=1E(Ti)). We remark that this path intersects exactly one edge of
{xy ∈ E(Pn) : x ∈ V (Tj), y ∈ V (Tj+1)}, for each j ∈ [n − 1]. By a similar reasoning,
since
∏
T ∈L Φ(T ) = +1, there exists a path L with endvertices u2n+1 and u4n whose ver-
tex set is ∪4ni=2n+1{ui, vi}, such that it contains all the edges inM1∩(∪2ni=n+1E(Ti)). Once
again, this path intersects exactly one edge of {xy ∈ E(Pn) : x ∈ V (Tj), y ∈ V (Tj+1)},
for each j ∈ {n+ 1, . . . , 2n− 1}. These two paths, together with the edges a and d form
the required Hamiltonian cycle of Pn containing M1, proving our claim. We remark that
this shows that there exists a perfect matching M2 of Pn such that M2 ∩ X = {a, d},
M1 ∩ M2 = ∅ and with M1 ∪ M2 inducing a Hamiltonian cycle of Pn. One can sim-
ilarly show that there exists a perfect matching M ′2 of Pn such that M ′2 ∩ X = {b, c},
M1 ∩M ′2 = ∅ and with M1 ∪M ′2 inducing a Hamiltonian cycle of Pn.
Case 2: One of L and R has an odd number of asymmetric 2-chains with respect to M1.
Without loss of generality, assume that R has an odd number of asymmetric 2-chains
with respect to M1, that is,
∏
T ∈R Φ(T ) = −1. Let M2 be the perfect matching of Pn
such that |M2∩X | = 4. We claim thatM1∪M2 induces a Hamiltonian cycle of Pn. Since∏
T ∈R Φ(T ) = −1, by appropriately concatenating paths as in Remark 3.7 we can deduce
that M1 ∪M2 contains the edge set of two disjoint paths R1 and R2, such that:
(i) |V (R1)| = |V (R2)| = 2n;
(ii) V (R1) ∪ V (R2) = ∪2ni=1{ui, vi};
364 Ars Math. Contemp. 23 (2023) #P3.01 / 349–366
(iii) the endvertices of R1 are u1 and v2n−1; and
(iv) the endvertices of R2 are v2 and u2n.
Next, we consider two subcases depending on the value of
∏
T ∈L Φ(T ). We shall be using
the fact that {u1u4n, v2n−1v4n−1, v2v2n+2, u2nu2n+1} = {a, b, c, d} = X ⊂M2.
Case 2a:
∏
T ∈L Φ(T ) = −1
As above, by Remark 3.7, we can deduce that M1 ∪M2 contains the edge set of two
disjoint paths L1 and L2, such that:
(i) |V (L1)| = |V (L2)| = 2n;
(ii) V (L1) ∪ V (L2) = ∪4ni=2n+1{ui, vi};
(iii) the endvertices of L1 are u2n+1 and v4n−1; and
(iv) the endvertices of L2 are v2n+2 and u4n.
The concatenation of the following paths and edges gives a Hamiltonian cycle of Pn con-
taining M1:
R1v2n−1v4n−1L1u2n+1u2nR2v2v2n+2L2u4nu1.
Case 2b:
∏
T ∈L Φ(T ) = +1.
Once again, by Remark 3.7 we can deduce that M1 ∪M2 contains the edge set of two
disjoint paths L1 and L2, such that:
(i) |V (L1)| = |V (L2)| = 2n;
(ii) V (L1) ∪ V (L2) = ∪4ni=2n+1{ui, vi};
(iii) the endvertices of L1 are u2n+1 and u4n; and
(iv) the endvertices of L2 are v2n+2 and v4n−1.
The concatenation of the following paths and edges gives a Hamiltonian cycle of Pn con-
taining M1:
R1v2n−1v4n−1L2v2n+2v2R2u2nu2n+1L1u4nu1.
This completes the proof.
3.2 The unbalanced case r < ℓ and final remarks
By following the proofs in Section 2, the results obtained for balanced papillon graphs are
now extended to unbalanced papillon graphs.
Theorem 3.9. The unbalanced papillon graph Pr,ℓ is PMH if and only if r and ℓ are both
even.
Proof. This is an immediate consequence of Proposition 3.5 and Theorem 3.8. In partic-
ular, when at least one of r and ℓ is odd, Pr,ℓ is not PMH because the perfect matching
∪r+ℓi=1{u2i−1u2i, v2i−1v2i} of Pr,ℓ (illustrated in Figure 11) cannot be extended to a Hamil-
tonian cycle.
M. Abreu et al.: Perfect matchings, Hamiltonian cycles and edge-colourings . . . 365
b b
b b
b b
b b
b b
b b
b b
b b
b b
b b
b b
b b
b
bb
b b b
b b
b b
b b
b b
b b
b
bb
b
Figure 11: P1,3 and P3,4: unbalanced papillon graphs are not always PMH. The above
perfect matchings do not extend to a Hamiltonian cycle.
Corollary 3.10. The papillon graph Pr,ℓ is PMH if and only if r and ℓ are both even.
Finally, we remark that since Pn is PMH for every even n ∈ N, balanced papillon
graphs provide us with examples of non-bipartite PMH cubic graphs which are cyclically
4-edge-connected and have girth 4 such that their order is a multiple of 16. Additionally,
by considering unbalanced papillon graphs, say P2,ℓ, for some even ℓ > 2, we can obtain
non-bipartite PMH cubic graphs having the above characteristics (that is, cyclically 4-edge-
connected and having girth 4) such that their order is 8ν, for odd ν ≥ 3.
It would also be very compelling to see whether there exist other 4-poles instead of the
C4-poles that can be used as building blocks when constructing papillon graphs and which
yield non-bipartite PMH or just E2F cubic graphs.
ORCID iDs
Marién Abreu https://orcid.org/0000-0003-3992-1029
John Baptist Gauci https://orcid.org/0000-0001-6584-8473
Domenico Labbate https://orcid.org/0000-0003-2597-7574
Federico Romaniello https://orcid.org/0000-0003-1166-3179
Jean Paul Zerafa https://orcid.org/0000-0002-3159-2980
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P3.02 / 367–390
https://doi.org/10.26493/1855-3974.2724.83d
(Also available at http://amc-journal.eu)
The search for small association schemes with
noncyclotomic eigenvalues*
Allen Herman † , Roghayeh Maleki ‡
Department of Mathematics and Statistics, University of Regina,
Regina Saskatchewan S4S 0A2, Canada
Received 9 November 2021, accepted 4 November 2022, published online 11 January 2023
Abstract
In this article we determine feasible parameter sets for (what could potentially be) com-
mutative association schemes with noncyclotomic eigenvalues that are of smallest possible
rank and order. A feasible parameter set for a commutative association scheme corresponds
to a standard integral table algebra with integral multiplicities that satisfies all of the pa-
rameter restrictions known to hold for association schemes. For each rank and involution
type, we generate an algebraic set for which any suitable integral solution corresponds to
a standard integral table algebra with integral multiplicities, and then try to find the small-
est suitable solution. The main results of this paper show the eigenvalues of association
schemes of rank 4 and nonsymmetric association schemes of rank 5 will always be cyclo-
tomic. In the rank 5 cases, the results rely on calculations done by computer for Gröbner
bases or for bases of rational vector spaces spanned by polynomials. We give several ex-
amples of feasible parameter sets for small symmetric association schemes of rank 5 that
have noncyclotomic eigenvalues.
Keywords: Association schemes, table algebras, character tables.
Math. Subj. Class. (2020): 05E30, 13P15
1 Introduction
This paper investigates the Cyclotomic Eigenvalue Question for commutative association
schemes that was posed by Simon Norton at Oberwolfach in 1980 [3]. This question asks if
*We would like to thank anonymous referees for carefully reading the manuscript and for their insightful
comments
†This author’s work was supported by an NSERC Discovery Grant.
‡Corresponding author.
E-mail addresses: allen.herman@uregina.ca (Allen Herman), rmaleki@uregina.ca (Roghayeh Maleki)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
368 Ars Math. Contemp. 23 (2023) #P3.02 / 367–390
the eigenvalues of all the adjacency matrices of relations in the scheme lie in a cyclotomic
number field, or equivalently if every entry of the character table (i.e., first eigenmatrix) of a
commutative association scheme is cyclotomic. Showing this is a straightforward exercise
for association schemes of rank 2 and 3. For commutative Schurian association schemes,
this property is a consequence of the character theory of Hecke algebras and the fact that
Morita equivalent algebras have isomorphic centers (see [9]). For commutative association
schemes that are both P - and Q-polynomial, it follows from the fact that the splitting field
of the scheme is quadratic extension of the rationals, a key ingredient of Bang, Dubickas,
Koolen, and Moulten’s proof of the Bannai-Ito conjecture ([2], see also [14]). Herman
and Rahnamai Barghi proved it for commutative quasi-thin schemes [12], which were later
shown by Muzychuk and Ponomarenko to always be Schurian [16]. Herman and Rahnamai
Barghi also showed the cyclotomic eigenvalue property holds for commutative association
schemes whose elements have valency ≤ 2 except for possibly one element of valency 3
and/or one element of valency > 4 [12, Theorem 3.3].
For association schemes in general we do not know if the character values have to be
cyclotomic, but we do have noncommutative examples for which the eigenvalues are not
cyclotomic – the smallest examples are two noncommutative Schurian association schemes
of order 26, and three noncommutative Schur rings of order 32 (in the latter case the corre-
sponding graphs are Cayley graphs on a nonabelian group of order 32).
In this article we investigate the cyclotomic eigenvalue question from a smallest coun-
terexample perspective. For a given rank and involution type, our approach will be to gen-
erate an algebraic set in a multivariate polynomial ring in variables corresponding to the
intersection numbers and character table parameters of such an association scheme. Each
suitable integer point in this algebraic set corresponds to a standard integral table algebra
with integral multiplicities (SITAwIM) that has the corresponding intersection matrices and
character table via its regular representation. We use the algebraic set to search for small
SITAwIMs of the given type that have some noncyclotomic eigenvalues.
The algebraic sets themselves are not easy to work with, as they are not monomial
and the number of variables and polynomial generators is too large for available computer
algebra systems to do efficient Gröbner basis calculations. After manually reducing the
algebraic sets with all available linear substitutions, we search for solutions by specifying
values for sufficiently many remaining parameters that the resulting algebraic set can be
resolved with a Gröbner basis calculation. Using this approach, we are able to show the an-
swer to the cyclotomic eigenvalue question is yes for all association schemes of rank 4 and
for both involution types of nonsymmetric association schemes of rank 5. For commutative
association schemes, the noncyclotomic eigenvalue property implies the Galois group of
the splitting field will be non-Abelian, and so there must be an orbit of size at least 3 in
its action on irreducible characters. We will say the Galois group acts k-point transitively
if the size of its largest orbit on irreducible characters is k. So for symmetric association
schemes of rank 5, noncylotomic eigenvalues can only occur when the Galois group of the
splitting field is 3- or 4-point transitive. When this action is 4-point transitive, the associ-
ation scheme will be pseudocyclic. This greatly reduces the number of cases we need to
consider, and our searches have been able to produce six feasible examples of orders less
than 1000, the smallest having order 249. When the action is 3-point transitive, the scheme
is not pseudocyclic, so the search space is much larger. We have been able to generate all
examples of order less than 100 and a few more with order less than 250, ten of which
satisfy all available feasibility criteria. The smallest of these feasible examples have order
A. Herman et al.: The search for small association schemes with noncyclotomic eigenvalues 369
35, 45, 76, and 93. From the partial classification of association schemes of order 35, we
know the order 35 example cannot be realized. The status of the larger feasible examples
is open.
2 Preliminaries
In this section, we review some background results that are needed in this work. Recall that
an involution ϕ of a finite-dimensional algebra A is a map ϕ : A → A such that ϕ◦ϕ = idA.
2.1 SITA parameters
An integral table algebra (A,B) is a finite-dimensional complex algebra A with distin-
guished basis B = {bi | i ∈ I = {0, 1, . . . , r − 1}} such that
(i) 1 ∈ B,
(ii) A has an involution ∗ : A → A that is additive, reverses multiplication, and acts as
complex conjugation on scalars,
(iii) B is ∗−invariant,
(iv) B produces non-negative integer structure constants (see 2.2),
(v) B satisfies the pseudo-inverse condition: for all bi, bj ∈ B, the coefficient of 1 in
bib
∗
j is positive if and only if bj = b
∗
i .
Note that since B∗ = B, the involution ∗ is a permutation of {0, 1, . . . , r − 1}. There-
fore, the action of the involution ∗ can be defined by (bi)∗ = bi∗ for all bi ∈ B.
In order to consider A as an algebra of square matrices over C, we identify the elements
of B with their left regular matrices in the basis B. The basis B is called standard when,
for all bi ∈ B, the coefficient of 1 in bib∗i is equal to the maximal eigenvalue of the regular
matrix bi. We refer to r = |B| as the rank of the table algebra, when the basis B is standard
we say that (A,B) is a standard integral table algebra, or SITA. The action of the involution
∗ on the basis B determines the involution type of the table algebra of a given rank.
The adjacency algebra of an association scheme is the prototypical example of a SITA,
as the defining basis of adjacency matrices is a standard basis. Conversely, the structure
constants determined by the basis of adjacency matrices of an association scheme deter-
mine a standard integral table algebra that is realizable as an association scheme. Many
open problems concerning missing combinatorial objects correspond to standard integral
table algebras that satisfy all the known conditions on their parameters for being realized
by an association scheme, but are yet to be actually constructed. We call such standard
integral table algebras (or their parameter sets) feasible.
Let P = (χi(bj))i,j be the character table of A with respect to the distinguished basis
B, whose rows are indexed by the irreducible characters of A and columns are indexed
by the basis B. As we can restrict ourselves to the commutative table algebras in this
paper, P will be an r × r matrix. We order the irreducible characters so that the entries
P0,j = χ0(bj) = δj , j = 0, 1, . . . , r − 1 are equal to the Perron-Frobenius eigenvalues of
the basis matrices (i.e., the degrees of standard basis elements, or in the association scheme
case, the valencies of the scheme relations). The order of a standard integral table algebra
370 Ars Math. Contemp. 23 (2023) #P3.02 / 367–390
is the sum of its degrees; that is, n =
∑r−1
j=0 δj . The multiplicity mi of each irreducible
character χi can be computed by the following formula [4]
r−1∑
j=0
|Pij |2
δj
=
n
mi
, for i = 0, 1, . . . , r − 1.
For table algebras, the multiplicity mi corresponds to the coefficient of χi when the stan-
dard feasible trace map ρ(
∑r−1
j=0 αjbj) = nα0 is expressed as a (positive) linear combina-
tion of the irreducible characters of A. We always have m0 = 1, but the other multiplicities
mi for i = 1, . . . , r−1 are only required to be positive real numbers. When the SITA is re-
alized by an association scheme, the standard feasible trace is the character corresponding
to the standard representation of the SITA, so the mi’s will be positive integers. This is just
one of the feasibility conditions for the parameters of an association scheme. In this way,
each feasible parameter set for association schemes determines a standard integral table
algebra with integral multiplicities, i.e., a SITAwIM.
A SITA is called pseudocyclic if its multiplicities mi for i > 0 are all equal to the
same positive constant m. By a result of Blau and Xu [18], pseudocyclic SITAs are also
homogeneous, that is, all degrees δi for i > 0 are equal to the same positive constant.
2.2 General conditions on SITA parameters
Let B = {b0, b1, ..., br−1} be the standard basis of a SITA (A,B). Denote the structure
constants relative to the basis B by (λijk)r−1i,j,k=0, so
bibj =
r−1∑
k=0
λijkbk, for all i, j ∈ {0, 1, . . . , r − 1}.
Let χ0(bi) = δi be the degree (or valency) of the basis element bi ∈ B, for all i ∈
{0, 1, . . . , r − 1}. When the algebra has a standard basis, we have δi = δi∗ = λi∗i0 (see
[5, Definition 1.3]).
Associativity of A and the pseudo-inverse condition on the standard basis can be used
to prove two general properties of the structure constants relative to B.
Lemma 2.1. For all i, j, k ∈ {0, 1, . . . , r − 1},
(i) λjki∗δi = λkij∗δj = λijk∗δk, and
(ii)
∑r−1
k=0 λjki = δj .
Proof. (i) By the associativity of multiplication we have the following condition on the
structure constants for all i, j, k, ℓ,m ∈ {0, 1, . . . , r − 1},∑
ℓ
λijℓλℓkm =
∑
ℓ
λiℓmλjkℓ
Now, fix k and let m = 0. Using the pseudo-inverse condition on B we have
λjki∗δi = λkij∗δj = λijk∗δk.
A. Herman et al.: The search for small association schemes with noncyclotomic eigenvalues 371
For (ii), we have that for all i, j ∈ {0, 1, . . . , r − 1}, bj∗bi =
∑r−1
k=0 λj∗ikbk =∑r−1
k=0 λi∗jk∗bk. Since χ0(b
∗
j ) = χ0(bj) and the degree map is an algebra homomorphism
from A to C, we have
δjδi = χ0(bj∗bi) =
r−1∑
k=0
λi∗jk∗δk,
which is equal by (i) to
∑r−1
k=0 λjkiδi. So, (ii) follows.
Note that Lemma 2.1(ii) tells us that every row sum of the left regular matrix of bj ∈ B
is equal to the constant δj .
Next, we consider restrictions on the parameters of a SITA imposed by its fusions. If
I ⊆ {1, . . . , r − 1}, we let bI =
∑
i∈I bi. When Λ = {{0}, I1, . . . , Is−1} is a partition of
{0, 1, . . . , r−1} for which BΛ = {b0, bI1 , . . . , bIs−1} is the basis of a table algebra (which
will automatically be the standard basis of a SITA in this case), then we say that BΛ is a
fusion of B, and conversely say that B is a fission of BΛ. The next lemma shows that every
SITA admits a rank 2 fusion.
Lemma 2.2. Every table algebra (A,B) with standard basis B = {b0, b1, . . . , br−1} of
rank r ≥ 3 has the trivial rank 2 fusion B{{0},{1,...,r−1}} = {b0, b1 + · · ·+ br−1}.
Proof. Let B+ =
∑r−1
j=0 bj . By [1] we have (B
+)2 = χ0(B
+)B+ = nB+. It follows that
((B−{b0})+)2 = nB+−2B++b0 = (n−2)B++b0 = (n−2)(B+−{b0})+(n−1)b0.
This implies {b0,B+ − b0} is a ∗-invariant subset of B that generates a 2-dimensional
subalgebra of A. The lemma follows.
The conditions imposed by fusion on the parameters of a commutative association
scheme were studied by Bannai and Song in [4]. For structure constants the conditions
are straightforward, for character table parameters the existence of a fusion imposes cer-
tain identities on partial row and column sums of P . Let Λ = {{0}, J1, . . . , Jd−1} be
the partition inducing the fusion BΛ = {b̃0, b̃J1 , . . . , b̃Jd−1} of our standard integral table
algebra basis B. If E = {e0, e1, . . . , er−1} is the basis of primitive idempotents of A,
then there is a (dual) partition Λ∗ = {{0},K1, . . . ,Kd−1} of {0, 1, . . . , r − 1}, unique
to the fusion, such that, if ẽ0 = e0 and ẽKi =
∑
k∈Ki ek for i = 1, . . . , d − 1, then
Ẽ = {ẽ0, ẽK1 , . . . , ẽKd−1} is the basis of primitive idempotents of the algebra CBΛ.
Let P̃ be the character table of the fusion BΛ, so the rows of P̃ are indexed by the
irreducible characters χ̃I for I ∈ Λ∗, and the columns of P̃ are indexed by the basis
elements bJ for J ∈ Λ. Let δ̃ = χ̃0 and δ = χ0 be the respective degree maps. Let
δ̃(bJ) = k̃J for all J ∈ Λ, and δ(bj) = kj for all j ∈ {1, . . . , r − 1}. Let m̃I and mi
denote the multiplicities of χ̃I and χi, respectively. Then we have the following identities
on partial row and column sums.
Theorem 2.3 (Theorem 1.4, [4]). Let J ∈ Λ and I ∈ Λ∗.
(i) For all j ∈ J ,
∑
i∈I miPi,j =
kjm̃I
k̃J
P̃I,J .
(ii) For all i ∈ I , then P̃I,J =
∑
j∈J Pi,j .
372 Ars Math. Contemp. 23 (2023) #P3.02 / 367–390
Proof. (i) We are assuming ẽI =
∑
i∈I ei. Using the formula for primitive idempotents in
a standard table algebra [1],
ẽI =
m̃I
n
∑
J
P̃I,J
k̃J
b̃∗J =
∑
J
∑
j∈J
m̃I P̃I,J
nk̃J
b∗j .
On the other hand,
ẽI =
∑
i∈I
ei =
∑
i∈I
mi
n
∑
j
Pi,j
kj
b∗j
=
∑
J
∑
j∈J
∑
i∈I
miPi,j
nkj
b∗j .
Therefore, for all j ∈ J ,
∑
i∈I miPi,j =
kjm̃I
k̃J
P̃I,J , as required.
(ii) When χi(b0) = 1, we have bjei = Pi,jei for all bj ∈ B. On the one hand,
b̃J ẽI = P̃I,J ẽI =
∑
i∈I
P̃I,Jei,
and on the other hand, assuming χi(b0) = 1 for all i ∈ I ,
b̃J ẽI =
∑
j∈J
∑
i∈I
bjei =
∑
i∈I
∑
j∈J
Pi,jei.
Therefore, P̃I,J =
∑
j∈J Pi,j for all i ∈ I .
We remark that the fusion condition (i) on partial column sums holds without change
for noncommutative table algebras. Note that standard character considerations tell us∑
j∈J mjχj(b0) = m̃J . Condition (ii) on partial row sums holds for the rows of P̃ indexed
by the χ̃I for which χi(b0) = 1 for all i ∈ I .
2.3 The splitting field and its Galois group
If (A,B) is a commutative integral table algebra with standard basis B = {b0, b1, . . . , br−1}
then the splitting field of (A,B) is the field K obtained by adjoining all the eigenvalues of
the regular matrices of elements of B to the rational field Q, or equivalently, the smallest
field K for which the character table P lies in Mr(K), the algebra of r × r matrices over
the field K. As each bj in B is a nonnegative integer matrix, K is also the unique mini-
mal Galois extension of Q that splits the characteristic polynomials of every bj ∈ B. Let
G = Gal(K/Q) be the Galois group of this splitting field. Since the irreducible characters
of A are also irreducible representations of A in the commutative case, G will act faithfully
on the set of irreducible characters of A via χσi (bj) = (χi(bj))
σ , for all χi ∈ Irr(A),
bj ∈ B, and σ ∈ G. In this way G permutes the rows of the character table P , as well
as the corresponding multiplicities. For SITAwIMs this means G can only permute sets of
irreducible characters with the same multiplicity.
By the Kronecker-Weber theorem, a necessary and sufficient condition for (A,B) to
be a standard integral table algebra with noncyclotomic character values is for this Galois
group G to be non-abelian. If G is non-abelian, the fact that the action of G on irreducible
characters of A is faithful forces there to be at least one orbit of size 3 or more.
A. Herman et al.: The search for small association schemes with noncyclotomic eigenvalues 373
Theorem 2.4 ([15]). Let (A,B) be an integral table algebra (possibly noncommutative).
Let H be the subset of G = Gal(K/Q) consisting of elements σ ∈ G whose action on the
character table P = (χ(b))χ,b can be realized by a permutation of the basis, that is, for all
b ∈ B there exists bσ ∈ B such that for all χ ∈ Irr(A), (Pχ,b)σ = χ(bσ) = Pχ,bσ . Then
H is a central subgroup of G.
Proof. To see that H is a subgroup of G, let σ, τ ∈ H , χ ∈ Irr(A), and b ∈ B. Then,
(Pχ,b)
στ = ((Pχ,b)
σ)τ = (Pχ,bσ )
τ = (Pχ,bστ ).
Therefore, στ ∈ H . Since G is finite, H is a subgroup. To see that H is central, let τ ∈ H ,
σ ∈ G, χ ∈ Irr(A), and b ∈ B. Then,
(Pχ,b)
στ = (Pχσ,b)
τ = (Pχσ,bτ ) = (Pχ,bτ )
σ = ((Pχb)
τ )σ = (Pχ,b)
τσ.
As the action of G on the rows of P is faithful, this implies στ = τσ, so H is contained in
Z(G).
Note that the above theorem always applies to commutative table algebras that are not
symmetric.
Corollary 2.5. Suppose (A,B) is a commutative table algebra that is not symmetric. Then
the restriction of complex conjugation to K is a nonidentity element of the center of G.
Proof. Commutative table algebras that are not symmetric always have at least one irre-
ducible character that is not real-valued. If otherwise, the identity χi(b∗j ) = χi(bj), for all
χi ∈ Irr(A) and bj ∈ B, would imply the character table P would not be invertible. For the
irreducible characters that are not real-valued, the restriction of complex conjugation to K
will be a non-identity element of G that is realized by the permutation of B corresponding
to the involution. By Theorem 2.4, this element lies in the center of G.
2.4 Algebraic sets for SITAwIMs of a given rank and involution type.
As indicated in the introduction, we will obtain our results by searching for suitable non-
negative integer points in an algebraic set (i.e., the solution set to a system of polynomial
equations) that is determined by the parameters of SITAwIMs of a given rank and invo-
lution type. To illustrate how the generating sets for the ideals corresponding to these
algebraic sets are produced, we give the type 4A1 case as an example. This is the algebraic
set corresponding to rank 4 SITAwIMs whose basis B contains one asymmetric pair, i.e.,
B = {b0, b1, b2, b∗2}. Using the properties of the involution, the row sum property, com-
mutativity of the algebra, and the fact that bibj =
∑3
k=0 λijkbk, the general form of the
regular matrices for the nontrivial elements of this basis is
374 Ars Math. Contemp. 23 (2023) #P3.02 / 367–390
b1 =

0 k1 0 0
1 k1 − 2x1 − 1 x1 x1
0 k1 − x2 − x3 x2 x3
0 k1 − x2 − x3 x3 x2
 ,
b2 =

0 0 0 k2
0 x1 k2 − x1 − x4 x4
1 x2 k2 − x2 − x5 − 1 x5
0 x3 k2 − x3 − x5 x5
 ,
b∗2 =

0 0 k2 0
0 x1 x4 k2 − x1 − x4
0 x3 x5 k2 − x3 − x5
1 x2 x5 k2 − x2 − x5 − 1
 .
Identifying entries in the matrix equations resulting from the identities that define the regu-
lar representation gives several linear and quadratic identities in the variables x1, . . . , x5, k1,
k2, each of which corresponds to a multivariate polynomial equaling 0. For example, iden-
tifying entries on both sides of the matrix equation
b1b2 = x1b1 + x2b2 + x3b
∗
2
gives a list of 8 polynomials:
−x2k2 + x4k1,
−x1k1 − x3k2 − x4k1 + k1k2,
−x1x3 − x2x4 + x23 − x3x4 + 2x3x5 − x3k2 + x4k1,
x1x3 − x1k1 + x2x4 − x2k2 − x23 + x3x4 − 2x3x5 − x4k1 + k1k2,
−x1x2 + x2x3 − x2x4 − x3x4 + 2x3x5 − x3k2 + x4k1 + x3,
x1x2 − x1k1 − x2x3 + x2x4 − x2k2 + x3x4 − 2x3x5 − x4k1 + k1k2 − x3,
−x21 + x1x3 − 2x1x4 + 2x1x5 − x2x4 + x3x4 − x3k2 + x4k1 − x4 + k2, and
x21 − x1x3 + 2x1x4 − 2x1x5 − x1k1 + x2x4 − x2k2 − x3x4 − x4k1 + k1k2 + x4 − k2.
We get similar lists of polynomials from the defining identities for b21, b1b
∗
2, b
2
2, b2b
∗
2, and
(b∗2)
2, and possibly still more from the commuting identities b1b2 = b2b1, b1b∗2 = b
∗
2b1,
and b2b∗2 = b
∗
2b2. In the type 4A1 case, up to sign, this process produces 16 distinct
polynomials.
When we add the integral multiplicities condition, it leads to extra trace identities that
can be added to our list. For each choice of multiplicities mi ∈ Z+, i = 1, . . . , r − 1, we
have an identity satisfied by our character table P resulting from the column orthogonality
relation:
kj +
r−1∑
i=1
miPi,j = 0.
In light of assumptions we can make regarding the Galois group, certain rows of P
will be Galois conjugate, and the sums of Pi,j’s corresponding to these rows have to be
rational algebraic integers, and thus integers. The multiplicities corresponding to Galois
conjugate rows are the same. Summing these rows of P gives the rational character table,
an integer matrix satisfying certain column and row orthogonality conditions. The entries
A. Herman et al.: The search for small association schemes with noncyclotomic eigenvalues 375
in each column of this matrix are bounded in terms of the first entry kj of the column, so
we can search for the possible rational character tables for a given choice of multiplicities.
For each possible rational character table, we can add linear trace identities
tr(bj) = kj +
r−1∑
i=1
Pi,j , j = 1, . . . , r − 1,
to our list of polynomials.
Let S be the set of polynomials produced by this process. Let I be the ideal gener-
ated by S, and let V(I) be the corresponding algebraic set. The regular matrices of any
SITAwIM of type 4A1 with the given choice of multiplicities corresponds naturally to a
point in V(I) with x1, . . . , x5 ∈ N and k1, k2 ∈ Z+. We will refer to this as a suitable in-
tegral point in the algebraic set. Conversely, any suitable integral point in V(I) corresponds
to a SITAwIM of this rank, involution type, and choice of multiplicities.
For example, if we assume m1 = m2 = m3 in the type 4A1 case, it adds the trace
identities tr(bj) = kj − 1 for j = 1, 2, 3, all of which reduce to x1 = x2. Since this pseu-
docyclic assumption implies the SITA is homogeneous, we also get m1 = k1 = k2. Other
linear identities, or ones that become linear after cancelling one of our nonzero degrees kj ,
can also be used to reduce the number of variables we need to consider. For example, in
the type 4A1 case, one of the elements of S is k2(k2 − 1− x2 − 2x5), so we can substitute
x2 = k2 − 1 − 2x5 and reduce the number of variables by one. After we reduce by all
available linear substitutions in the type 4A1 case, only one polynomial remains:
f(x5, k1) = 36x
2
5 − 24x5k1 + 4k21 + 32x5 − 11k1 + 7.
Putting this together with our linear substitutions, we can conclude that any pseudocyclic
SITAwIM of type 4A1 corresponds, via the above regular matrices, to an integer point
(x1, x2, x3, x4, x5, k1, k2) for which f(x5, k2) = 0, x5 ≥ 0, k1 = k2 > 0, x1 = x2 =
x4 = k1 − 2x5 − 1 ≥ 0, and x3 = 4x5 − k1 + 2 ≥ 0. This is an effective formula to
generate pseudocyclic SITAwIMs of type 4A1.
We refer the readers to [10] for the GAP implementation that produces the defining list
of polynomials for rank 4 and 5 SITAwIMs of each involution type.
3 Rank 4 SITAwIMs have cyclotomic eigenvalues
In this section we show that rank 4 SITAwIMs have cyclotomic eigenvalues. In this case
there are two involution types to consider: type 4A1 and type 4S.
Proposition 3.1. Rank 4 SITAwIMs with one asymmetric pair of standard basis elements
have cyclotomic eigenvalues. In fact, their eigenvalues lie in quadratic number fields.
Proof. Suppose (A,B) is a SITAwIM of rank 4 with B = {b0, b1, b2, b∗2}. If there were
nonidentity elements of B with noncyclotomic eigenvalues, the Galois group G of the
splitting field K would have to be 3-point transitive; i.e., a transitive subgroup of the group
Sym({χ1, χ2, χ3}). Since G would have to be non-abelian, it would have to be isomorphic
to S3. But |Z(G)| > 1 by Corollary 2.5, so this is a contradiction.
Since there are no 3-transitive groups with a central element of order 2, we can conclude
that G is cyclic of order 2, and therefore K is a quadratic extension of Q.
Theorem 3.2. Symmetric rank 4 SITAwIMs have cyclotomic eigenvalues.
376 Ars Math. Contemp. 23 (2023) #P3.02 / 367–390
Proof. Suppose (A,B) is a symmetric SITAwIM of rank 4 that has noncyclotomic eigen-
values. If G is the Galois group of its splitting field K, then as in the rank 4 one asymmetric
pair case, G must act as the full symmetric group on the set {χ1, χ2, χ3}. In particular, this
implies these three characters have the same multiplicity m. Therefore, n = 1 + 3m and
the character table P of (A,B) has the form
b0 b1 b2 b3 multiplicities
χ0 1 δ1 δ2 δ3 1
χ1 1 α1 β1 γ1 m
χ2 1 α2 β2 γ2 m
χ3 1 α3 β3 γ3 m
where {δ1, α1, α2, α3}, {δ2, β1, β2, β3}, and {δ3, γ1, γ2, γ3} are the eigenvalues of b1, b2,
and b3, respectively. If we apply Theorem 2.3(i) to the column of P labeled by b1, we get
m(α1 + α2 + α3) =
δ1
n− 1
(n− 1)(−1), so α1 + α2 + α3 =
−δ1
m
.
Since α1 + α2 + α3 is an algebraic integer, we must have that m divides δ1. Similarly m
divides δ2 and δ3. Since δ1 + δ2 + δ3 = n− 1 = 3m we must have δ1 = δ2 = δ3 = m.
Assume α1 is a noncyclotomic eigenvalue of b1. Since δ1 is an integral eigenvalue of b1,
the minimal polynomial µα1(x) of α1 in Q[x] will be a divisor of (x−α1)(x−α2)(x−α3).
If the degree of µα1(x) is 1 or 2, it would follow that α1 is rational or lies in a quadratic
extension of Q, which runs contrary to our assumption that it is not cyclotomic. So (x −
α1)(x−α2)(x−α3) is the minimal polynomial of α1 in Q[x]. This implies Q(α1, α2, α3)
is the splitting field of α1 over Q. Since α1 is not cyclotomic, this has to be an extension
of Q with [Q(α1, α2, α3) : Q] = 6. Since Q(α1, α2, α3) ⊆ K and [K : Q] = |G| = 6, we
must have K = Q(α1, α2, α3).
Now consider the left regular matrices of b1, b2, b3 in the basis B. For convenience we
write these in this form:
b1 =

0 m 0 0
1 u x1 x4
0 v x2 x5
0 w x3 x6
 , b2 =

0 0 m 0
0 x1 u
′ x7
1 x2 v
′ x8
0 x3 w
′ x9
 , b3 =

0 0 0 m
0 x4 x7 u
′′
0 x5 x8 v
′′
1 x6 x9 w
′′
 ,
where the u, v, and w entries are determined by the row sum criterion. Applying the
structure constant identities which define the left regular matrices produces one polynomial
identity in the variables x1, . . . , x9,m for each entry of the product bibj for i, j ∈ {1, 2, 3}.
Since B is pseudocyclic, we have three more trace identities. On the one hand, we
have tr(b1) = u + x2 + x6 = (m − 1 − x1 − x4) + x2 + x6, and on the other, tr(b1) =
δ1 + α1 + α2 + α3 = m + α1 + α2 + α3 = m − 1, so we can restrict our algebraic set
by adding the polynomial x2 + x6 − x1 − x4 to our list. Similar identities coming from
tr(b2) = m− 1 and tr(b3) = m− 1 show we can add the polynomials x1 + x9 − x2 − x8
and x4 + x8 − x6 − x9 to our list.
A. Herman et al.: The search for small association schemes with noncyclotomic eigenvalues 377
Next, we reduce our list of polynomials using all available linear substitutions and
obtain
x1 = v = m− x2 − x5 x6 = u′′ = m− x4 − x7
x2 = u
′ = m− x1 − x7 x8 = w′ = m− x3 − x9
x3 = x5 = x7 x9 = v
′′ = m− x5 − x8.
x4 = w = m− x3 − x6
This implies the matrix of b1 is
b1 =

0 m 0 0
1 u x1 x4
0 x1 x2 x3
0 x4 x3 x6
 ,
so by the row sum criterion x1 + x2 + x3 = x4 + x3 + x6 = m, which implies x1 + x2 =
x4 + x6. Since the identity we obtained by considering tr(b1) was x1 + x4 = x2 + x6, we
must conclude that x4 = x2 and hence x6 = x1. Similarly, we see that the matrix of b2 is
b2 =

0 0 m 0
0 x1 x2 x3
1 x2 v
′ x8
0 x3 x8 x9
 ,
therefore, x3 + x8 + x9 = m and we must have x1 + x2 = x8 + x9. Comparing this to
the identity x1 + x9 = x2 + x8 obtained by considering tr(b2), we see that x9 = x2 and it
then follows that x8 = x1.
Therefore, we have
b1 =

0 m 0 0
1 x3 − 1 x1 x2
0 x1 x2 x3
0 x2 x3 x1
 , b2 =

0 0 m 0
0 x1 x2 x3
1 x2 x3 − 1 x1
0 x3 x1 x2
 , and
b3 =

0 0 0 m
0 x2 x3 x1
0 x3 x1 x2
1 x1 x2 x3 − 1
 .
If we take Q to be the permutation matrix
Q =

1 0 0 0
0 0 1 0
0 0 0 1
0 1 0 0
 ,
then we have Q−1b1Q = b2, Q−1b2Q = b3, and Q−1b3Q = b1. It follows that the regular
matrices of b1, b2, and b3 have the same characteristic polynomial, and that the Galois group
G has a nontrivial central element of order 3 that permutes the corresponding columns in
the character table. But this is contrary to G being isomorphic to S3. We conclude that for
symmetric SITAwIMs of rank 4, the eigenvalues of basis elements must be cyclotomic.
Corollary 3.3. All association schemes of rank 4 have cyclotomic eigenvalues.
378 Ars Math. Contemp. 23 (2023) #P3.02 / 367–390
4 Rank 5 SITAwIMs
For rank 5 SITAwIMs we have three involution types to consider: type 5S, type 5A1, and
type 5A2.
4.1 Type 5A2
Theorem 4.1. Every rank 5 SITAwIM (A,B) with B = {b0, b1, b∗1, b3, b∗3} has cyclotomic
eigenvalues.
Proof. Let B = {b0, b1, b∗1, b3, b∗3} be the standard basis of a SITAwIM of rank 5, with
character table P , splitting field K, and Galois group G = Gal(K/Q). As the table
algebra is not symmetric, we know by Corollary 2.5 that G has a central element of order
2. If the character table P has a noncyclotomic entry, then G must also be a 3- or 4-point
transitive non-Abelian subgroup of Sym({χ1, χ2, χ3, χ4}), so the only possibility is for
G ≃ D4, the dihedral group of order 8. This implies the action of G on the last 4 rows of
P is 4-transitive, and so we must have that the multiplicities m1, m2, m3, and m4 are all
equal to the same positive integer m. So as in the symmetric rank 4 case, this implies the
table algebra is homogeneous: δ1 = δ2 = δ3 = δ4 = m.
This implies our regular matrices of B will have this pattern:
b1 =

0 0 m 0 0
1 x1 m− 1− x1 − x5 − x9 x5 x9
0 x2 m− x2 − x6 − x10 x6 x10
0 x3 m− x3 − x7 − x11 x7 x11
0 x4 m− x4 − x8 − x12 x8 x12
 ,
b∗1 =

0 m 0 0 0
0 m− x2 − x6 − x10 x2 x10 x6
1 m− 1− x1 − x5 − x9 x1 x9 x5
0 m− x4 − x8 − x12 x4 x12 x8
0 m− x3 − x7 − x11 x3 x11 x7
 ,
b3 =

0 0 0 0 m
0 x5 x10 x13 m− x5 − x10 − x13
0 x6 x9 x14 m− x6 − x9 − x14
1 x7 x12 x15 m− 1− x7 − x12 − x15
0 x8 x11 x16 m− x8 − x11 − x16
 ,
b∗3 =

0 0 0 m 0
0 x9 x6 m− x6 − x9 − x14 x14
0 x10 x5 m− x5 − x10 − x13 x13
0 x11 x8 m− x8 − x11 − x16 x16
1 x12 x7 m− 1− x7 − x12 − x15 x15
 .
In addition to the set of polynomial identities in the variables x1, . . . , x16,m we obtain
by applying the structure constant identities to these regular matrices, we again have the ad-
ditional trace identities coming from tr(b1) = tr(b3) = m− 1, which adds the polynomial
identities
x1 + x7 + x12 + 1 = x2 + x6 + x10 and x5 + x9 + x15 + 1 = x8 + x11 + x16
to our list. The result is a list of 13 distinct polynomial generators, up to sign, for an ideal
A. Herman et al.: The search for small association schemes with noncyclotomic eigenvalues 379
of Q[x1, . . . , x16,m]. The available linear substitutions are:

x16 = 2m− 6x1 − x2 − 2,
x15 = x1,
x14 = x8 = 3x1 + x2 + x3 + 1−m,
x13 = x11 = 2x1 − x3 + 1,
x12 = x9 = x7 = x5 =
m− 1
2
− x1,
x10 = x3,
x6 = x4 = m− x1 − x2 − x3,
(4.1)
so the reduced ideal now lies in Q[x1, x2, x3, x4,m]. With the above substitutions, the
regular matrices have this pattern:
b1 =

0 0 m 0 0
1 x1 x1 x5 x5
0 x2 x1 x4 x3
0 x3 x5 x5 x11
0 x4 x5 x8 x5
 , b3 =

0 0 0 0 m
0 x5 x3 x11 x5
0 x4 x5 x8 x5
1 x5 x5 x1 x1
0 x8 x11 x16 x1
 .
When we substitute x16 = x2 + y for an extra variable y, then reduce using the identities
in (4.1) and calculate the Gröbner basis for the resulting ideal with respect to an ordering
of variables with y maximal, we find that y2 is one of the elements of the basis.
Therefore, x16 must be equal to x2 for all points in our algebraic set. Substituting x2
for x16 in the first equation of (4.1) gives x2 = m− 3x1 − 1, substituting this into the last
equation makes x4 = 2x1 − x3 + 1 = x11, and substituting x2 = m − 3x1 − 1 into the
third equation gives us x8 = x3. Hence b1 and b3 have the same characteristic polynomial,
so they have the same eigenvalues. Consequently, b∗1 and b
∗
3 have the same four eigenvalues
as b1. This implies the Galois group of the splitting field will act transitively on the last
four columns of the character table, hence the Galois group will be Abelian. It follows that
any rank 5 SITAwIM whose standard basis has two distinct asymmetric pairs must have
cyclotomic eigenvalues.
4.2 Type 5A1
Theorem 4.2. Every SITAwIM (A,B) of involution type 5A1 has cyclotomic eigenvalues.
Proof. Let B = {b0, b1, b2, b3, b∗3} be the basis of a SITAwIM of type 5A1. By Corol-
lary 2.5, complex conjugation will be realized by a central element of the Galois group G
of the splitting field K of QB. If the character table P has an entry which is not cyclo-
tomic, then as in the type 5A2 case, we must have that G ≃ D4 and acts 4-transitively on
{χ1, χ2, χ3, χ4}. It follows that our SITAwIM (A,B) is both pseudocyclic and homoge-
neous.
380 Ars Math. Contemp. 23 (2023) #P3.02 / 367–390
This implies that the pattern for our regular matrices in this case will be:
b1 =

0 m 0 0 0
1 m− 1− x1 − 2x5 x1 x5 x5
0 m− x2 − 2x6 x2 x6 x6
0 m− x3 − x7 − x8 x3 x7 x8
0 m− x4 − x8 − x7 x4 x8 x7
 ,
b2 =

0 0 m 0 0
0 x1 m− x1 − 2x9 x9 x9
1 x2 m− 1− x2 − 2x10 x10 x10
0 x3 m− x3 − x11 − x12 x11 x12
0 x4 m− x4 − x11 − x12 x12 x11
 ,
b3 =

0 0 0 0 m
0 x5 x9 x13 m− x5 − x9 − x13
0 x6 x10 x14 m− x6 − x10 − x14
1 x7 x11 x15 m− 1− x7 − x11 − x15
0 x8 x12 x16 m− x8 − x12 − x16
 ,
b∗3 =

0 0 0 m 0
0 x5 x9 m− x5 − x9 − x13 x13
0 x6 x10 m− x6 − x10 − x14 x14
0 x8 x12 m− x8 − x12 − x16 x16
1 x7 x11 m− 1− x7 − x11 − x15 x15
 .
In addition to the polynomial identities obtained by applying the structure constant iden-
tities to these regular matrices, we again have three extra trace identities coming from
tr(b1) = tr(b2) = tr(b3) = m− 1:
x1+2x5 = x2+2x7, x1+2x11 = x2+2x10, and x5+x10+x15+1 = x8+x12+x16.
In addition to these, the other available linear substitutions, including those that become
linear after we cancel m > 0, are:
x16 = m− x8 − x12 − x15
x14 = x12 = m− x6 − x10 − x11
x13 = x8 = m− x5 − x6 − x7
x9 = x6 = x4 = x3 =
m
2
− x1
x2 = x1.
Since we have the identity x3 = m2 − x1, integrality of x3 and x1 implies m = 2k is
even. Making as many substitutions as possible, we can leave ourselves with a set of 11
nonlinear polynomials in Q[x1, x5, x15,m]. Using a computer, we calculate the Gröbner
basis of the ideal generated by these 11 polynomials, with m and x15 of highest weight.
If we set y = x15, the first polynomial in this Gröbner basis is the following element of
Q[m, y]:
W (y,m) =
1
5184
(5184y4 − 5184y3m+ 1944y2m2 − 324ym3 + 81
4
m4 + 7776y3
− 6160y2m+ 1622ym2 − 142m3 + 4292y2 − 2392ym+ 330m2
+ 1032y − 304m+ 91).
A. Herman et al.: The search for small association schemes with noncyclotomic eigenvalues 381
This means 5184 ·W (y,m) is an integer polynomial that must have a nonnegative solution
with y an integer and m an even integer. But when we substitute m = 2k, 5184 ·W (y, 2k)
has the form 2Q(y, k) + 1 for some polynomial Q(y, k) ∈ Z[y, k], and it is impossible
for Q(y, k) = − 12 to have an integral solution. This implies there are no pseudocyclic
SITAwIMs of involution type 5A1. In particular this means we can conclude that all rank
5 SITAwIMs whose standard basis has exactly one asymmetric pair will have cyclotomic
eigenvalues.
Corollary 4.3. The cyclotomic eigenvalue property holds for every nonsymmetric rank 5
association scheme.
4.3 Type 5S
If (A,B) is a symmetric rank 5 SITAwIM with noncyclotomic eigenvalues, the action of
the Galois group G = Gal(K/Q) of the splitting field K on the irreducible characters of
A will either be 3- or 4-point transitive. We begin with the 4-point transitive case.
4.3.1 Type 5S with 4-point transitive Galois group
Again in this case we deduce that (A,B) is pseudocyclic and homogeneous from G being
4-point transitive. In addition to the polynomial identities obtained by applying the struc-
ture constant identities to our regular matrices, we also have four trace identities coming
from tr(b1) = tr(b2) = tr(b3) = tr(b4) = m − 1. Altogether our initial list consists
of 124 polynomials in 25 variables. By applying all available linear substitutions, we can
reduce to a list of 21 polynomials in Q[x1, x2, x3, x5, x7, x14, x15,m]. Along the way our
first trace identity reduces to
2(x3 + x5 + x14 − x23) = m,
so we can conclude that m must be even. The Gröbner basis of this ideal generated by these
21 polynomials can be calculated in a few hours on our desktop implementation of GAP
[6], but is too complicated for any easy interpretation. Instead, reducing to a basis of the
rational span of these 21 polynomials leaves us with just 6 polynomials. Using these, we
run a search for suitable nonnegative integer solutions, letting m run over increasing even
integers and x1, x2, and x3 over the sets of three nonnegative integers that sum to at most
m. With these specifications, a Gröbner basis calculation solves for the possible values of
the four remaining variables efficiently. When a suitable nonnegative integer solution is
identified, we substitute its values back into our regular matrices and compute the factors
of their characteristic polynomials. Noncyclotomic eigenvalues are detected by applying
GAP’s GaloisType command [6] to irreducible factors of degree 3 or 4. Our searches
have found there is only one example with noncyclotomic eigenvalues with m ≤ 62. We
found more examples by carrying out a narrow search with the values of x1, x2, and x3
set to within a 10% error of m4 for 64 ≤ m ≤ 250. Up to permutation equivalence, we
have found six symmetric rank 5 SITAwIMs with 4-point transitive Galois group that have
noncyclotomic eigenvalues. In all of these cases the Galois group is isomorphic to S4.
(Here we give the factorizations of the characteristic polynomials of their basis elements,
from these it is possible to recover the character table P numerically, and from that their
other parameters.)
382 Ars Math. Contemp. 23 (2023) #P3.02 / 367–390
Noncyclotomic SITAwIMs of type 5S: 4-point transitive examples
n = 249 : (x− 62)(x4 + x3 − 93x2 − 57x+ 12),
(x− 62)(x4 + x3 − 93x2 − 306x+ 261),
(x− 62)(x4 + x3 − 93x2 − 306x− 237),
(x− 62)(x4 + x3 − 93x2 − 140x+ 925)
n = 321 : (x− 80)(x4 + x3 − 120x2 − 341x− 242),
(x− 80)(x4 + x3 − 120x2 − 20x+ 2968),
(x− 80)(x4 + x3 − 120x2 − 301x− 400),
(x− 80)(x4 + x3 − 120x2 + 301x+ 1042)
n = 473 : (x− 118)(x4 + x3 − 177x2 − 266x+ 279),
(x− 118)(x4 + x3 − 177x2 − 266x+ 3117),
(x− 118)(x4 + x3 − 177x2 + 680x− 667),
(x− 118)(x4 + x3 − 177x2 + 207x+ 4536)
n = 633 : (x− 158)(x4 + x3 − 237x2 − 356x+ 10897),
(x− 158)(x4 + x3 − 237x2 − 145x+ 11108),
(x− 158)(x4 + x3 − 237x2 + 1754x− 3451),
(x− 158)(x4 + x3 − 237x2 − 778x+ 5411)
n = 785 : (x− 196)(x4 + x3 − 294x2 − 1619x− 1524),
(x− 196)(x4 + x3 − 294x2 − 49x+ 20456),
(x− 196)(x4 + x3 − 294x2 + 1521x+ 3186),
(x− 196)(x4 + x3 − 294x2 + 736x+ 7896)
n = 993 : (x− 248)(x4 + x3 − 372x2 + 931x− 128),
(x− 248)(x4 + x3 − 372x2 + 931x+ 9802),
(x− 248)(x4 + x3 − 372x2 + 2917x− 6086),
(x− 248)(x4 + x3 − 372x2 + 1924x+ 7816).
For all of these examples, the noncyclotomic character table demands a certain alge-
braic structure of the Wedderburn decomposition of QB. If the character table of (A,B) is
P = (Pi,j)
4
i,j=0 = (χi(bj))
4
i,j=0, then
• for all j ∈ {1, 2, 3, 4}, the four 4-dimensional primitive extension fields Q(P1,j),
Q(P2,j), Q(P3,j), and Q(P4,j) are pairwise distinct and Galois conjugate over Q;
• for all i ∈ {1, 2, 3, 4}, the four primitive extension fields Q(Pi,1), Q(Pi,2), Q(Pi,3),
and Q(Pi,4) are equal; and
• for all i, j ∈ {1, 2, 3, 4}, QB ≃ Q⊕Q(Pi,j) as Q-algebras.
Another interesting fact is that the field of Krein parameters will be equal to the splitting
field K, this is the minimal field of realization for the dual intersection matrices.
In the last section we explain how to verify that these six SITAwIMs satisfy all the
known feasibility conditions for being an association scheme. The first one is the smallest
rank 5 example with 4-point transitive Galois group, we present its parameters in detail
here.
Theorem 4.4. The smallest symmetric rank 5 SITAwIM with noncyclotomic eigenvalues
for which the Galois group of the splitting field is 4-point transitive has order 249. Up to
permutation equivalence, its standard basis is given by:
A. Herman et al.: The search for small association schemes with noncyclotomic eigenvalues 383
B =
b0, b1 =

0 62 0 0 0
1 15 14 12 20
0 14 16 17 15
0 12 17 18 15
0 20 15 15 12
 , b2 =

0 0 62 0 0
0 14 16 17 15
1 16 18 16 11
0 17 16 11 18
0 15 11 18 18
 ,
b3 =

0 0 0 62 0
0 12 17 18 15
0 17 16 11 18
1 18 11 18 14
0 15 18 14 15
 , b4 =

0 0 0 0 62
0 20 15 15 12
0 15 11 18 18
0 15 18 14 15
1 12 18 15 16

 .
The character table of (A,B) is shown below. The roots of the degree 4 polynomials
above have been approximated to six significant digits using Wolfram|Alpha [17].
P =

1 62 62 62 62
1 9.45706 −4.83450 −8.21429 2.59173
1 0.165779 −7.32957 10.6401 −4.47634
1 −0.777430 10.45989 −2.18457 −8.49789
1 −9.84541 0.704180 −1.24127 9.38250
 .
Since this algebra is self-dual, the second eigenmatrix is obtained by setting Qi,j = Pj,i for
i = 1, 2, 3, 4 and leaving the first row and column alone. The nontrivial dual intersection
matrices are as follows, with irrational entries approximated to six significant digits:
L∗1 =

0 62 0 0 0
1 16.2247 17.5718 15.3191 11.8843
0 17.5718 10.8695 18.0841 15.4745
0 15.3191 18.0841 13.9307 14.6661
0 11.8843 15.4745 14.6661 19.9751
 ,
L∗2 =

0 0 62 0 0
0 17.5718 10.8695 18.0841 15.4745
1 10.8695 18.3339 16.0173 15.7793
0 18.0841 16.0173 11.1233 16.7753
0 15.4745 15.7793 16.7753 13.9710
 ,
L∗3 =

0 0 0 62 0
0 15.3191 18.0841 13.9307 14.6661
0 18.0841 16.0173 11.1233 16.7753
1 13.9307 11.1233 17.5255 18.4206
0 14.6661 16.7753 18.4206 12.1381
 ,
L∗4 =

0 0 0 0 62
0 11.8843 15.4745 14.6661 19.9751
0 15.4745 15.7793 16.7753 13.9710
0 14.6661 16.7753 18.4206 12.1381
1 19.9751 13.9710 12.1381 14.9159
 .
Remark 4.5. One might ask if there are metric association schemes of rank 5 with noncy-
clotomic splitting fields that have 4-point transitive Galois groups. With our method, this
can be resolved by setting x4, x5, x9, x10, x17 = 0, calculating the Gröbner basis, and us-
ing known intersection array restrictions to bound tridiagonal entries of b1. This approach
allows one to make the same conclusion as Blau and Xu obtain for pseudocyclic metric
association schemes in general, that the intersection array has to be [2, 1, 1, 1; 1, 1, 1, 1]
[18, Theorem 5.4]. But the splitting field of this association scheme has a 3-point transitive
abelian Galois group, so the answer is no.
384 Ars Math. Contemp. 23 (2023) #P3.02 / 367–390
4.3.2 Type 5S with 3-point transitive Galois group
The other possibility for a symmetric SITAwIM of rank 5 with noncyclotomic eigenvalues
is the case where the Galois group of the splitting field is non-abelian and acts 3-point
transitively, so must be isomorphic to S3. Let (A,B) be such a SITAwIM, with splitting
field K and Galois group G, and suppose the orbits of G on the irreducible characters of
A are {χ0}, {χ1}, and {χ2, χ3, χ4}. In this situation the table algebra is not necessarily
pseudocyclic, nor does it have to be homogeneous, so we do not have as many linear
substitutions available to reduce our algebraic set initially. Instead, to find the SITAwIMs of
a given order, we can first make a list of possible rationalized character tables for SITAwIMs
of that order. The rationalized character table is an integer matrix with columns indexed by
B and rows are indexed by the sums of irreducible characters of A up to Galois conjugacy
over Q. In our 3-point transitive case, it takes this form:
b0 b1 b2 b3 b4 multiplicities
χ0 1 δ1 δ2 δ3 δ4 1
χ1 1 a1 a2 a3 a4 m1
χ2 + χ3 + χ4 3 t1 t2 t3 t4 3m2
The rows and columns of the rationalized character table satisfy orthogonality relations
induced by those of the usual character table. In our case the orthogonality relations give
the following identities:
• δ1 + δ2 + δ3 + δ4 = m1 + 3m2 = n− 1;
• a1 + a2 + a3 + a4 = −1;
• t1 + t2 + t3 + t4 = −3;
• 1 + a
2
1
δ1
+
a22
δ2
+
a23
δ3
+
a24
δ4
= nm1 ;
• 3 + a1t1δ1 +
a2t2
δ2
+ a3t3δ3 +
a4t4
δ4
= 0;
• δ1 +m1a1 +m2t1 = 0;
• δ2 +m1a2 +m2t2 = 0;
• δ3 +m1a3 +m2t3 = 0; and
• δ4 +m1a4 +m2t4 = 0.
These identities are subject to the restrictions 1 ≤ m1,m2, δ1, δ2, δ3, δ4, and −δi ≤
ai ≤ δi for i = 1, 2, 3, 4, and −3δi ≤ ti ≤ 3δi for i = 1, 2, 3, 4. So a straightfor-
ward search will produce all the rationalized character tables possible whose associated
SITAwIM would have degree n.
Given a rationalized character table, we get four linear trace identities tr(bi) = δi +
ai + ti, i = 1, 2, 3, 4 that can be added to our list of polynomial generators. This helps
us to reduce our search space enough to allow the search and Gröbner basis calculations
techniques to uncover suitable nonnegative solutions to the system and produce regular
matrices for a SITAwIM with this rationalized character table. This approach has two
A. Herman et al.: The search for small association schemes with noncyclotomic eigenvalues 385
computational barriers, which have limited our ability to guarantee a complete account
only for orders up to 100. First, since we must consider every possibility for m1 and m2
with 1 +m1 + 3m2 = n, the number of possible rational character tables of a given order
can be very large and time-consuming to generate, and for almost all of these we find no
SITAwIM. Secondly, the values of the xi’s are not as limited as they are in the homogeneous
case, so when the minimum δi is large, the search space for all the values of x1, x2, and x3
we need to check grows in size exponentially.
Our complete search for orders up to 100 found six examples. Their multiplicities and
factorizations of the characteristic polynomials of their basis elements are as follows:
Noncyclotomic SITAwIMs of type 5S: 3-point transitive examples
n = 35 : m1 = 4,m2 = 10, µbi (x) = (x− 4)(x+ 1)(x
3 − 6x+ 2), (x− 6)2(x+ 1)3,
(x− 12)(x+ 3)(x3 − 12x− 2), (x− 12)(x+ 3)(x3 − 12x+ 12);
n = 45 : m1 = 8,m2 = 12, µbi (x) = (x− 4)
2(x+ 1)3, (x− 8)(x+ 1)(x3 − 12x+ 14),
(x− 8)(x+ 1)(x3 − 12x+ 4), (x− 24)(x+ 3)(x3 − 18x+ 18);
n = 76 : m1 = 18,m2 = 19, µbi (x) = (x− 3)
2(x+ 1)3, (x− 18)(x+ 1)(x3 − 27x− 18),
(x− 18)(x+ 1)(x3 − 27x− 42), (x− 36)(x+ 2)(x3 − 36x− 48);
n = 88a : m1 = 66,m2 = 7, µbi (x) = (x− 3)
4(x+ 1), (x− 14)(x)(x3 + 2x2 − 72x− 16),
(x− 35)(x)(x3 + 5x2 − 120x− 360), (x− 35)(x)(x3 + 5x2 − 120x+ 80);
n = 88b : m1 = 66,m2 = 7, µbi (x) = (x− 3)
4(x+ 1), (x− 21)(x)(x3 + 3x2 − 96x− 384),
(x− 21)(x)(x3 + 3x2 − 96x− 472), (x− 42)(x)(x3 + 6x2 − 120x− 784); and
n = 93 : m1 = 2,m2 = 30, µbi (x) = (x− 12)(x+ 6)(x
3 − 15x+ 2), (x− 20)(x+ 10)
(x3 − 21x− 16), (x− 30)(x+ 15)(x3 − 24x+ 8), (x− 30)2(x+ 1)3.
Narrow searches of orders 101 to 250, the first with δ1 ≤ 4 and at least two of δ2, δ3,
and δ4 equal, and the second with δ1 ≤ 12, a1 = k1, and at least two of δ2, δ3, and δ4 equal
produced a few more examples:
n = 116 : m1 = 58,m2 = 19, µbi (x) = (x− 1)
4(x+ 1), (x− 19)(x)(x3 + x2 − 48x+ 72)
(x− 19)(x)(x3 + x2 − 48x− 44), (x− 76)(x)(x3 + 4x2 − 72x− 32);
n = 129 : m1 = 86,m2 = 14, µbi (x) = (x− 2)
4(x+ 1), (x− 28)(x)(x3 + 2x2 − 99x+ 150)
(x− 28)(x)(x3 + 2x2 − 99x− 108), (x− 70)(x)(x3 + 5x2 − 135x− 75);
n = 165 : m1 = 32,m2 = 44, µbi (x) = (x− 4)
2(x+ 1)3, (x− 32)(x+ 1)(x3 − 48x− 32),
(x− 32)(x+ 1)(x3 − 48x− 112), (x− 96)(x+ 3)(x3 − 72x− 144);
n = 189 : m1 = 20,m2 = 56, µbi (x) = (x− 8)
2(x+ 1)3, (x− 20)(x+ 1)(x3 − 30x− 20),
(x− 80)(x+ 4)(x3 − 75x+ 70), (x− 80)(x+ 4)(x3 − 75x− 200);
n = 190 : m1 = 18,m2 = 57, µbi (x) = (x− 9)
2(x+ 1)3, (x− 36)(x+ 2)(x3 − 48x+ 32),
(x− 36)(x+ 2)(x3 − 48x+ 112), (x− 108)(x+ 6)(x3 − 72x+ 144);
n = 217 : m1 = 30,m2 = 62, µbi (x) = (x− 6)
2(x+ 1)3, (x− 60)(x+ 2)(x3 − 75x− 100),
(x− 60)(x+ 2)(x3 − 75x− 170), (x− 90)(x+ 3)(x3 − 90x− 180);
n = 231a : m1 = 32,m2 = 66, µbi (x) = (x− 6)
2(x+ 1)3, (x− 32)(x+ 1)(x3 − 48x− 96),
(x− 96)(x+ 3)(x3 − 96x− 352), (x− 96)(x+ 3)(x3 − 96x− 128);
n = 231b : m1 = 32,m2 = 66, µbi (x) = (x− 6)
2(x+ 1)3, (x− 32)(x+ 1)(x3 − 48x+ 16),
(x− 96)(x+ 3)(x3 − 96x− 128), (x− 96)(x+ 3)(x3 − 96x+ 208).
386 Ars Math. Contemp. 23 (2023) #P3.02 / 367–390
Example 4.6. The smallest noncyclotomic symmetric rank 5 SITAwIM with order n = 35
has regular matrices b0,
b1 =

0 4 0 0 0
1 0 0 0 3
0 0 0 2 2
0 0 1 2 1
0 1 1 1 1
 , b2 =

0 0 6 0 0
0 0 0 3 3
1 0 5 0 0
0 1 0 2 3
0 1 0 3 2
 , b3 =

0 0 0 12 0
0 0 3 6 3
0 2 0 4 6
1 2 2 4 3
0 1 3 3 5
 , and
b4 =

0 0 0 0 12
0 3 3 3 3
0 2 0 6 4
0 1 3 3 5
1 1 2 5 3
 .
Its first and second eigenmatrices (with irrationals approximated to six significant dig-
its) are as follows:
P =

1 4 6 12 12
1 −1 6 −3 −3
1 −2.60168 −1 −0.167055 2.768734
1 0.339877 −1 3.54461 −3.88448
1 2.26180 −1 −3.37755 1.11575
 , and
Q =

1 4 10 10 10
1 −1 −6.50420 0.849692 5.65451
1 4 −5/3 −5/3 −5/3
1 −1 −0.139212 2.95384 −2.81463
1 −1 2.30728 −3.23707 0.929791
 .
Its dual intersection matrices, again with irrational entries approximated to six signifi-
cant digits, are: L∗0 = b0,
L∗1 =

0 4 0 0 0
1 3 0 0 0
0 0 2/3 5/3 5/3
0 0 5/3 2/3 5/3
0 0 5/3 5/3 2/3
 ,
L∗2 =

0 0 10 0 0
0 0 5/3 25/6 25/6
1 2/3 0.0541562 2.59972 5.67949
0 5/3 2.59972 3.51139 20/9
0 5/3 5.67946 20/9 0.431651
 ,
L∗3 =

0 0 0 10 0
0 0 25/6 5/3 25/6
0 5/3 2.59972 3.51139 20/9
1 2/3 3.51139 2.50545 2.31644
0 5/3 20/9 2.31169 3.79463
 ,
L∗4 =

0 0 0 0 10
0 0 25/6 25/6 5/3
0 5/3 5.67946 20/9 0.431651
0 5/3 20/9 2.31649 3.79463
1 2/3 0.431651 3.79463 4.10706
 .
5 Checking feasibility
In this section we review the feasibility checks we have applied to the parameters of the
noncyclotomic symmetric rank 5 SITAwIMs identified in the previous section. The pa-
rameters include the regular (a.k.a. intersection) matrices bi (i ∈ {0, 1, . . . , r − 1}), the
A. Herman et al.: The search for small association schemes with noncyclotomic eigenvalues 387
character table (first eigenmatrix) P , the dual character table (second eigenmatrix) Q, and
the dual intersection matrices (Krein parameters) L∗i = (κijk)
r−1
k,j=0 (i ∈ {0, 1, . . . , r−1}).
For commutative association schemes, we consider these to be equivalent since knowledge
of any one of these determines the other.
We have tested our examples on the following feasibility conditions, which apply to
general symmetric association schemes:
• the handshaking lemma: for i, j ∈ {1, . . . , r − 1}, if i ̸= j, then (bi)i,jkj must be
even (see [11, Lemma 7]);
• realizability of all closed subsets and quotients;
• the triangle count condition: for j = 1, . . . , r − 1, 1
6
r−1∑
i=0
miP
3
i,j = t ∈ N;
• the absolute bound condition: for i ∈ {0, . . . , r − 1}
∑
k;qijk ̸=0
mk ≤
{
mimj i ̸= j(
mi+1
2
)
i = j;
• nonnegativity of Krein parameters: (L∗i )j,k ≥ 0 for i, j, k ∈ {0, . . . , r − 1}; and
• Martin and Kodalen’s Gegenbauer polynomial criterion (see [13, Theorem 3.7 and
Corollary 3.8]).
We are aware of one more feasibility condition for symmetric association schemes, the for-
bidden quadruple condition described in [7, Corollary 4.2]. Our 4-point transitive examples
do not have any nontrivial Krein parameters equal to zero, so they satisfy this condition vac-
uously. This is not the case for our 3-point transitive examples, to date these have not been
tested for this condition.
We have ordered these feasibility conditions according to the ease we are able to check
them. Since our algorithms require the multiplicities as part of the input and produce the
intersection matrices, we have to compute P , then Q, then the dual intersection matrices in
order from there. As our objective is only to report the examples that pass all conditions,
once an example fails one of our conditions below it is removed and its status for subsequent
conditions is not reported.
We will indicate our examples from the previous section by Galois group action and
order: 3pt35, 3pt45, etc. Recall that 3pt35 means the 3-point transitive example of order
35.
5.1 Handshaking lemma condition:
Only five of our examples have nontrivial basis elements of odd degree, of these five, three
of them fail the handshaking lemma condition: 3pt88a, 3pt88b, and 3pt116. 3pt76 and
3pt190 pass despite having a nontrivial basis element of odd degree.
388 Ars Math. Contemp. 23 (2023) #P3.02 / 367–390
5.2 Realizability of closed subsets and quotients:
All of our 4-point transitive examples are primitive, so there are no closed subsets or quo-
tients to consider. On the other hand, all of the remaining 3-point transitive examples have
a unique nontrivial closed subset of rank 2. For all but one of these, the quotient also has
rank 2. The exception is 3pt129, for which the quotient has rank 4. Since this quotient
table algebra has an element of non-integral degree, it is not realizable as an association
scheme.
5.3 Triangle count condition.
All of our examples pass.
5.4 Absolute bound condition.
All of our 4-point transitive examples pass. We can see from the multiplicities that 3pt45,
3pt76, and 3pt165 will pass. 3pt35 could potentially fail for i = j = 1 but passes because
κ1,1,k = 0 for k = 2, 3, 4. 3pt93 and 3pt129 also pass because enough nontrivial Krein
parameters are 0.
5.5 Nonnegative Krein parameter condition.
For all of our 3- and 4-point transitive examples, we have calculated the dual intersection
matrices and found them to be nonnegative.
5.6 Gegenbauer polynomial condition.
We check that Gmiℓ (
1
mi
L∗i ) is a nonnegative matrix for all ℓ ≥ 1 and i = 1, . . . , 4 using the
approach of [13, §3.3].
We illustrate the process of checking this condition with 3pt35. In the case m1 = 4, it
is not possible to find an ℓ∗ satisfying the conditions of [13, Corollary 3.16]. However, L∗1
is a block matrix, and the upper left 2× 2 block
[
0 4
1 3
]
is the dual intersection matrix cor-
responding to the association scheme generated by the complete graph of order 5, in which
it also occurs with nontrivial multiplicity 4. It follows that the first column of Gmiℓ (
1
mi
L∗i )
will always be nonnegative for all ℓ ≥ 1, so the result follows by [13, Corollary 3.8] and
the remark following it.
In the cases m2 = m3 = m4 = 10, we find that the minimum ℓ∗ required for [13,
Corollary 3.16] is ℓ∗ = 6, and we can check that G10ℓ (
1
10L
∗
i ) has nonnegative entries for
all ℓ ∈ {1, . . . , 7} and all i = 2, 3, 4. So, 3pt35 passes all the feasibility conditions, with
the possible exception of the forbidden quadruple condition.
In all of the remaining 3-transitive examples, B∗ contains a rank 2 closed subset of
order mi + 1 for one i. So, a similar argument as in the 3pt35 case applies for this mi.
For the other mi a suitable ℓ∗ can be found. After evaluating the appropriate Gegenbauer
polynomials at 1miL
∗
i , we found the result to be a nonnegative matrix.
All of our 4-point transitive examples pass the Gegenbauer polynomial test. In each
case we have found a value of ℓ∗ and shown all of the required evaluations result in non-
negative matrices.
A. Herman et al.: The search for small association schemes with noncyclotomic eigenvalues 389
In summary, we have verified that the six 4-point transitive examples pass all of the
feasibility conditions, and ten of the 3-point transitive examples pass them: 3pt35, 3pt45,
3pt76, 3pt93, 3pt165, 3pt189, 3pt190, 3pt217, 3pt231a, and 3pt231b. Note that by the
partial classification of association schemes of order 35 and rank 5 in [8], we know 3pt35
cannot be realized.
ORCID iDs
Allen Herman https://orcid.org/0000-0001-9841-636X
Roghayeh Maleki https://orcid.org/0000-0003-1803-4316
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P3.03 / 391–402
https://doi.org/10.26493/1855-3974.2882.c5e
(Also available at http://amc-journal.eu)
Comparing Wiener, Szeged and revised Szeged
index on cactus graphs
Stefan Hammer *
Graz University of Technology, Rechbauerstraße 12, Graz, Austria
Received 9 May 2022, accepted 1 November 2022, published online 11 January 2023
Abstract
We show that on cactus graphs the Szeged index is bounded above by twice the Wiener
index. For the revised Szeged index the situation is reversed if the graph class is further
restricted. Namely, if all blocks of a cactus graph are cycles, then its revised Szeged index
is bounded below by twice its Wiener index. Additionally, we show that these bounds
are sharp and examine the cases of equality. Along the way, we provide a formulation of
the revised Szeged index as a sum over vertices, which proves very helpful, and may be
interesting in other contexts.
Keywords: Wiener index, (Revised) Szeged index, cactus graphs.
Math. Subj. Class. (2020): 05C09
1 Introduction
Presumably the first topological graph index, the Wiener index, was invented in 1947 by the
chemist Wiener [25], and is used to correlate physicochemical properties to the structure
of chemical compounds [3, 11]. Since then it was and still is thoroughly studied, see e.g.
[1, 4, 5, 6, 7, 17] for only some of the latest results. Over time many more topological graph
indices were devised and investigated. One such topological graph index is the Szeged
index that came up as an extension of a formula for the Wiener index of trees. It was first
introduced in [10] without proper name. By its construction it has meaningful connections
to the Wiener index. However, Randić found that the Szeged index is lacking something
for chemical applications in comparison to the Wiener index, and thus introduced in [21]
a slightly adapted variant of the Szeged index, the later so-called revised Szeged index.
It produces better correlations in chemistry than the normal Szeged index [21] and both
Szeged indices combined can be used to provide a measure of bipartivity of graphs [20].
*Stefan Hammer acknowledges the support of the Austrian Science Fund (FWF): W1230.
E-mail address: stefan.hammer@tugraz.at (Stefan Hammer)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
392 Ars Math. Contemp. 23 (2023) #P3.03 / 391–402
It is rather easy to see that the Wiener index and the (revised) Szeged index coincide
on trees. Furthermore, in 1994 some conjectures about the relation of the Wiener and the
Szeged index on connected graphs were made by Dobrynin and Gutman [8, 10]. A year
later already Dobrynin and Gutman proved that the Wiener index and the Szeged index are
equal if and only if every block of the graph is complete [9]. Another year later Klavžar
et al. showed that the Szeged index is at least as big as the Wiener index [16]. Since
then many more authors investigated the relation of the Wiener and the Szeged index, see
[2, 13, 14, 19] and references therein. This research has been extended to the revised Szeged
index [15, 29, 30] and to certain graph classes [12, 18, 24], with the most recent work on
cactus graphs dating from this year. For further current research in the context of comparing
graph indices with the Wiener index, we refer the interested reader to [26, 27, 28].
In this paper, we want to show new relations between Wiener, Szeged and revised
Szeged index for the special case of cactus graphs. Namely, we prove that the Szeged
index is bounded above by twice the Wiener index. In case of the revised Szeged index
the situation is more complex. For bipartite cacti the revised Szeged is equal to the Szeged
index, but if we limit the class of cactus graphs to those that have only cycles as blocks,
we can reverse the above statement. That is, the revised Szeged index is bounded below by
twice the Wiener index. Additionally, we show that these bounds are sharp and examine
the cases of equality. Along the way, we provide a formulation of the revised Szeged index
as a sum over vertices, which proves very helpful, and may be interesting in other contexts.
The paper is organized as follows. In Section 2, we first introduce the main definitions
and directly afterwards show how the revised Szeged index can be written as a sum over
vertices (Theorem 2.1). Then we introduce some auxiliary results needed in the following
sections. The relation of the Szeged index and the Wiener index on cactus graphs is the
main topic of Section 3. We show that the Szeged index is bounded above by twice the
Wiener index (Theorem 3.1), and also look at equality cases. Section 4 starts with an
example showing that arbitrary cactus graphs can have a revised Szeged index equal to
twice its Wiener index. As a consequence, we look at a subclass of the cactus graphs to
prove a reverse relation for the revised Szeged and the Wiener index (Theorem 4.2).
2 Preliminaries and the revised Szeged index as vertex sum
If not otherwise mentioned, we are working with a finite, simple and connected graph G,
that has vertex set V (G) and edge set E(G). Let u, v be vertices of G. Then we denote
with dG(u, v) the distance of u and v in G, that is, the length of the shortest path connecting
u and v in G. For a path P , we use |P | for its length. Furthermore, we write nG(u, v) for
the number of vertices closer to u than to v, and oG(u, v) = oG(v, u) for the number of
vertices with equal distance to u and v. With this, the Wiener index, the Szeged index, and
the revised Szeged index are defined respectively by
W (G) =
∑
{u,v}⊆V (G)
dG(u, v) =
1
2
∑
u,v∈V (G)
dG(u, v),
Sz(G) =
∑
{s,t}∈E(G)
nG(s, t)nG(t, s),
Sz∗(G) =
∑
{s,t}∈E(G)
(
nG(s, t) +
1
2
oG(s, t)
)(
nG(t, s) +
1
2
oG(t, s)
)
.
S. Hammer: Comparing Wiener, Szeged and revised Szeged index on cactus graphs 393
Note, that the Wiener index is a sum over all unordered pairs of vertices, whereas the
(revised) Szeged index is a sum over all edges.
In [22], Simić et al. introduced for vertices u, v and an edge {s, t} the function
µu,v({s, t}) =

1 if

dG(u, s) < dG(u, t) and dG(v, s) > dG(v, t),
or
dG(u, s) > dG(u, t) and dG(v, s) < dG(v, t),
0 otherwise.
This can be considered an indicator function that is 1 if and only if the vertices u and v
contribute to nG(s, t)nG(t, s). Bonamy et al. [2] used µu,v to rewrite the Szeged index in
the following way:
Sz(G) =
∑
{u,v}⊆V (G)
∑
{s,t}∈E(G)
µu,v({s, t}).
With this reformulation, the Szeged index is also a sum over all unordered pairs of vertices.
Additionally, Bonamy et al. called all edges e satisfying µu,v(e) = 1, ‘good’ for {u, v}, and
referenced this again to Simić et al. [22]. However, Simić et al. used the term ‘good edge’
for a completely different concept. Because of this, and the fact that the term ‘good’ is not
descriptive, we decided to use a different notation. We call edges e satisfying µu,v(e) =
1, (u, v)-distance-disparate, and denote with disG(u, v) the number of (u, v)-distance-
disparate edges in G. Hence, we can write for the Szeged index,
Sz(G) =
∑
{u,v}⊆V (G)
disG(u, v) =
1
2
∑
u,v∈V (G)
disG(u, v).
Since the revised Szeged index may not even be an integer, there cannot be a single
indicator function as there is for the Szeged index. So it seems difficult to formulate the
revised Szeged index as sum over vertices. Still a rather similar approach works. The first
step is to consider an equivalent of µu,v for single vertices and edges having end points
with the same distance to the vertex. Namely, we define for a vertex v and an edge {s, t},
νv({s, t}) =
{
1 if dG(v, s) = dG(v, t),
0 otherwise,
an indicator function that is 1 if and only if the end points of the edge have the same distance
to v. Now, similar to before, we call edges e satisfying νu(e) = 1 and νv(e) = 1 for vertices
u and v, (u, v)-distance-equal, and denote with deqG(u, v) the number of (u, v)-distance-
equal edges in G. These are the ingredients necessary to write the revised Szeged index as
sum over vertices.
Theorem 2.1. The revised Szeged index of a graph G can be written as sum over vertices
in the following form:
Sz∗(G) =
1
2
∑
u,v∈V (G)
(
disG(u, v) + deqG(u, u)−
1
2
deqG(u, v)
)
.
394 Ars Math. Contemp. 23 (2023) #P3.03 / 391–402
Proof. Let n be the number of vertices in G. Use that n = nG(s, t) + nG(t, s) + oG(s, t)
for all edges {s, t} to rewrite the revised Szeged index:
Sz∗(G) =
∑
{s,t}∈E(G)
(
nG(s, t) +
1
2
oG(s, t)
)(
nG(t, s) +
1
2
oG(t, s)
)
=
∑
{s,t}∈E(G)
(
nG(s, t)nG(t, s) +
1
2
oG(s, t)(n− oG(s, t)) +
1
4
oG(s, t)
2
)
= Sz(G) +
1
2
n
∑
{s,t}∈E(G)
oG(s, t)−
1
4
∑
{s,t}∈E(G)
oG(s, t)
2.
(2.1)
Since a vertex v is counted in oG(s, t) if and only if dG(v, s) = dG(v, t), we can rewrite
the second sum to∑
{s,t}∈E(G)
oG(s, t) =
∑
u∈V (G)
∑
e∈E(G)
νu(e) =
∑
u∈V (G)
deqG(u, u).
For the third term notice that vertices u and v are involved in oG(s, t) · oG(s, t) if and only
if dG(u, s) = dG(u, t) and dG(v, s) = dG(v, t), that is {s, t} is counted in deqG(u, v).
Thus, we can reformulate this sum as well:∑
{s,t}∈E(G)
oG(s, t)
2 =
∑
u,v∈V (G)
deqG(u, v).
Insert the reformulations and the Szeged index written as vertex sum in Equation (2.1) and
write for n the sum over all vertices to get the desired result:
Sz∗(G) =
1
2
∑
u,v∈V (G)
disG(u, v) +
1
2
n
∑
u∈V (G)
deqG(u, u)−
1
4
∑
u,v∈V (G)
deqG(u, v)
=
1
2
∑
u,v∈V (G)
(
disG(u, v) + deqG(u, u)−
1
2
deqG(u, v)
)
.
A noteworthy consequence of the above result is that the difference between the Szeged
and the revised Szeged index can be nicely described.
Corollary 2.2. The difference between the Szeged and the revised Szeged index of a graph
G on n vertices satisfies
Sz∗(G)− Sz(G) = 1
2
∑
{s,t}∈E(G)
(
n · oG(s, t)−
1
2
oG(s, t)
2
)
=
1
2
n
∑
u∈V (G)
deqG(u, u)−
1
4
∑
u,v∈V (G)
deqG(u, v).
Before we come to the comparison of the Wiener index and the (revised) Szeged in-
dex on cactus graphs, we need some general results about graphs. The first is about the
connection of disG and dG on cycles.
S. Hammer: Comparing Wiener, Szeged and revised Szeged index on cactus graphs 395
Lemma 2.3. Let u and v be two distinct vertices of a cycle C of length n. Then
disC(u, v) =
{
2 dC(u, v) if n is even,
2 dC(u, v)− 1 if n is odd.
Proof. To make things easier, we think of a suitable embedding of C in the plane and say
right for counterclockwise, and left for clockwise. For some vertex w in C, let Pr(w) be
the path starting at w and going ⌊n/2⌋ edges to the right, and Pl(w) the path starting at w,
going ⌊n/2⌋ edges to the left. We denote the terminal vertices of Pr(w) and Pl(w) with wr
and wl, respectively.
Let e be an edge in C. It is clear that if e is in Pr(u), then the left vertex of e is closer
to u, and vice versa, if e is in Pl(u), then the right vertex of e is closer to u. For v the
situation is the same. Thus, e is (u, v)-distance-disparate if and only if it is contained in the
path Pr(u) ∩ Pl(v), or in the path Pl(u) ∩ Pr(v).
Without lost of generality, we can assume v is in Pr(u), see Figure 1 for an exemplary
illustration of the situation. In this case, Pr(u) ∩ Pl(v) is a shortest path from u to v, and
Pl(u) ∩ Pr(v) is a shortest path from ul to vr. So we have
disC(u, v) = dC(u, v) + dC(ul, vr). (2.2)
By inclusion–exclusion principle, the distance from ul to vr can be determined by
dC(ul, vr) = |Pl(u) ∩ Pr(v)|
= |Pr(u) ∩ Pl(v)|+ |Pl(u)|+ |Pr(v)| − |E(C)|
= dC(u, v) + 2 ⌊n/2⌋ − n.
Now considering even and odd n respectively, and inserting dC(ul, vr) in (2.2) completes
the proof.
u
ur = ul
v
vr = vl
Pr(u)Pl(u)
Pr(v)
Pl(v) u
urul
v
vr
vl
Pr(u)Pl(u)
Pr(v)
Pl(v)
Figure 1: Cycle C of even length left and of odd length right, with vertices u, v, and the
paths going right and left including their terminal vertices.
The next result is about splitting distances in a block-cut-vertex decomposition of the
given graph. Recall, a block is a maximal 2-connected subgraph, and in a block-cut-vertex
decomposition blocks only overlap at cut vertices. More information on blocks and the
block-cut-vertex decomposition can be found in [23].
396 Ars Math. Contemp. 23 (2023) #P3.03 / 391–402
Proposition 2.4. Let u and v be vertices of a graph G with set of blocks B obtained by
the block-cut-vertex decomposition for G. For a block B in B, denote by uB and vB the
vertices in B closest to u and v, respectively. Then
dG(u, v) =
∑
B∈B
dG(uB , vB). (2.3)
Proof. Let P be a shortest path from u to v and B′ ⊆ B the set of blocks visited by
P . Every block B in B′ is entered by uB and left by vB , so P can be decomposed into
subpaths PB , where for a block B the subpath PB starts at uB and ends at vB . Since every
subpath of a shortest path is a shortest path itself, it follows that
dG(u, v) = |P | =
∑
B∈B′
|PB | =
∑
B∈B′
dG(uB , vB). (2.4)
Now consider a block B not visited by P . Since we have a block-cut-vertex decompo-
sition, there is a unique vertex w in B minimizing the distance from the block B to the path
P . This vertex is also a cut vertex and thus it minimises the distance from B to any vertex
of the path P . Hence, it follows that uB = vB = w, and dG(uB , vB) = 0. This finishes
the proof.
Note, in the proof of Proposition 2.4 we do not use that blocks are two-connected. That
means, instead of blocks, we could split the graph into arbitrary subgraphs that only overlap
at cut vertices.
In the remaining two sections, we apply the above tools to the so called cactus graphs.
These are connected graphs where every two distinct cycles have at most one common
vertex. Alternatively, the graph consists of a single vertex, or every block is either an edge
or a cycle.
3 Comparing Wiener and Szeged index on cactus graphs
Since every edge on a shortest path from u to v is clearly (u, v)-distance-disparate, formu-
lating the Szeged index as sum over vertices gives the first part of the following inequality:
W (G) ≤ Sz(G) ≤ Sz∗(G).
Already Simić et al. used the indicator function µu,v to show additionally that equality
holds in the first part if and only if every block of G is complete, see [22, Theorem 2.1].
The inequality of the second part is clear by definition, whereas equality holds if and only
if G is bipartite. This was shown by Pisanski and Randić, see [20, Theorem 1]. Besides, it
follows from Corollary 2.2.
Here, we want to show a different inequality, true for the special class of cactus graphs.
Theorem 3.1. Let G be a cactus graph, then
Sz(G) ≤ 2W (G),
with equality if and only if every block of G is a cycle of even length.
A special case of this result was already given in [18]. There, Li and Zhang showed
that Theorem 3.1 holds for unicyclic graphs.
S. Hammer: Comparing Wiener, Szeged and revised Szeged index on cactus graphs 397
Proof. Let u, v be vertices in G and e be an edge in a block B. With uB as in Propo-
sition 2.4 every shortest path from e to u uses uB . The same is true for v and vB as
in Proposition 2.4, respectively. Thus e is (u, v)-distance-disparate if and only if it is
(uB , vB)-distance-disparate. Hence, with B as set of blocks, we can write
disG(u, v) =
∑
B∈B
disG(uB , vB). (3.1)
Suppose that every block is a cycle of even length. Then by Lemma 2.3 and Proposi-
tion 2.4,
Sz(G) =
1
2
∑
u,v∈V (G)
disG(u, v) =
1
2
∑
u,v∈V (G)
∑
B∈B
disG(uB , vB)
=
1
2
∑
u,v∈V (G)
∑
B∈B
2dG(uB , vB) =
∑
u,v∈V (G)
dG(u, v)
= 2W (G).
(3.2)
Now if there is at least one odd cycle C, then again by Lemma 2.3, there is a strict
inequality instead of the third equality in the above formula. Finally, if there is a block
consisting of only a single edge {s, t}, then disG(s, t) = 1 = dG(s, t), and thus also
Sz(G) < 2W (G).
Note, blocks consisting of two vertices connected with two edges considered as cycles
of length two can be allowed in Theorem 3.1. Clearly, this is not a characterisation of
graphs G satisfying Sz(G) ≤ 2W (G), since every complete graph Kn satisfies Sz(Kn) =
W (Kn). Unfortunately, it is also not a characterisation of graphs satisfying Sz(G) =
2W (G). Below, we give an example of a graph satisfying the equation that is not a cactus
graph.
Example 3.2. Let G consist of three paths of length two joined at their end points. Attach
on one side of the end points of the paths two edges by their end points and on the other
side three edges. See Figure 2 for an exemplary drawing. It can be checked via a computer,
or even easily by hand that
Sz(G) = 192 = 2 · 96 = 2W (G).
Figure 2: A bipartite non-cactus graph G satisfying Sz(G) = 2W (G).
398 Ars Math. Contemp. 23 (2023) #P3.03 / 391–402
By generalizing the graph in Example 3.2 to have k paths instead of only 3, more
example graphs satisfying the equality can be found. Not for every k a suitable number of
edges can be attached, but it seems there is no cap for k. The biggest example graph G
we found has 783 paths of length 2, 28 edges on one and 656 009 edges on the other side
attached. It satisfies
Sz(G) = 862 902 435 600 = 2 · 431 451 217 800 = 2W (G).
This suggests that also if the cyclomatic number, which is just |E(G)| − |V (G)| + 1 for
connected graphs, is large, Sz(G) = 2W (G) can still hold for non-cactus graphs.
4 Comparing Wiener and revised Szeged index on cactus graphs
From the last section, we can conclude that Sz∗(G) ≤ 2W (G) holds for bipartite cactus
graphs G. But in case of non-bipartite cactus graphs the situation becomes more compli-
cated. There are even cactus graphs G satisfying Sz∗(G) = 2W (G), where not every
block is a cycle of even length as the following example shows.
Example 4.1. Take a cycle of length 13, a cycle of length 11, six edges and join them at a
single vertex to obtain a cactus graph G, as depicted in Figure 3. It can be checked that
Sz∗(G) = 3636 = 2 · 1818 = 2W (G).
Figure 3: A cactus graph G satisfying Sz∗(G) = 2W (G).
With this in mind, it seems difficult to make any concrete statements about the con-
nection of the revised Szeged and the Wiener index in the case of cactus graphs. Hence,
we focused on a subclass of cactus graphs and found the following relation, which is in
contrast to Theorem 3.1.
S. Hammer: Comparing Wiener, Szeged and revised Szeged index on cactus graphs 399
Theorem 4.2. Suppose every block of a graph G is a cycle. Then
2W (G) ≤ Sz∗(G),
with equality if and only if every cycle in G has even length.
Note, clearly a graph where every block is a cycle is a cactus graph.
Proof. Let u, v be vertices in G and e be an edge in a block B. Again, we use the notation
of Proposition 2.4 with uB and vB for the vertices in B closest to u and v, respectively.
Since uB is on every shortest path from u to e, and the same is true for vB and v, it is
evident that deqB(u, v) = deqB(uB , vB). Furthermore, the set of blocks B of G induces a
partition of the edge set. Hence,
deqG(u, v) =
∑
B∈B
deqB(u, v) =
∑
B∈B
deqB(uB , vB).
Thus, with Theorem 2.1 we can formulate the revised Szeged index of G as
Sz∗(G) =
1
2
∑
u,v∈V (G)
(
disG(u, v) + deqG(u, u)−
1
2
deqG(u, v)
)
=
1
2
∑
u,v∈V (G)
∑
B∈B
(
disG(uB , vB) + deqB(uB , uB)−
1
2
deqB(uB , vB)
)
.
(4.1)
Next we distinguish two cases, whereby the second case has two sub-cases, to show
that for any vertices uB and vB in a block B,
2 dG(uB , vB) ≤ disG(uB , vB) + deqB(uB , uB)−
1
2
deqB(uB , vB). (4.2)
Case 1: Suppose that B is a cycle of even length. Then,
deqB(uB , uB) = 0 = deqB(uB , vB),
and by Lemma 2.3,
disG(uB , vB) = 2 dG(uB , vB).
Case 2: Suppose that B is a cycle of odd length.
Case 2.1: If uB ̸= vB , then
deqB(uB , uB) = 1, deqB(uB , vB) = 0,
and again by Lemma 2.3
disG(uB , vB) = 2 dG(uB , vB)− 1.
Case 2.2: If uB = vB , then
deqB(uB , uB) = 1 = deqB(uB , vB),
disG(uB , vB) = 0 = 2 dG(uB , vB).
400 Ars Math. Contemp. 23 (2023) #P3.03 / 391–402
So in Case 1 and Case 2.1, we have equality in (4.2), and in Case 2.2, (4.2) is fulfilled
with a strict inequality. Therefore by Proposition 2.4 and (4.1),
2W (G) =
1
2
∑
u,v∈V (G)
∑
B∈B
2 dG(uB , vB) ≤ Sz
∗(G),
with equality if and only if every cycle in G has even length.
ORCID iDs
Stefan Hammer https://orcid.org/0000-0002-8064-8733
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P3.04 / 403–416
https://doi.org/10.26493/1855-3974.2913.35e
(Also available at http://amc-journal.eu)
Component (edge) connectivity of
pancake graphs*
Xiaohui Hua † , Lulu Yang
School of Mathematics and Information Science, Henan Normal University, Xinxiang,
Henan 453007, P. R. China
Received 25 June 2022, accepted 17 November 2022, published online 17 January 2023
Abstract
The l-component (edge) connectivity of a graph G, denoted by cκl(G) (cλl(G)), is
the minimum number of vertices (edges) whose removal from G results in a disconnected
graph with at least l components. The pancake graph Pn is a popular underlying topology
for distributed systems. In the paper, we determine the cκl(Pn) and cλl(Pn) for 3 ≤ l ≤ 5.
Keywords: Component connectivity, component edge connectivity, pancake graphs, fault tolerance.
Math. Subj. Class. (2020): 05C40, 05C75
1 Introduction
Multiprocessor systems are always built according to a graph which is called its intercon-
nection network (network, for short). In a network, vertices correspond to processors, and
edges correspond to communicating links between pairs of vertices. Since failures of pro-
cessors and links are inevitable in multiprocessor systems, fault tolerance is an important
issue in interconnection networks. Fault tolerance of interconnection networks becomes
an essential problem and has been widely studied, such as, structure connectivity and sub-
structure connectivity of hypercubes [20], extra connectivity of bubble sort star graphs [10],
g-extra conditional diagnosability of hierarchical cubic networks [21], g-good-neighbor
connectivity of graphs [25], conditional connectivity of Cayley graphs generated by uni-
cyclic graphs [26].
Given a connected graph G = (V,E), where V is the set of processors and E is the
set of communication links between processors. The connectivity κ(G) of a graph G is
the minimum number of vertices of G, if any, whose deletion disconnect G. The edge
*The authors are grateful to anonymous referees for helpful remarks and suggestions.
†Corresponding author.
E-mail addresses: xhhua@htu.edu.cn (Xiaohui Hua), yllyanglulu@126.com (Lulu Yang)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
404 Ars Math. Contemp. 23 (2023) #P3.04 / 403–416
connectivity λ(G) of a graph G is the minimum number of edges of G, if any, whose
deletion disconnect G. The g-extra connectivity of G, denoted by κg(G), is the minimum
number of vertices whose removal separates G such that each component of the remaining
graph has at least g + 1 vertices.
The classic parameter is the connectivity κ(G) and edge connectivity λ(G). In general,
the larger κ(G) or λ(G) is, the more stable the network is. The l-component connectivity
of a graph was first introduced by Chartrand [8] and Sampathkumar [22], independently.
Note that cκ2(G) = κ(G) and cλ2(G) = λ(G) for any graph if it is not a complete
graph. Therefore, the l-component (edge) connectivity can be regarded as a generalization
of the classical (edge) connectivity. The two parameters have been investigated in several
interconnection networks. See for example [3, 7, 12, 16, 23, 27, 28, 29]. Recently, the
relationship between extra connectivity and component connectivity of general networks
has been investigated by Li et al. [14], while the relationship between extra edge connec-
tivity and component edge connectivity of regular networks has been suggested by Hao et
al. [15] and Guo et al. [13], independently.
The pancake graph, denoted by Pn, is one of alternative interconnection networks for
multiprocessor systems, and it poses some attractive topological properties, such as (n −
1)-regular, node-symmetric, bipartite and recursive [1]. The pancake graph has drawn
considerable attention, such as, structure connectivity and substructure connectivity [6],
super connectivity [19] and neighbor connectivity [9, 24] had been considered. For more
examples, see [1, 5, 11, 17, 18, 30] and references therein.
The rest of the paper is organized as follows. Section 2 formally gives the definition of
pancake graphs. In addition, we introduce some preliminary results. Section 3 determines
the l-component connectivity of Pn for l = 3, 4, 5. Section 4 determines the l-component
edge connectivity of Pn for l = 3, 4, 5. Concluding remarks are covered in Section 5.
2 Preliminaries
In this paper, graph-theoretical terminology and notation not defined here mostly follow
[2].
For any two graphs G1 and G2, G1 ∩ G2 = (V (G1) ∩ V (G2), E(G1) ∩ E(G2)).
For any sets A and B, A − B = {x : x ∈ A but x /∈ B} and we sometimes write
A − B as A \ B if B ⊆ A. For X,Y ⊆ V (G), [X,Y ] represents the edge set of G in
which one end is in X and the other is in Y . The distance of two vertices u and v in a
graph G, denote by disG(u, v), is the length of a shortest path between u and v in G. Set
NG(u) = {v : disG(u, v) = 1}, and set NG(U) =
⋃
u∈U NG(u) − U . For any vertex v,
denote by E(v) the edges incident to v. A k-cycle, denoted by Ck, is a cycle on k vertices,
and a k-path u1u2...uk, is a path on k vertices. Let ⟨n⟩ = {1, 2, · · · , n}.
Definition 2.1 ([1]). The n-dimensional pancake graph is denoted by Pn. The vertex set
V (Pn) = {u = u1u2 · · ·un|ui ∈ ⟨n⟩, ui ̸= uj for i ̸= j}, the edge set E(Pn) = {uv|u =
u1u2 · · ·uk · · ·un, v = uk = ukuk−1 · · ·u2u1uk+1 · · ·un−1un and 2 ≤ k ≤ n}, where
uk denotes the unique k-neighbour of u, the edge uuk is called k-edge.
Clearly, Pn consists of (n− 1) kinds of edges. The pancake graphs P2, P3 and P4 are
shown in Figure 1. The pancake graphs are Cayley graphs with having the hierarchical (re-
cursive) structure. The removal of all n-edges from Pn results in n connected components
P 1n , P
2
n , · · · , Pnn , where P in is the subgraph of Pn induced by {u = u1u2 · · ·un ∈ V (Pn) :
X. Hua and L. Yang: Component (edge) connectivity of pancake graphs 405
Figure 1: The pancake graphs P2, P3 and P4.
un = i}. Clearly, P in is isomorphic to the (n − 1)-dimensional pancake graph Pn−1 [17].
We call P in a sub-pancake of Pn. For any vertex u ∈ V (P in), just one vertex in N(u) is not
in V (P in), we define this vertex to be out-neighbor of u. For i ̸= j ∈ ⟨n⟩, an edge is called
a cross-edge if its two terminal vertices are in P in and P
j
n, respectively.
Lemma 2.2 ([1, 5, 17, 18]). An n-dimensional pancake graph Pn has the following com-
binatorial properties.
(1) Pn has n! vertices, (n− 1)n!/2 edges, (n− 1)-regular.
(2) The girth of Pn is 6 for n ≥ 3. Let the 6-cycle be presented as u1u2u3u4u5u6. Then
u1u2, u3u4, u5u6 are 2-edges and u2u3, u4u5, u1u6 are 3-edges.
(3) For any i ̸= j, the number of cross edges between P in and P jn is (n− 2)!.
Remark 2.3. One and the same path of length 2 cannot be contained in two 6-cycles.
Lemma 2.4 ([30]). Let F be a set of faulty vertices in Pn with |F | ≤ 2n− 4 for n ≥ 5. If
Pn −F is disconnected, then it has exactly two components, one of which is a singleton or
a single edge.
In [4], Chen and Tan proposed the family of interconnection networks SPn. It is obvi-
ously that Pn is one of the network of SPn.
Lemma 2.5 ([11]). Let F be a set of faulty vertices in Pn with |F | ≤ 2n− 5 for n ≥ 3. If
Pn − F is disconnected, then it has exactly two components, one of which is a singleton.
Lemma 2.6 ([11, 30]). Let F be a set of faulty vertices in Pn with |F | ≤ 3n−8 for n ≥ 5.
If Pn − F is disconnected, then it either has two components, one of which is a singleton
or a single edge, or has three components, two of which are singletons.
Lemma 2.7 ([11]). Let F be a set of faulty vertices in Pn with |F | ≤ 4n − 11 for n ≥ 6.
If Pn − F is disconnected, then Pn − F satisfies one of the following conditions:
(1) Pn−F has two components, one of which is a singleton or a single edge or a 3-path;
406 Ars Math. Contemp. 23 (2023) #P3.04 / 403–416
(2) Pn − F has three components, two of which are singletons;
(3) Pn−F has three components, two of which are a singleton and an edge, respectively;
(4) Pn − F has four components, three of which are singletons.
Lemma 2.8 ([4, 11]). κ1(Pn) = 2n − 4 for n ≥ 3, κ2(Pn) = 3n − 7 for n ≥ 5 and
κ3(Pn) = 4n− 10 for n ≥ 6.
Hereafter, we suppose that F is a vertex cut or an edge cut in Pn. For each i ∈ ⟨n⟩, let
Fi = F ∩ V (P in) or Fi = F ∩ E(P in), and fi = |Fi|. Let I = {i ∈ ⟨n⟩|fi ≥ n − 2},
P In =
⋃
i∈I P
i
n, FI =
⋃
i∈I Fi, and let J = ⟨n⟩ \ I , P Jn =
⋃
j∈J P
j
n, FJ =
⋃
j∈I fj . Also,
we Let H be the union of smaller components of Pn − F and let c(H) be the number of
components of H .
3 The component connectivity of Pn
Lemma 3.1. Let S be an independent set of V (Pn) for n ≥ 4. Then the following asser-
tions hold.
(1) If |S| = 2, then |N(S)| ≥ 2n− 3.
(2) If |S| = 3, then |N(S)| ≥ 3n− 6.
(3) If |S| = 4, then |N(S)| ≥ 4n− 8.
Proof. For (1), let S = {v1, v2}. By Lemma 2.2, Pn contains no 4-cycle. Thus, v1 and v2
have at most one common neighbor, and |N(S)| = |N(v1)|+|N(v2)|−|N(v1)∩N(v2)| ≥
2n− 3.
For (2), let S = {v1, v2, v3}. By Lemma 2.2, Pn contains 6-cycle, there exists at
most three common neighbors among these three singletons. Thus, we have |N(S)| ≥∑3
i=1 |N(vi)| − 3 ≥ 3n− 6.
For (3), let S = {v1, v2, v3, v4}. Since Pn contains 8-cycle, and in order to make
these four singletons contain as many common vertices as possible, we may assume that
the 8-cycle is presented as v1u1v2u2v3u3v4u4. Then there exists four common neighbors
among these four singletons. If there exists five common neighbors among these four
singletons, then it forms a cycle of length less than 6 or two 6-cycles with common 2-
path, contradicting Lemma 2.2(2) and Remark 2.3, respectively. Thus, we have |N(S)| ≥∑4
i=1 |N(vi)| − 4 ≥ 4n− 8.
The following remark provides instances that attain the bounds for the assertions of
Lemma 3.1.
Remark 3.2. Let x = 123 · · ·n, y = (x2)3, z = (y2)3, w = (y2)n, o = (w2)3. Clearly,
{x, y, z} is an independent set of Pn and {x, y, z} lie on a 6-cycle in the subgraph of Pn,
{x, y, w, o} is an independent set of Pn and {x, y, w, o} lie on a 8-cycle in the subgraph of
Pn. Clearly, if S = {x, y}, then |N(S)| = 2n − 3. Since Pn − F has three components,
we have cκ3(Pn) ≤ 2n − 3. Similarly, if S = {x, y, z}, then |N(S)| = 3n − 6. Since
Pn − F has four components, we have cκ4(Pn) ≤ 3n− 6. Also, if S = {x, y, w, o}, then
|N(S)| = 4n− 8. Since Pn − F has five components, we have cκ5(Pn) ≤ 4n− 8.
Theorem 3.3. For n ≥ 3, cκ3(Pn) = 2n− 3.
X. Hua and L. Yang: Component (edge) connectivity of pancake graphs 407
Proof. It is true if n = 3. From Remark 3.2, we obtain the upper bound cκ3(Pn) ≤ 2n− 3
for n ≥ 4. It suffices to show cκ3(Pn) ≥ 2n − 3. Suppose on the contrary that there is a
vertex cut F with |F | ≤ 2n− 4, and Pn − F has at least three components.
We first consider that n = 4. Since |F | ≤ 4, it is clear that |I| ≤ 2. If |I| = 1, let
I = {i}, then fi ∈ {2, 3, 4}. If |I| = 2, let I = {i, j}, then fi = fj = 2. No matter
which case, it’s not hard to prove that P4−F has at most two components, a contradiction.
We now consider that n ≥ 5. By Lemma 2.4, Pn − F has exactly two components, a
contradiction.
Thus, cκ3(Pn) ≥ 2n− 3.
Next, we give a lemma which is used by Theorem 3.5 and 3.6.
Lemma 3.4. For n ≥ 5, if |I| ≤ 3, then P Jn − FJ is connected.
Proof. By the definition of J , we have |J | = n − |I| ≥ n − 3 ≥ 2 for n ≥ 5 and
fj ≤ n − 3 for j ∈ J . Since each subgraph P jn is isomorphic to Pn−1, by Lemma 2.2,
we have κ(P jn) = n − 2. Thus, for each j ∈ J , P jn − Fj is connected. For distinct
j, k ∈ J , by Lemma 2.2, the number of cross edges between P jn and P kn is (n− 2)!, since
(n − 2)! > 2(n − 3) for n ≥ 5, we have P jn − Fj is connected to P kn − Fk. Therefore,
P Jn − FJ is connected.
Theorem 3.5. For n ≥ 4, cκ4(Pn) = 3n− 6.
Proof. Remark 3.2 acquires the upper bound cκ4(Pn) ≤ 3n − 6 for n ≥ 4. It suffices to
show cκ4(Pn) ≥ 3n − 6. Suppose that there is a vertex cut F with |F | ≤ 3n − 7, and
Pn − F has at least four components.
We first consider that n = 4. By Theorem 3.3, cκ3(P4) = 5 and |F | ≤ 3n− 7 = 5, we
have know Pn − F has at most three components, a contradiction.
Next, Let n ≥ 5. Lemma 2.6 shows that the removal of a vertex cut with no more that
3n − 8 vertices in Pn results in a disconnected graph with at most three components, a
contradiction. To complete the proof, we need to show result holds when |F | = 3n − 7.
Partition Pn into n disjoint copies P 1n , P
2
n , · · · , Pnn of Pn−1 along dimension n. Recall that
I = {i ∈ ⟨n⟩ : fi ≥ n − 2}. Since |F | = 3n − 7, it is clear that |I| ≤ 2. By Lemma 3.4,
P Jn − FJ is connected. If |I| = 0, then Pn − F = P Jn − FJ is connected, a contradiction.
Consider the following cases.
Case 1: |I| = 1. Let I = {i}.
Case 1.1: n− 2 ≤ fi ≤ 3(n− 1)− 8.
Since each subgraph P in is isomorphic to Pn−1, by Lemma 2.6, P
i
n − Fi has at most
three components, and all small components contain at most two vertices in total. Since
(n−1)!−2 > 3n−7 for n ≥ 5, the large component of P in−Fi is connected to P Jn −FJ .
This implies that |V (H)| ≤ 2. It is clear that c(H) ≤ |V (H)| ≤ 2, a contradiction.
Case 1.2: 3n− 10 ≤ fi ≤ 3n− 7.
In this case, we have FJ = |F | − fi ≤ (3n− 7)− (3n− 10) = 3. Since every vertex
of H has exactly one out-neighbor, we have |V (H)| ≤ 3. If |V (H)| = 3, then c(H) ≤ 2.
Otherwise, c(H) = 3 and it implies that H is a set of three singletons. By Lemma 3.1, we
have |NPn(V (H))| ≥ 3n− 6 > 3n− 7 = |F |, a contradiction. If |V (H)| ≤ 2, it is clear
that c(H) ≤ |V (H)| ≤ 2, a contradiction.
Case 2: |I| = 2.
408 Ars Math. Contemp. 23 (2023) #P3.04 / 403–416
Let I = {i, j}. Without loss of generality, assume fi ≤ fj . Since |F | = 3n − 7,
we have n − 2 ≤ fi ≤ fj ≤ (3n − 7) − (n − 2) = 2n − 5. If fi = 2n − 5, then
fi + fj = 2(2n− 5) = 4n− 10 > 3n− 7 for n ≥ 5. Thus, it requires that fi ≤ 2n− 6.
Case 2.1: n− 2 ≤ fi ≤ fj ≤ 2n− 6.
For l ∈ {i, j}, if P ln − Fl is disconnected, by Lemma 2.4, P ln − Fl has at most two
components, one of which is a singleton or an edge. Since (n−1)!− (n−2)!−2 > 3n−7
for n ≥ 5 and l ∈ {i, j}, then the large component of P ln − Fl is connected to P Jn − FJ . It
implies that c(H) ≤ 2, a contradiction.
Case 2.2: fj = 2n− 5, and fi = n− 2.
Since |F | = 3n − 7, we have FJ = |F | − fi − fj = 0. Thus, at most two vertices in
P in ∪ P jn − (Fi ∪ Fj) cannot connect with P Jn − FJ in Pn − F , and the two vertices form
an edge. Thus, c(H) ≤ 1, a contradiction.
Theorem 3.6. For n ≥ 6, cκ5(Pn) = 4n− 8.
Proof. Remark 3.2 acquires the upper bound cκ5(Pn) ≤ 4n − 8. It suffices to show
cκ5(Pn) ≥ 4n− 8.
Suppose that there is a vertex cut F with |F | ≤ 4n − 9, and Pn − F has at least
five components. Lemma 2.7 shows that the removal of a vertex cut with no more that
4n − 11 vertices in Pn results in a disconnected graph with at most four components, a
contradiction. To complete the proof, we need to show result holds when 4n− 10 ≤ |F | ≤
4n − 9. Partition Pn into n disjoint copies P 1n , P 2n , · · · , Pnn of Pn−1 along dimension n.
Recall that I = {i ∈ ⟨n⟩ : fi ≥ n − 2}. Since |F | ≤ 4n − 9, it is clear that |I| ≤ 3. By
Lemma 3.4, P Jn − FJ is connected. If |I| = 0, then Pn − F = P Jn − FJ is connected, a
contradiction.
Consider the following cases.
Case 1: |I| = 1. Let I = {i}.
Case 1.1: n− 2 ≤ fi ≤ 4(n− 1)− 11.
Since each subgraph P in is isomorphic to Pn−1, by Lemma 2.7, P
i
n − Fi has at most
four components, and all small components contain at most three vertices in total. Since
(n−1)!−3 > 4n−9 for n ≥ 6, the large component of P in−Fi is connected to P Jn −FJ .
This implies that |V (H)| ≤ 3. It is clear that c(H) ≤ |V (H)| ≤ 3, a contradiction.
Case 1.2: 4n− 14 ≤ fi ≤ 4n− 9.
In this case, we have FJ = |F | − fi ≤ (4n− 9)− (4n− 14) = 5. Since every vertex
of H has exactly one out-neighbor, we have |V (H)| ≤ 5. If |V (H)| = 5, then c(H) ≤ 3.
Otherwise, c(H) ≥ 4 and it implies that H contains five singletons or three singletons
together with an edge. In the former case, let H = H0 ∪ {x}, where H0 is a set of four
singletons and x is a singleton. By Lemma 3.1(3), we have |NPn(V (H0))| ≥ 4n − 8.
Clearly, |NPn(V (H))| = |NPn(V (H0))|+ |NPn(x)| − |NPn(V (H)) ∩NPn(x)| ≥ 4n−
8 + (n − 1) − 4 = 5n − 13 > 4n − 9 ≥ |F | for n ≥ 6, a contradiction. In the latter
case, let H = H0 ∪ {u, v}, where H0 is a set of three singletons and uv is an edge.
Then, we have |NPn(V (H0))| ≥ 3n− 6 by Lemma 3.1(2) and |NPn({u, v})| = 2n− 4 by
Lemma 2.8. Also, the girth of Pn is 6 and it follows that |NPn(V (H))∩NPn({u, v})| ≤ 3.
Thus |NPn(V (H))| = |NPn(V (H0))|+ |NPn({u, v})| − |NPn(V (H))∩NPn({u, v})| ≥
3n − 6 + (2n − 4) − 3 = 5n − 13 > 4n − 9 ≥ |F | for n ≥ 6, a contradiction. If
|V (H)| = 4, then c(H) ≤ 3. Otherwise, H contains four singletons. By Lemma 3.1(3),
X. Hua and L. Yang: Component (edge) connectivity of pancake graphs 409
we have |NPn(V (H))| ≥ 4n − 8 > 4n − 9 ≥ |F |, a contradiction. Also, if |V (H)| ≤ 3,
it is clear that c(H) ≤ |V (H)| ≤ 3, a contradiction.
Case 2: |I| = 2.
Let I = {i, j}. Without loss of generality, assume fi ≤ fj . Since |F | ≤ 4n − 9,
we have n − 2 ≤ fi ≤ fj ≤ (4n − 9) − (n − 2) = 3n − 7. If fi ≥ 3n − 10, then
fi + fj ≥ 2(3n− 10) = 6n− 20 > 4n− 9 for n ≥ 6. Thus, it requires that fi ≤ 3n− 11.
Case 2.1: n− 2 ≤ fi ≤ fj ≤ 3n− 11.
For each l ∈ {i, j}, if P ln − Fl is disconnected, by Lemma 2.6, P ln − Fl has at most
three components and all smaller components contain at most two vertices in total. Since
(n− 1)!− (n− 2)!− 2 > 4n− 9 for n ≥ 6, the large component of P ln − Fl is connected
to P Jn − FJ . Thus, |V (H)| ≤ 2|I| = 4. If |V (H)| = 4, then c(H) ≤ 3. Otherwise, H
contains four singletons. By Lemma 3.1(3), we have |NPn(V (H))| ≥ 4n − 8 > 4n −
9 ≥ |F |, a contradiction. Also, if |V (H)| ≤ 3, it is clear that c(H) ≤ |V (H)| ≤ 3, a
contradiction.
Case 2.2: 3n− 10 ≤ fj ≤ 3n− 7, and n− 2 ≤ fi ≤ 4n− 9− (3n− 10) = n+ 1.
P in − Fi has at most two components, one of which is a singleton. If fj = 3n − 10,
by Theorem 3.5, P jn − Fj has at most three components. Then FJ = |F | − fi − fj ≤
4n− 9− (n− 2)− (3n− 10) = 3. If 3n− 9 ≤ fj ≤ 3n− 7, then FJ = |F | − fi − fj ≤
4n− 9− (n− 2)− (3n− 9) = 2. No matter which case, c(H) ≤ 3, a contradiction.
Case 3: |I| = 3.
Let I = {i, j, k}. Without loss of generality, assume fi ≤ fj ≤ fk. Since |F | ≤ 4n−9,
we have n − 2 ≤ fi ≤ fj ≤ fk ≤ (4n − 9) − 2(n − 2) = 2n − 5. If fi ≥ 2n − 6, then
fi+fj+fk ≥ 3(2n−6) = 6n−18 > 4n−9 for n ≥ 6. Thus, it requires that fi ≤ 2n−7.
If fj ≥ 2n− 6, then fi + fj + fk ≥ n− 2 + 2(2n− 6) = 5n− 14 > 4n− 9 for n ≥ 6.
Thus, it requires that fj ≤ 2n− 7.
Case 3.1: n− 2 ≤ fi ≤ fj ≤ fk ≤ 2n− 7.
For each l ∈ {i, j, k}, if P ln − Fl is disconnected, by Lemma 2.5, P ln − Fl has at most
two components, one of which is a singleton. Since (n− 1)!− 2(n− 2)!− 1 > 4n− 9 for
n ≥ 6, the large component of P ln−Fl is connected to P Jn −FJ . Thus, |V (H)| ≤ 3|I| = 3.
It is clear that c(H) ≤ |V (H)| ≤ 3, a contradiction.
Case 3.2: n− 2 ≤ fi ≤ fj ≤ 2n− 7 < fk ≤ 2n− 5.
For each l ∈ {i, j}, if P ln − Fl is disconnected, by Lemma 2.5, P ln − Fl has at
most two components, one of which is a singleton. By a similar argument as Case 3.1,
the large component of P ln − Fl is connected to P Jn − FJ . Since |fk| ≤ 2n − 5 ≤
3(n−1)−8 for n ≥ 6, by Lemma 2.6, either P kn −Fk is connected or P kn −Fk has at most
three components and all smaller components contain at most two vertices in total. Since
(n − 1)! − 2(n − 2)! − 2 > 4n − 9 ≥ |F | for n ≥ 6, the large component of P kn − Fk is
connected to P Jn − FJ . Thus, |V (H)| ≤ 4. Then, an argument similar to Case 2.1 shows
that c(H) ≤ 3, a contradiction.
4 The edge component connectivity of Pn
Theorem 4.1. For n ≥ 3, cλ3(Pn) = 2n− 3.
Proof. Take an edge e = xy and F = E(x)∪E(y). Then |F | = 2n−3 and Pn−F has at
least three components. Hence cλ3(Pn) ≤ 2n− 3. It suffices to show cλ3(Pn) ≥ 2n− 3.
410 Ars Math. Contemp. 23 (2023) #P3.04 / 403–416
We consider an inductive proof as follows. The statement of theorem holds for n = 3.
We assume that the result holds for Pn−1, and prove that it also holds for Pn, where n ≥ 4.
Suppose that there is an edge set F with |F | ≤ 2n − 4, and Pn − F has at least three
components. Consider n disjoint copies P 1n , P
2
n , · · · , Pnn . Since I = {i ∈ ⟨n⟩ : fi ≥
n− 2}, and |F | ≤ 2n− 4, it is clear that |I| ≤ 2.
Consider the following cases.
Case 1: |I| = 0.
Each P in − Fi is connected for i ∈ ⟨n⟩. For distinct i, j ∈ ⟨n⟩, by Lemma 2.2, the
number of cross edges between P in and P
j
n is (n − 2)!. Since (2n − 4) < 3(n − 2)! for
n ≥ 4, there are at most two [P in, P jn]’s which are contained in F for distinct i, j ∈ ⟨n⟩.
Thus Pn − F is connected, a contradiction.
Case 2: |I| = 1. Let I = {i}.
Case 2.1: n− 2 ≤ fi ≤ 2(n− 1)− 4.
If each P in − Fi is connected for i ∈ ⟨n⟩, since (2n − 4) − (n − 2) < 2(n − 2)! for
n ≥ 4, then there is at most one [P in, P jn] which is contained in F for distinct i, j ∈ ⟨n⟩.
Thus, Pn − F is connected, a contradiction. Hence, there exists i such that P in − Fi is not
connected. By the inductive hypothesis, P in − Fi has at most two components.
Since (2n− 4)− (n− 2) < 2(n− 2)! for n ≥ 4, there is at most one [P jn, P kn ] which
is contained in F for distinct j, k ∈ ⟨n⟩ \ {i}. Thus P Jn − FJ is connected. Furthermore,
|[P in, P Jn −FJ ]| = (n−1)! > 2n−4−(n−2) for n ≥ 4. At least one component of P in−Fi
is connected to P Jn − FJ . Hence Pn − F has at most two components, a contradiction.
Case 2.2: 2n− 5 ≤ fi ≤ 2n− 4.
In this case, we have |F |−fi ≤ (2n−4)− (2n−5) = 1. Then P Jn −FJ is connected.
Note that at most one vertex of P in −Fi is disconnected to P Jn −FJ . Hence Pn −F has at
most two components, a contradiction.
Case 3: |I| = 2.
Let I = {i, j}. Then fi = fj = n − 2 and |F | − fi − fj = 0. Thus P ln − Fl has
at most two components for any l ∈ {i, j} and P Jn − FJ is connected. And so either any
component of P ln − Fl is connected to P Jn − FJ or two singletons are connected and the
other component of P ln − Fl is connected to P Jn − FJ if both P in − Fi and P jn − Fj have a
singleton, respectively. Thus Pn − F has at most two components, a contradiction.
Theorem 4.2. For n ≥ 3, cλ4(Pn) = 3n− 5.
Proof. Take a 3-path xyz and F = E(x)∪E(y)∪E(z). Then |F | = 3n−5 and Pn−F has
at least four components. Hence cλ4(Pn) ≤ 3n−5. It suffices to show cλ4(Pn) ≥ 3n−5.
We consider an inductive proof as follows. The statement of theorem holds for n = 3.
We assume that the result holds for Pn−1, and prove that it also holds for Pn, where n ≥ 4.
Suppose that there is an edge set F with |F | ≤ 3n − 6, and Pn − F has at least four
components. Consider n disjoint copies P 1n , P
2
n , · · · , Pnn . Since I = {i ∈ ⟨n⟩ : fi ≥
n− 2}, and |F | ≤ 3n− 6, it is clear that |I| ≤ 3.
Consider the following cases.
Case 1: |I| = 0.
Similar to the proof of Case 1 of Theorem 4.1, we can show that Pn − F is connected
for n ≥ 5, a contradiction. Consider that n = 4. Since (4−2)! = 2 and |F | ≤ 3n− 6 = 6,
X. Hua and L. Yang: Component (edge) connectivity of pancake graphs 411
there are at most three [P i4, P
j
4 ]’s which are contained in F . Hence P4 −F has at most two
components, a contradiction.
Case 2: |I| = 1. Let I = {i}.
Case 2.1: n− 2 ≤ fi ≤ 3(n− 1)− 6.
Similar to the proof of Case 2.1 of Theorem 4.1, we can show that Pn − F has at most
three components for n ≥ 5, a contradiction. Consider that n = 4. Then 2 ≤ fi ≤ 3. If
fi = 2, then P i4 − Fi has at most two components, and |F | − fi ≤ (3n− 6)− 2 = 4. It is
not hard to prove that P4 −F has at most two components, a contradiction. If fi = 3, then
P i4 − Fi has at most three components, and |F | − fi ≤ (3n− 6)− 3 = 3. It is not hard to
prove that P4 − F has at most three components, a contradiction.
Case 2.2: 3n− 8 ≤ fi ≤ 3n− 6.
In this case, we have |F | − fi ≤ (3n − 6) − (3n − 8) = 2. Furthermore, P Jn − FJ is
connected. Note that at most two vertices of P in −Fi are disconnected to P Jn −FJ . Hence
Pn − F has at most three components, a contradiction.
Case 3: |I| = 2.
Let I = {i, j}. Without loss of generality, assume fi ≤ fj . Then fj ≤ 3n − 6 −
(n− 2) = 2n− 4.
Case 3.1: n− 2 ≤ fj ≤ 2(n− 1)− 4.
In this case, we have n− 2 ≤ fi ≤ fj ≤ 2(n− 1)− 4. By Theorem 4.1, both P in − Fi
and P jn − Fj have at most two components.
Consider that n = 4. Then fi = fj = 2, implying that P l4 − Fl has at most two
components for l ∈ {i, j}, and |F | − fi − fj ≤ (3n − 6) − 2 − 2 = 2. It is not hard to
prove that P4 − F has at most three components, a contradiction.
Consider that n ≥ 5. Since |[P kn , P ln]| = (n − 2)! > 3n − 6 − 2(n − 2) for n ≥ 5
and k, l ∈ ⟨n⟩ \ {i, j}, P Jn −FJ is connected. Furthermore, |[P in, P Jn −FJ ]| = (n− 1)!−
(n− 2)! > 3n− 6− 2(n− 2) for n ≥ 5. At least one component of P in − Fi is connected
to P Jn −FJ . Similarly, at least one component of P jn−Fj is connected to P Jn −FJ . Hence
Pn − F has at most three components, a contradiction.
Case 3.2: fj = 2n− 5.
Then n− 2 ≤ fi ≤ 3n− 6− (2n− 5) = n− 1.
If fi = n − 2, P in − Fi has at most two components. Then |F | − fi − fj ≤ 1, and
so P Jn − FJ is connected. Thus Pn − F has at most three components, a contradiction. If
fi = n− 1, then |F | − fi − fj = 0 and P Jn − FJ is connected. Thus Pn − F has at most
three components, a contradiction.
Case 3.3: fj = 2n− 4.
Then fi = n − 2 and |F | − fi − fj = 0. Thus P in − Fi has at most two components
and P Jn − FJ is connected. Thus Pn − F has at most two components, a contradiction.
Case 4: |I| = 3.
Let I = {i, j, k}. Then fi = fj = fk = n − 2 and |F | − fi − fj − fk = 0. Thus
P ln − Fl has at most two components for any l ∈ {i, j, k}. Thus Pn − F has at most two
components, a contradiction.
Theorem 4.3. For n ≥ 3, cλ5(Pn) = 4n− 7.
Proof. Take a 4-path xyzw and F = E(x) ∪ E(y) ∪ E(z) ∪ E(w). Then |F | = 4n − 7
and Pn − F has at least five components. Hence cλ5(Pn) ≤ 4n − 7. It suffices to show
cλ5(Pn) ≥ 4n− 7.
412 Ars Math. Contemp. 23 (2023) #P3.04 / 403–416
We consider an inductive proof as follows. The statement of theorem holds for n = 3.
We assume that the result holds for Pn−1, and prove that it also holds for Pn, where n ≥ 4.
Suppose that there is an edge set F with |F | ≤ 4n − 8, and Pn − F has at least five
components. Consider n disjoint copies P 1n , P
2
n , · · · , Pnn . Since I = {i ∈ ⟨n⟩ : fi ≥
n− 2}, and |F | ≤ 4n− 8, it is clear that |I| ≤ 4.
Consider the following cases.
Case 1: |I| = 0.
Similar to the proof of Case 1 of Theorem 4.2, we can show that Pn − F is connected
for n ≥ 5 and P4 − F has at most two components, a contradiction.
Case 2: |I| = 1. Let I = {i}.
Case 2.1: n− 2 ≤ fi ≤ 4(n− 1)− 8.
Similar to the proof of Case 2.1 of Theorem 4.1, we can show that Pn − F has at
most three components for n ≥ 5, a contradiction. Consider that n = 4. Then 2 ≤
fi ≤ 4 and (4 − 2)! = 2. If fi = 2, then P i4 − Fi has at most two components, and
|F | − fi ≤ (4n − 8) − 2 = 6. It is not hard to prove that P4 − F has at most three
components, a contradiction. If fi = 3, then P i4 − Fi has at most three components, and
|F | − fi ≤ (4n − 8) − 3 = 5. It is not hard to prove that P4 − F has at most three
components, a contradiction. If fi = 4, then P i4 −Fi has at most four components, three of
which are singletons, and |F | − fi ≤ (4n− 8)− 4 = 4. It is not hard to prove that P4 −F
has at most four components, a contradiction.
Case 2.2: 4n− 11 ≤ fi ≤ 4n− 8.
In this case, we have |F | − fi ≤ (4n − 8) − (4n − 11) = 3. Since 3 < 2(n − 2)!
for n ≥ 4, there is at most one [P jn, P kn ] which is contained in F for j, k ∈ ⟨n⟩ \ {i}, and
so P Jn − FJ is connected. Note that at most three vertices of P in − Fi are disconnected to
P Jn − FJ . Hence Pn − F has at most four components, a contradiction.
Case 3: |I| = 2.
Let I = {i, j}. Without loss of generality, assume fi ≤ fj . Then fj ≤ 4n− 8− (n−
2) = 3n− 6.
Case 3.1: n− 2 ≤ fj ≤ 2(n− 1)− 4.
Similar to the proof of Case 3.1 of Theorem 4.2, we can show that Pn − F has at most
three components, a contradiction.
Case 3.2: 2n− 5 ≤ fj ≤ 3n− 9.
Consider that n = 4. Then fj = 3 and 2 ≤ fi ≤ 3. Then P j4 − Fj has at most three
components. If fi = 2, then P i4 − Fi has at most two components, and |F | − fi − fj ≤
(4n− 8)− 2− 3 = 3. It is not hard to prove that P4 − F has at most four components, a
contradiction. If fi = 3, then P i4 − Fi has at most three components, and |F | − fi − fj ≤
(4n−8)−3−3 = 2. Suppose either P i4−Fi or P
j
4 −Fj contains no singleton, it is not hard
to prove that P4 −F has at most three components, a contradiction. Suppose both P i4 −Fi
and P j4 −Fj contain singletons, then P4−F has at most four components, a contradiction.
Otherwise, P4−F have five components, four of which are singletons. If two singletons of
P l4 are not an edge of P
l
4 for l ∈ {i, j}, then fl ≥ 4, a contradiction. Thus, two singletons
form an edge of P i4 and the other two singletons form an edge of P
j
4 , implying that the four
singletons form a 4-cycle, contradicting Lemma 2.2(2).
X. Hua and L. Yang: Component (edge) connectivity of pancake graphs 413
Consider that n ≥ 5. If fi ≥ 2n − 3, then fi + fj ≥ 2(2n − 3) > 4n − 8 ≥ |F |,
a contradiction. Thus n − 2 ≤ fi ≤ 2n − 4. Note that 2n − 5 ≤ fj ≤ 3n − 9. By
Theorem 4.2, P jn − Fj has at most three components. Since |[P kn , P ln]| = (n − 2)! >
4n− 8− (n− 2)− (2n− 5) for k, l ∈ ⟨n⟩ \ {i, j}, P Jn − FJ is connected. Furthermore,
|[P in, P Jn − FJ ]| = (n − 1)! − (n − 2)! > 4n − 8 − (n − 2) − (2n − 5). At least one
component of P in − Fi is connected to P Jn − FJ . Similarly, at least one component of
P jn − Fj is connected to P Jn − FJ .
If n− 2 ≤ fi ≤ 2n− 6, by Theorem 4.1, P in −Fi has at most two components. Hence
Pn −F has at most four components, a contradiction. If fi = 2n− 5, and assume first that
fj = 2n−5. Then |F |−fi−fj = 4n−8− (2n−5)− (2n−5) = 2. By Theorem 4.1, we
have P ln − Fl has three components for l ∈ {i, j}. Similar to the case of fi = 3 in the first
paragraph of Case 3.2, Pn −F has at most four components, a contradiction. Now assume
that 2n−4 ≤ fj ≤ 2n−3, then |F |−fi−fj ≤ 4n−8−(2n−5)−(2n−4) = 1. It is not
hard to prove that Pn − F has at most three components, a contradiction. If fi = 2n − 4,
then fj = fi = 2n − 4 and |F | − fi − fj = 4n − 8 − 2(2n − 4) = 0. By Theorem 4.2,
P in − Fi has at most three components. Hence Pn − F has at most three components, a
contradiction.
Case 3.3: fj = 3n− 8.
It follows that n − 2 ≤ fi ≤ n. If fi = n − 2, by Theorem 4.1, P in − Fi has at most
two components. Then |F | − fi − fj ≤ 4n − 8 − (3n − 8) − (n − 2) = 2. It is not
hard to prove that Pn − F has at most four components, a contradiction. If fi = n− 1, by
Theorem 4.2, n− 1 < 3(n− 1)− 5 for n ≥ 4, then P in−Fi has at most three components,
and |F | − fi − fj ≤ 4n − 8 − (3n − 8) − (n − 1) = 1. Then Pn − F has at most
four components, a contradiction. If fi = n, then |F | − fi − fj = 0. By Theorem 4.2,
both P in − Fi and P jn − Fj have at most four components. Then Pn − F has at most four
components, a contradiction.
Case 3.4: fj = 3n− 7.
Similar to the proof of Case 3.3 of Theorem 4.3, we can show that Pn − F has at most
three components, a contradiction.
Case 3.5: fj = 3n− 6.
Then fi = n − 2 and |F | − fi − fj = 0. By Theorem 4.1, P in − Fi has at most two
components. Thus Pn − F has at most two components, a contradiction.
Case 4: |I| = 3.
Let I = {i, j, k}. Without loss of generality, assume fi ≤ fj ≤ fk. Then fk ≤
4n − 8 − 2(n − 2) = 2n − 4. Consider n = 4. Then fi = 2, fj = 2, fk = 2, or
fi = 2, fj = 2, fk = 3, or fi = 2, fj = 2, fk = 4, or fi = 2, fj = 3, fk = 3. No matter
which case, it’s not hard to prove that P4−F has at most four components, a contradiction.
Next, we consider n ≥ 5. If fj ≥ 2n − 5, then fi + fj + fk ≥ n − 2 + 2(2n − 5) =
5n− 12 > 4n− 8 ≥ |F | for n ≥ 5, a contradiction. Then fj ≤ 2n− 6.
Case 4.1: n− 2 ≤ fi ≤ fj ≤ fk ≤ 2n− 6.
By Theorem 4.1, P ln − Fl has at most two components for any l ∈ {i, j, k}. Since
|[P xn , P yn ]| = (n− 2)! > 4n− 8− 3(n− 2) for n ≥ 5 and x, y ∈ ⟨n⟩ \ {i, j, k}, P Jn − FJ
is connected. Furthermore, |[P ln, P Jn − FJ ]| = (n− 1)!− 2(n− 2)! > 4n− 8− 3(n− 2)
for n ≥ 5. At least one component of P ln − Fl is connected to P Jn − FJ . Hence Pn − F
has at most four components, a contradiction.
Case 4.2: n− 2 ≤ fi ≤ fj ≤ 2n− 6 < fk ≤ 2n− 4.
414 Ars Math. Contemp. 23 (2023) #P3.04 / 403–416
By Theorem 4.1, P ln − Fl has at most two components for any l ∈ {i, j}, and by
Theorem 4.2, P kn − Fk has at most three components. Since |F | − fi − fj − fk ≤
4n − 8 − 2(n − 2) − (2n − 5) = 1, P Jn − FJ is connected, and at most four vertices
of P in−Fi, P jn−Fj and P kn −Fk are disconnected to P Jn −FJ . If four vertices of P in−Fi,
P jn −Fj and P kn −Fk are disconnected to P Jn −FJ , then two of which forms an edge, and
then Pn −F has at most four components, a contradiction. Otherwise, Pn −F has at most
four components, a contradiction.
Case 5: |I| = 4.
Let I = {i, j, k, p}. Then fi = fj = fk = fp = n−2 and |F |−fi−fj −fk−fp = 0.
Thus P ln − Fl has at most two components for any l ∈ {i, j, k, p}. Thus Pn − F has at
most three components, a contradiction.
5 Concluding remarks
In this paper, we study the l-component (edge) connectivity of Pn for 3 ≤ l ≤ 5. We
have know that the l-component connectivity of Pn are cκ3(Pn) = 2n − 3 for n ≥ 3,
cκ4(Pn) = 3n − 6 for n ≥ 4, cκ5(Pn) = 4n − 8 for n ≥ 6. Also, for n ≥ 3, we have
know the l-component edge connectivity of Pn are cλ3(Pn) = 2n− 3, cλ4(Pn) = 3n− 5,
cλ5(Pn) = 4n− 7. We study the larger component (edge) connectivity of Pn in the future
work.
ORCID iDs
Xiaohui Hua https://orcid.org/0000-0002-1215-3616
Lulu Yang https://orcid.org/0000-0002-7801-4862
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P3.05 / 417–440
https://doi.org/10.26493/1855-3974.2930.8e4
(Also available at http://amc-journal.eu)
Hamilton cycles in primitive graphs
of order 2rs*
Shaofei Du † , Yao Tian , Hao Yu
Capital Normal University, School of Mathematical Sciences,
Bejing 100048, People’s Republic of China
Received 24 July 2022, accepted 4 November 2022, published online 24 January 2023
Abstract
After long term efforts, it was recently proved by Du, Kutnar and Marušič in 2021
that except for the Petersen graph, every connected vertex-transitive graph of order rs has
a Hamilton cycle, where r and s are primes. A natural topic is to solve the hamiltonian
problem for connected vertex-transitive graphs of 2rs. This topic is quite nontrivial, as
the problem is still unsolved even for that of r = 3 and 5. In this paper, it is shown that
except for the Coxeter graph, every connected vertex-transitive graph of order 2rs contains
a Hamilton cycle, provided the automorphism group acts primitively on vertices.
Keywords: Vertex-transitive graph, Hamilton cycle, primitive group, automorphism group, orbital
graph.
Math. Subj. Class. (2020): 05C25, 05C45
1 Introduction
Throughout this paper graphs are finite, simple and undirected, and groups are finite. Given
a graph X , by V (X), E(X) and Aut(X) we denote the vertex set, the edge set and the
automorphism group of X , respectively. A graph X is vertex- or arc-transitive if Aut(X)
acts transitively on vertices or arcs, respectively.
Given a transitive group G on Ω, a subset B of Ω is called a block of G if, for any
g ∈ G, we have either B = Bg or B ∩ Bg = ∅. Clearly, G has blocks Ω and {α} for any
α ∈ Ω, which are said to be trivial. Then G is said to be primitive if it has no nontrivial
blocks. Moreover, a vertex-transitive graph X is said to be primitive if Aut(X) is primitive
on vertices.
*The authors would like to thank the referees for their helpful suggestions. This work is partially supported by
the National Natural Science Foundation of China (12071312 and 11971248). All authors declare that this paper
has no conflict of interest.
†Corresponding author.
E-mail addresses: dushf@mail.cnu.edu.cn (Shaofei Du), tianyao202108@163.com (Yao Tian),
3485676673@qq.com (Hao Yu)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
418 Ars Math. Contemp. 23 (2023) #P3.05 / 417–440
A simple path (resp. cycle) containing all vertices of a graph is called a Hamilton path
(resp. cycle) of this graph. For convenience, a Hamilton-cycle (resp. path) is usually abbre-
viated by a H-cycle (resp. H-path). A graph containing a Hamilton cycle will be sometimes
referred as a hamiltonian graph.
In 1970, Lovász asked in [1] that
Does every finite connected vertex-transitive graph have a Hamilton path?
Up to now, this question remains unresolved and no connected vertex-transitive graph with-
out a Hamilton path is known to exist. Moreover, only four (families) of connected vertex-
transitive graphs on at least three vertices not having a Hamilton cycle are known, which are
Petersen graph, Coxeter graph and triangle-replaced graphs from them. Since all of these
graphs are not Cayley graph, we may ask if every connected Cayley graph has a Hamilton
cycle.
It has been shown that connected vertex-transitive graphs of orders kp, k ≤ 6, 10p
(p ≥ 11), pj (j ≤ 5) and 2p2, where p is a prime contain a Hamilton path, see [2, 5, 18, 19,
20, 25, 26, 27, 28, 31]. Furthermore, for all of these families, except for the graphs of order
6p and 10p and that four exceptions, they contain a Hamilton cycle. With the exception of
the Petersen graph, Hamilton cycles are also known to exist in connected vertex-transitive
graphs whose automorphism groups contain a transitive subgroup with a cyclic commutator
subgroup of prime-power order (see [6] and also [9, 17, 24]).
So far we know that Cayley graphs of the following groups contain a Hamilton cycle:
nilpotent groups of odd order, with cyclic commutator subgroups (see [6, 11, 12]); dihedral
groups of order divisible by 4 (see [3]); and arbitrary p-groups (see [30]). A Hamilton path
and in some cases even a Hamilton cycle was proved to exist in cubic Cayley graphs arising
from (2, s, 3)-generated groups (see [13, 14, 15]).
Recently, Kutnar, Marusic and the first author proved that vertex transitive graphs of
order rs have a Hamilton cycle, except for the Petersen graph (see [7, 8]). This work took
many years, because of a difficult case, which is a primitive graph with automorphism
group PSL(2, p) and a point-stabilizer Dp−1. A natural question is to consider hamiltonian
problem for vertex-transitive graphs of order 2rs. As mentioned above, some special cases
have been solved such as that of graphs of order 4p, 6p, 10p and 2p2, where p is a prime
(Hamilton path or cycle). To solve the general case, a necessary step is to deal with all
primitive graphs of such order. The main result of this paper is the following theorem.
Theorem 1.1. Except for Coxeter graph, every connected vertex-transitive graph of order
2rs contains a Hamilton cycle provided the automorphism group acts primitively on its
vertices, where r and s are primes.
After this introductory section, some notations, basic definitions and useful facts will
be given in Section 2 and Theorem 1.1 will be proved in Section 3.
2 Preliminaries
By ⌊a⌋ and ⌈a⌉, we denote the largest integer that is smaller than a and smallest integer
that is larger than a, respectively. For a prime q, a finite field of order q will be denoted by
Fq . Set F∗q = Fq \ {0}, S = {t2
∣∣ t ∈ Fq}, S∗ = S ∩ F∗q and N = F∗q \ S∗. Then the
elements in S and N are called to be squares and non-squares, respectively. By Zn and
D2n we denote a cycle group of order n and dihedral group of order 2n, respectively. For
S. Du et al.: Hamilton cycles in primitive graphs of order 2rs 419
a group G and L ⊂ G, by CG(L) and NG(L) we denote the centralizer and normalizer
of L in G, respectively. A semi-product of K and H is denoted by K ⋊ H , where K is
normal. Let G be a group with a normal subgroup N , we denote the image of g ∈ G under
the natural homomorphism of G to G/N by g. For a group G and its subgroup H , [G : H]
denotes the set of right cosets of H in G; HgH denotes the orbit containing Hg under the
action of H . Recall that the socle of G which is denoted by soc(G) is defined to be the
product of all minimal normal subgroups of G.
Let G act on some set Ω. For some α ∈ Ω and g ∈ G, set αG = {αg
∣∣ g ∈ G}. For
α ∈ Ω, set H = Gα. Then the action of G on Ω is equivalent to its right multiplication
action on right cosets [G : H] relative to H . For a subset ∆ of Ω, by G(∆) and G{∆}, we
denote the pointwise and setwise stabilizer of ∆ in G, respectively.
In a graph X , let a ∈ V (X) and B ⊂ V (X), by d(a, B) we denote the number of
neighbors of a in B. Given A, B ⊂ V (X), if d(a, B) = d(a′, B) for any a, a′ ∈ A, then
we denote d(a, B) by d(A, B). Moreover, set d(B) = d(B,B). The neighborhood of any
vertex a in the graph X is denoted by X1(a).
In what follows we recall some definitions related to orbital graphs and semiregular
automorphisms.
Let G be a transitive permutation group on Ω. Then G induces a natural action on
Ω × Ω. We call the orbits of G on Ω × Ω the orbitals of G, and in particular the trivial
orbital is referred to {(α, α) | α ∈ Ω}. The orbital digraph X(G, Γ) relative to an
orbital Γ is defined to be the directed graph with vertex set Ω and edge set Γ. Each orbital
Γ has an associated paired orbital Γ′ defined by Γ′ = {(β, α)
∣∣ (α, β) ∈ Γ}, and of
course, Γ is said to be self-paired if Γ = Γ′ in which case X(G, Γ) can be viewed as
an undirected graph (orbital graph). The G-arc-transitive graphs with vertex-set Ω are
precisely the orbital graphs X(G, Γ) for the nontrivial self-paired orbitals Γ. In addition,
take a point α ∈ Ω, the orbits of the stabilizer Gα on Ω are called suborbits of G relative
to α. There is a one-to-one correspondence between the suborbits and the orbitals of G.
Each orbital Γi corresponds to a suborbit ∆i = {β ∈ Ω | (α, β) ∈ Γi }. Conversely, each
suborbit ∆i corresponds to an orbital Γi = { (α, β)g | g ∈ G, β ∈ ∆i }. A suborbit of
G is said to be self-paired if the corresponding orbital is self-paired. Thus we often use
X(G, ∆i) and X(G, ∆i ∪∆′i) to denote graphs X(G, Γ) and X(G, Γ∪Γ′) respectively.
Let m ≥ 1 and n ≥ 2 be integers. An automorphism ρ of a graph X is called (m, n)-
semiregular (in short, semiregular) if as a permutation on V (X) it has a cycle decomposi-
tion consisting of m cycles of length n. If m = 1 then X is called a circulant; it is in fact a
Cayley graph of a cyclic group of order n. Let P be the set of orbits of ρ, that is, the orbits
of the cyclic subgroup ⟨ρ⟩ generated by ρ. We let the quotient graph corresponding to P
be the graph XP whose vertex set equals P with A, B ∈ P adjacent if there exist vertices
a ∈ A and b ∈ B, such that a ∼ b in X .
The following four results will be used later.
Proposition 2.1 ([29, page 167]). Let Fq be the finite field of odd prime order q. Then
|(S∗ + 1) ∩ (−S∗)| =
{
(q − 5)/4 q ≡ 1 (mod 4),
(q + 1)/4 q ≡ 3 (mod 4).
This implies that if q ≡ 1 (mod 4) then
|S∗∩ (S∗+1)| = (q−5)/4, |N ∩ (N +1)| = (q−1)/4, |S∗∩ (N ±1)| = (q−1)/4.
420 Ars Math. Contemp. 23 (2023) #P3.05 / 417–440
No. soc(G) 2rs Action Comment
1 PSL(2, q) q(q + 1)/2 Gα ∩ soc(G) = D2(q−1)/d d = (2, q − 1),
G = PGL(2, 11)
for q = 11
2 PSL(2, q) q(q − 1)/2 Gα ∩ soc(G) = D2(q+1)/d d = (2, q − 1)
3 PSL(2, 47) 2× 47× 23 S4
4 PSL(2, 17) 2× 17× 3 S4
5 PSL(2, 41) 2× 41× 7 A5
Table 1: Primitive groups of degree 2rs, where the socle PSL(2, q).
Proposition 2.2 ([16, Theorem 6] (Jackson’s Theorem)). Every 2-connected regular graph
of order n and valency at least n/3 contains a Hamilton cycle.
Proposition 2.3 ([4, Corollary 3]). If X is a connected Cayley graph of an abelian group
of order at least 3, then every edge of X lies in a hamiltonian cycle.
Lemma 2.4 ([27, Lemma 5]). Let X be a graph admitting an (m, p)-semiregular auto-
morphism ρ, where p is a prime. Let C be a cycle of length m in the quotient graph XP ,
where P is the set of orbits of ρ. Then, the lift of C either contains a cycle of length mp or
it consists of p disjoint m-cycles. In the latter case we have d(S, S′) = 1 for every edge
SS′ of C.
3 Proof of Theorem 1.1
To prove Theorem 1.1, let X be a connected vertex-transitive graph of order 2rs, where r
and s are primes. Set G = Aut(X). It has been proved that X contains a Hamilton cycle
if 2rs = 2p2 or 4p for a prime p, provided X is not the Coxeter graph which is of order 28.
Therefore, in what follows we assume that r < s. If G acts 2-transitively on V (X), then X
is a complete graph, which contains a H-cycle. Now we need to consider all the primitive
groups of degree 2rs of rank at least 3 from [10] (or [21]), where r and s are distinct odd
primes. Let H be a point stabilizer in soc(G). Checking [10], all the possible groups are
listed in Tables 1 and 2.
Table 1 gives the these groups with the socle PSL(2, q). The first two cases H = Dq−1
and H = Dq+1 will be dealt with in Subsections 3.1 and 3.2, respectively. With the help of
Magma, we can show that every vertex-transitive graph is hamiltonian, arising from other
three groups in Table 1.
Table 2 gives these groups whose socle is a classical simple group which is not PSL(2, q),
an alternative group or a sporadic simple group. These groups will be dealt with in Subsec-
tion 3.3.
3.1 soc(G) = PSL(2, q) and H = Dq−1
Let G = PSL(2, q) and H = Dq−1. Consider the action of G on the set [G : H] of cosets
of H in G, see row 1 of Table 1. Then the degree is q(q + 1)/2 = 2rs, thus q ≡ 3(mod 4)
and in particular −1 ∈ N , the set of non-squares. Set F∗q = ⟨θ⟩.
S. Du et al.: Hamilton cycles in primitive graphs of order 2rs 421
No. soc(G) 2rs Action Comment
1 PSL(4, q) q
3−1
q−1 (q
2 + 1) 2-spaces q = 3; or q = 5; or
q ≡ 11, 29(mod 30) and
q prime and q ≥ 59
2 PSL(5, q) q
5−1
q−1 (q
2 + 1) 2-spaces q ≡ −1(mod 10),
q prime and q ≥ 29
3 PΩ−(2m, q) (q
m+1)(qm−1−1)
q−1 on t.s. 1-spaces m even
4 PΩ+(2m, q) (q
m−1)(qm−1+1)
q−1 on t.s. 1-spaces m odd
5 PSL(3, 5) 2× 31× 3 on (1, 2)-dim. flags G = PSL(3, 5).2
6 Ac
c(c+1)
2 on 2-sets c ≥ 5
7 M11 66 S5
8 M12 66 M10 : 2
9 M23 506 A8
10 J1 266 PSL(2, 11)
Table 2: Primitive groups G of degree 2rs, where soc(G) ̸= PSL(2, q).
For any g ∈ SL(2, q), set g = gZ(SL(2, q)). In SL(2, q), set
u =
(
1 1
0 1
)
, u′ =
(
1 0
1 1
)
, l =
(
θ 0
0 θ−1
)
, t =
(
0 1
−1 0
)
.
Since PSL(2, q) has only one conjugacy class of subgroups isomorphic to Dq−1, we may
set H = ⟨l, t⟩. Let V be the row vector space so that the action of g ∈ GL(2, q) on a
vector (x, y) is just defined as (x, y) · g. Set yx = ⟨(x, y)⟩. Then all the projective points
are {∞, 0, 1, 2, · · · , q − 1}. The action of G on [G : H] is equivalent to its action on the
set of unordered pairs of distinct projective points, where H = G{0,∞}. Thus we have
u′ : {∞, 0} → {1, 0}, li : {j, j + 1} → {jθ−2i, (j + 1)θ−2i},
lit : {j, j + 1} → {−j−1θ2i, −(j + 1)−1θ2i}.
Then the all ⟨u⟩-orbits are
B∞ = {{∞, i}|i ∈ Fq}, Bj = {{i, i+ j}|i ∈ Fq}, j ∈ {1, 2, 3, · · · ,
q − 1
2
}.
Set B = {Bj
∣∣ j ∈ {1, 2, 3, · · · , q−12 }}. Considering the action of NG(⟨u⟩) = ⟨u⟩⋊ ⟨l⟩
on the vertices, we know that NG(⟨u⟩) fixes the block B∞ setwise and acts transitively on
other vertices. In particular, ⟨l⟩ fixes B∞ and acts regularly on q−12 remaining blocks Bj
in B.
The suborbits of G have been determined in [22] and an alternative description is given
below.
Lemma 3.1. Suppose q ≡ 3(mod 4). Then every nontrivial suborbit of G relative to H
can be written as {j, j + 1}H , where j ∈ Fq , with length q−12 and q − 1 if and only if
j2 + j ∈ N and j2 + j ∈ S, respectively. Moreover, {j, j + 1}H is self-paired if and
only if either j + 1 ∈ N or j ∈ S, and if it is non self-paired, then its paired suborbit is
{−j, −j − 1}H .
422 Ars Math. Contemp. 23 (2023) #P3.05 / 417–440
Proof. For i ∈ F∗q , direct computations show that {∞, i} belongs to {0, 1}H or {0, −1}H
depending on whether i ∈ S∗ or i ∈ N , respectively. Since ⟨l⟩ ≤ H acts regularly on B,
any other suborbits can also be written as {j, j + 1}H . The length of {j, j + 1}H is q−12
and q−1 if and only if the order of the stabilizer for {j, j+1} in H is 2 and 1, respectively.
But the former holds if and only if there exists some k ∈ Zq such that lkt fixes {j, j + 1},
i.e., j +1 = −j−1θ2k. Therefore we deduce that the length of the suborbit is q−12 or q− 1
depending on j2 + j ∈ N or j2 + j ∈ S, respectively.
Let ∆ = {j, j + 1}H . If j + 1 = 0, then ∆∗ = {0, 1}H . If j + 1 ̸= 0, then
∆∗ = { −jj+1 , −1}
H . Now, ∆ is self-paired if and only if there exists some element of H
mapping {j, j + 1} to { −jj+1 , −1}. From
{lk(j), lk(j + 1)} = {jθ−2k, (j + 1)θ−2k} = { −j
j + 1
, −1} and
{lkt(j), lkt(j + 1)} = {−j−1θ2k, −(j + 1)−1θ2k} = { −j
j + 1
, −1},
we know that such element of H exists if and only if j + 1 ∈ N or j ∈ S, as desired.
Suppose that ∆ is not self-paired and j + 1 ̸= 0. Then j + 1 = θ−2k ∈ S and lk maps
{ −jj+1 , −1} to {−j, −j − 1}, that is ∆
∗ = { −jj+1 , −1}
H = {−j, −j − 1}H .
Remark 3.2. By Lemma 3.1, it is easy to determine the number of nontrivial suborbits of
length q−12 or q − 1, and the number of nontrivial paired suborbits. But we do not need
these numbers in here.
Before going to prove the main result, we first give a technical lemma on number theory.
Lemma 3.3. Suppose that q is an odd prime. If a, b ∈ F∗q and a ̸= b. Then
|(S∗ + a) ∩ (S∗ + b) ∩N | ≤ ⌈ 18 (q + 11 + 2
√
q)⌉,
|(S∗ + a) ∩ (N + b) ∩N | ≤ ⌈18 (q + 11 + 2
√
q)⌉,
|(S∗ + a) ∩ (N + b) ∩ S∗| ≥ ⌊ 18 (q − 11− 2
√
q)⌋,
|(N + a) ∩ (N + b) ∩ S∗| ≥ ⌊18 (q − 11− 2
√
q)⌋.
Proof. Set η : F∗q → {±1} by assigning the elements of S∗ to 1 and that of N to −1 and
moreover, set η(0) = 0. This η is exactly that in [23, Example 5.10]. Also we need to
quote the following three results from [23, Theorems 5.4, 5.48, 5.41]:
(i)
∑
x∈Fq η(x) = 0;
(ii)
∑
x∈Fq η(x
2 + Ax + B) = q − 1 for A2 − 4B = 0 or −1 for otherwise, where
A,B ∈ Fq;
(iii) |m| ≤ 2√q, where m :=
∑
x∈Fq η(x(x− 1)(x− t)) and t ∈ Fq .
For four inequalities of the lemma, we have the same arguments and here we just prove
the first one. Set W = (S∗ + a) ∩ (S∗ + b) ∩N , that is
W = {x ∈ Fq | η(x− a) = η(x− b) = 1, η(x) = −1}.
S. Du et al.: Hamilton cycles in primitive graphs of order 2rs 423
Now let a, b ∈ S∗. Then by the above three formulas (i) – (iii), we have
|W | = 1
8
∑
x∈Fq\{0,a,b}
(1 + η(x− a))(1 + η(x− b))(1− η(x))
=
1
8
∑
x∈Fq\{0,a,b}
(1− η(x) + η(x− a) + η(x− b)− η(x(x− a))− η(x(x− b))
+ η((x− a)(x− b))− η((x− a)(x− b)x)
=
1
8
[(q − 3)− (−η(b)− η(a))− (η(−a) + η(b− a))− (η(−b) + η(a− b))
− (−1− ηb(b− a))− (−1− ηa(a− b)) + (−1− η(ab)) +m]
≤ ⌈1
8
(q + 11 + 2
√
q)⌉.
According to Lemma 3.1, we shall deal with the orbital graphs X = X(G, ∆) or
X = X(G, ∆∪∆∗), according to that ∆ is self-paired and of length q−12 , non self-paired
and of length q−12 , self-paired and of length q− 1, and non self-paired and of length q− 1,
respectively, in the following four lemmas.
Lemma 3.4. Suppose that ∆ is a self-paired suborbit of length q−12 . Then X(G, ∆) is
hamiltonian.
Proof. Let X = X(G, ∆), where ∆ is self-paired and of length q−12 . Let Y be the quotient
graph induced by ⟨u⟩, with vertices B ∪ {B∞}. Then by Lemma 3.1, we may set ∆ =
{j, j + 1}H , where j(j + 1) ∈ N , j + 1 ∈ N and j ∈ Fq. Then the neighborhood of
{0,∞} is:
X1({0, ∞}) = ∆ = {{jθ−2k, (j + 1)θ−2k}
∣∣ k ∈ Fq}.
Since |∆| = q−12 and ⟨l⟩ acts regularly on B, d(B∞, Bi) = 1 for any i = 1, 2, 3, · · · ,
q−1
2 .
The lemma will be proved by the following three steps:
Step 1: Show d(Bm, Bi) ≤ 2 for any i, m = 1, 2, 3, · · · , q−12 .
Since ⟨l⟩ is regular on B and {0, 1} ∈ B1, we may just consider d(B1, Bi) =
d({0, 1}, Bi) for any i = 1, 2, 3, · · · , q−12 . Since u
′ maps {∞, 0} to {0, 1}, we know
that
X1({0, 1}) = ∆u
′
= {{jθ−2k, (j + 1)θ−2k}
∣∣ k ∈ Fq}u′
= {{ jθ
−2k
1 + jθ−2k
,
(j + 1)θ−2k
1 + (j + 1)θ−2k
}
∣∣ k ∈ Fq}.
So a vertex in X1({0, 1}) is contained in Bi if and only if
{ jθ
−2k
1 + jθ−2k
,
(j + 1)θ−2k
1 + (j + 1)θ−2k
} = {t, t+ i} for some t,
if and only if one of the following two systems of equations has solutions:
jθ−2k
1 + jθ−2k
= t,
(j + 1)θ−2k
1 + (j + 1)θ−2k
= t+ i; (3.1)
424 Ars Math. Contemp. 23 (2023) #P3.05 / 417–440
and
jθ−2k
1 + jθ−2k
= t+ i,
(j + 1)θ−2k
1 + (j + 1)θ−2k
= t. (3.2)
Solving Equation (3.1), we get
ij(j + 1)u2 + (2ij + i− 1)u+ i = 0,
where u = θ−2k. This equation has solutions for u if and only if
δ1 := (2ij + i− 1)2 − 4i2j(j + 1) = i2 − (2 + 4j)i+ 1 ∈ S∗.
Suppose that the above equation has solutions, say u1 and u2. Since u1u2 = (j(j+1))−1,
a non-square, we know that u1, u2 ̸= 0, one of them is a non-square and the other one is a
square. Therefore, there exists exactly one solution for θ−2k = u if and only if δ1 ∈ S∗,
noting that every θ−2k gives a unique t, equivalently, a unique vertex in the block Bi.
Solving Equation (3.2), we get
ij(j + 1)u2 + (2ij + i+ 1)u+ i = 0,
where u = θ−2k. This equation has solutions for u if and only if
δ2 := (2ij + i+ 1)
2 − 4i2j(j + 1) = i2 + (2 + 4j)i+ 1 ∈ S∗.
Similarly, there exists exactly one solution for θ−2k if and only if δ2 ∈ S∗.
Summarizing Equation (3.1) and Equation (3.2), we get d({0, 1}, Bi) ≤ 2.
Step 2: Show that for a given j, there exists some i such that d(Bj , Bi) = 2.
It suffices to show d({0, 1}, Bi) = 2 for some i ̸= 0, equivalently, to show that the
number of Bi (i ̸= 1) such that d(B1, Bi) = 1 is less than q−12 − 1− 2 =
q−7
2 .
Now, d(B1, Bi) = 1 if and only if
δ1δ2 = (i
2 − (2 + 4j)i+ 1)(i2 + (2 + 4j)i+ 1) = y ∈ N,
that is
u2 + (2− (2 + 4j)2)u+ 1− y = 0, (3.3)
where u = i2. Note that for a given u ∈ S∗, i and −i give the same block Bi. Thus a
solution of u can provide at most one block Bi satisfying our conditions.
In what follows, we analyse the number of solutions for u.
Equation (3.3) has some solutions for u if and only if
δ := (2− (2 + 4j)2)2 − 4(1− y) ∈ S,
that is
y ∈ S + t, where t = −4j(j + 1) ∈ S.
Now y ∈ (S+ t)∩N . First suppose that 1− y ∈ N . Then y ∈ (S+ t)∩N ∩ (1+S). By
Lemma 3.3, we have at most ⌈ 18 (q+ 11+ 2
√
q)⌉+ 1 choices for y, and then for u as well.
S. Du et al.: Hamilton cycles in primitive graphs of order 2rs 425
Secondly, suppose that 1− y ∈ S. Then y ∈ (S + t) ∩N ∩ (1 +N). By Lemma 3.3, we
have at most ⌈ 18 (q + 11 + 2
√
q)⌉+ 1 choices for y. Since every y may give two solutions
for u, we have at most 2⌈ 18 (q + 11 + 2
√
q)⌉+ 2 solutions for u.
In summary, we have at most
⌈1
8
(q + 11 + 2
√
q)⌉+ 2⌈1
8
(q + 11 + 2
√
q)⌉+ 3
blocks Bi such that d(B0, Bi) = 1. Now
⌈1
8
(q + 11 + 2
√
q)⌉+ 2⌈1
8
(q + 11 + 2
√
q)⌉+ 3 ≤ q − 7
2
,
provided q > 169. In other words, if q > 169 there exists some i such that d(B0, Bi) = 2.
For 7 ≤ q ≤ 169, only the primes 19, 43, 67 and 163 satisfy q(q+1)2 = 2rs. For these
primes, we can get a Hamilton cycle by Magma.
Step 3: Show the existence of a H-cycle.
Let us come back to the proof of the lemma. Let Y1 = Y [B], the subgraph of Y induced
by B. Then Y1 is a Cayley graph on Z q−1
2
. Since the valency of X is q−12 , d(B1, B∞) = 1,
and d(B1, Bi) ≤ 2, it follows from
1
2
(
q − 1
2
− 1− 2) ≥ 1
3
· q − 1
2
that Y1 has at most two connected components. Since q−12 is odd, Y1 must be connected.
Now there are double edges between B1 and Bi for some i. By Proposition 2.3, Y1 contains
a cycle passing the edge B1Bi, say · · ·BjB1Bi · · · . In Y , replacing the edge BjB1 by the
path BjB∞B1, we get a H-cycle, say C for Y . By Proposition 2.4, C can be lifted to a
H-cycle of X .
Lemma 3.5. Suppose that ∆ is a non self-paired suborbit of length q−12 . Then X(G, ∆)
is hamiltonian.
Proof. Let X = X(G, ∆ ∪ ∆∗), where ∆ is non self-paired and of length q−12 . Let Y
be the quotient graph induced by B ∪ {B∞}. Then by Lemma 3.1, we may set ∆ =
{j, j + 1}H and ∆∗ = {−j, −j − 1}H where j(j + 1) ∈ N , j + 1 ∈ S, j ∈ N and
j ∈ Fq. Then the neighborhood of {0, ∞} is:
X1({0, ∞}) = ∆∪∆∗ = {{jθ−2k, (j+1)θ−2k}, {(−j)θ−2k, (−j−1)θ−2k}
∣∣ k ∈ Fq}.
Since |∆ ∪ ∆∗| = q − 1 and ⟨l⟩ acts regularly on B, d(B∞, Bi) = 2 for any i =
1, 2, · · · , q−12 .
The lemma will be proved by the following two steps:
Step 1: d(Bk, Bi) ∈ {0, 2, 4} for any i, k = 1, 2, · · · , q−12 .
Since ⟨l⟩ is regular on B and {0, 1} ∈ B1, we may just consider d(B1, Bi) =
d({0, 1}, Bi) for any i = 1, 2, 3, · · · , q−12 . Since u
′ maps {∞, 0} to {0, 1}, we know
426 Ars Math. Contemp. 23 (2023) #P3.05 / 417–440
that
X1({0, 1}) = {∆, ∆∗}u
′
= {{ jθ
−2k
1 + jθ−2k
,
(j + 1)θ−2k
1 + (j + 1)θ−2k
}, { (−j)θ
−2k
1 + (−j)θ−2k
,
(−j − 1)θ−2k
1 + (−j − 1)θ−2k
}
∣∣ k ∈ Fq}.
A vertex in X1({0, 1}) is contained in Bi if and only if some of the following four systems
of equations has solutions:
jθ−2k
1 + jθ−2k
= t,
(j + 1)θ−2k
1 + (j + 1)θ−2k
= t+ i; (3.4)
jθ−2k
1 + jθ−2k
= t+ i,
(j + 1)θ−2k
1 + (j + 1)θ−2k
= t; (3.5)
(−j)θ−2k
1 + (−j)θ−2k
= t,
(−j − 1)θ−2k
1 + (−j − 1)θ−2k
= t+ i; (3.6)
(−j)θ−2k
1 + (−j)θ−2k
= t+ i,
(−j − 1)θ−2k
1 + (−j − 1)θ−2k
= t. (3.7)
Solving Equation (3.4) and Equation (3.6), we get the respective equation
ij(j + 1)u2 ± (2ij + i− 1)u+ i = 0,
where u = θ−2k. For each of these two equations, it has solutions for u if and only if
δ1 := (2ij + i− 1)2 − 4i2j(j + 1) = i2 − (2 + 4j)i+ 1 ∈ S∗.
Since the product of two solutions u1 and u2 is (j(j + 1))−1, a non-square, we know that
either u1 ∈ S∗ or u2 ∈ S∗ if the above equation has solutions. Therefore, there exists
exactly one solution for θ−2k = u if and only if δ1 ∈ S∗, noting that every θ−2k gives a
unique t, equivalently, a unique vertex in the block Bi. Totally, two systems of equations
give two vertices in the Bi.
Solving Equation (3.5) and Equation (3.7), we get respective equation
ij(j + 1)u2 ± (2ij + i+ 1)u+ i = 0,
where u = θ−2k. This equation has solutions for u if and only if
δ2 := (2ij + i+ 1)
2 − 4i2j(j + 1) = i2 + (2 + 4j)i+ 1 ∈ S∗.
Similarly, there exists exactly one solution for θ−2k if and only if δ2 ∈ S∗. Totally, two
systems of equations give two vertices in the Bi.
In summary, d(B1, Bi) = 2 if and only if δ1δ2 ∈ N ; and d(B1, Bi) = 0 or 4 provided
δ1δ2 ∈ S.
S. Du et al.: Hamilton cycles in primitive graphs of order 2rs 427
Step 2: Show the existence of a H-cycle.
Let Y1 = Y [B], the subgraph of Y induced by B. Then Y1 is a Cayley graph on Z q−1
2
.
Since the valency of X is q − 1, d(B1, B∞) = 2, and d(B1, Bi) ≤ 4, it follows from
1
4
(q − 1− 4− 2) ≥ 1
3
· q − 1
2
that Y1 has at most two connected components. Then, using the same arguments in Step 3
of Lemma 3.4, one may get a H-cycle of X .
Lemma 3.6. Suppose that ∆ is a self-paired suborbit of length q − 1. Then X(G, ∆) is
hamiltonian.
Proof. Let X = X(G, ∆), where ∆ is self-paired and of length q − 1. Let Y be the
quotient graph induced by B∪ {B∞}. Then by Lemma 3.1, we may set ∆ = {j, j + 1}H
where j(j + 1) ∈ S∗ and either j + 1 ∈ N or j ∈ S∗. Then the neighborhood of {0, ∞}
is:
X1({0, ∞}) = ∆ = {{jθ−2k, (j + 1)θ−2k}, {(−j)θ−2k, (−j − 1)θ−2k}
∣∣ k ∈ Fq}.
Since |∆| = q− 1 and ⟨l⟩ acts regularly on B, d(B∞, Bi) = 2 for any i = 1, 2, · · · , q−12 .
The lemma will be proved by the following two steps:
Step 1: d(Bm, Bi) ≤ 4 for any i, m ∈ F∗q .
Since ⟨l⟩ is regular on B and {0, 1} ∈ B1, we may just consider d(B1, Bi) =
d({0, 1}, Bi) for any i ∈ F∗q . Now,
X1({0, 1}) = {∆}u
′
= {{ jθ
−2k
1 + jθ−2k
,
(j + 1)θ−2k
1 + (j + 1)θ−2k
},
{ (−j)θ
−2k
1 + (−j)θ−2k
,
(−j − 1)θ−2k
1 + (−j − 1)θ−2k
}
∣∣ k ∈ Fq}.
A vertex in X1({0, 1}) is contained in Bi if and only if one of the following four systems
of equations has solutions:
jθ−2k
1 + jθ−2k
= t,
(j + 1)θ−2k
1 + (j + 1)θ−2k
= t+ i; (3.8)
jθ−2k
1 + jθ−2k
= t+ i,
(j + 1)θ−2k
1 + (j + 1)θ−2k
= t; (3.9)
(−j)θ−2k
1 + (−j)θ−2k
= t,
(−j − 1)θ−2k
1 + (−j − 1)θ−2k
= t+ i; (3.10)
(−j)θ−2k
1 + (−j)θ−2k
= t+ i,
(−j − 1)θ−2k
1 + (−j − 1)θ−2k
= t. (3.11)
428 Ars Math. Contemp. 23 (2023) #P3.05 / 417–440
Solving Equation (3.8) and Equation (3.10) we get the respective equation
ij(j + 1)u2 ± (2ij + i− 1)u+ i = 0,
where u = θ−2k. Each of these two equations has solutions for u only if
δ1 := (2ij + i− 1)2 − 4i2j(j + 1) = i2 − (2 + 4j)i+ 1 ∈ S.
(1) δ1 ∈ S∗: Since the product of two solutions u1 and u2 is (j(j + 1))−1, a square, we
know that either u1, u2 ∈ S∗ or u1, u2 ∈ N∗. Therefore, there exist two solutions
for θ−2k = u only if δ1 ∈ S∗. Noting that every θ−2k gives a unique t, equivalently,
one vertex in the block Bi. Thus two systems of equations give two vertices in Bi.
(2) δ1 = 0: For these two equations, there is just one solution for u and it gives a unique
t. Thus two systems of equations give one vertice in Bi.
Solving Equation (3.9) and Equation (3.11), we get respective equation
ij(j + 1)u2 ± (2ij + i+ 1)u+ i = 0,
where u = θ−2k. This equation has solutions for u if and only if
δ2 := (2ij + i+ 1)
2 − 4i2j(j + 1) = i2 + (2 + 4j)i+ 1 ∈ S.
Similarly, if δ2 ∈ S∗, there exist exactly two solutions for θ−2k. Thus two equations give
two vertices in Bi. If δ2 = 0, there exists one solution for θ−2k. Thus we only get one
vertex in Bi.
In summary, d(B1, Bi) = 2 if and only if δ1δ2 ∈ N ; d(B1, Bi) = 0 or 4, provided
δ1δ2 ∈ S∗; and d(B1, Bi) = 1 or 3 if and only if δ1δ2 = 0.
Step 2: Show the existence of a H-cycle.
Let Y1 = Y [B] be the subgraph of Y induced by B. Then Y1 is a Cayley graph on
Z q−1
2
. Since the valency of X is q − 1, d(B1, B∞) = 2, and d(B1, Bi) ≤ 4, it follows
from
1
4
(q − 1− 4− 2) ≥ 1
3
· q − 1
2
.
Then we get a H-cycle, with the same arguments as in Step 3 of Lemma 3.4.
Lemma 3.7. Suppose that ∆ is a non self-paired suborbit of length q − 1. Then X(G, ∆)
is hamiltonian.
Proof. In this case, ∆ = {1, 0}H and ∆∗ = {−1, 0}H . Let X = X(∆ ∪∆∗) and Y the
quotient graph induced by B ∪ {B∞}. Then the neighborhood of {0, ∞} is:
X1({0, ∞}) = ∆ ∪∆∗ = {{0, θk}, {∞, θk}
∣∣ k ∈ Fq}.
By observing the vertices of block B∞, we get d(B∞) = q − 1, and since ⟨l⟩ is regular on
B, d(B∞, Bi) = 2 for any i = 1, 2, · · · , q−12 . Since u
′ maps {∞, 0} to {0, 1}, we know
that
X1({0, 1}) = {∆, ∆∗}u
′
= {{0, θ
k
1 + θk
}, {1, θ
k
1 + θk
}
∣∣ k ∈ Fq}.
S. Du et al.: Hamilton cycles in primitive graphs of order 2rs 429
A direct computation shows d(B1) = 2. Moreover, d(B1, Bi) is exactly the number of
union of solutions of the following two equations:
{0, θ
k
1 + θk
} = {t+ i, t} and {1, θ
k
1 + θk
} = {t+ i, t}.
Solving them, we get four solutions:
θk =
−i
1 + i
, t = −i; θk = i
1− i
, t = 0;
θk =
1− i
i
, t = 1− i; θk = −i− 1
i
, t = 1.
Therefore, d(B1, Bi) = 4.
Since ⟨l⟩ is regular on B, d(Bj , Bi) = d(B1, Bi′) for some i′ and d(Bi) = d(B1).
Then we conclude that d(Bi, Bj) = 4 and d(Bi) = 2. Thus the graph Y \ {B∞} is a
complete graph. As before, X is hamiltonian.
3.2 soc(G) = PSL(2, q) and H = Dq+1
Let G = PSL(2, q) and H = Dq+1. Consider the action of G on the set [G : H] of cosets
of H in G, see row 2 of Table 1. Then n = q(q−1)2 = 2rs. This implies that q ≡ 1(mod 4)
and both q and q−14 are primes. So r =
q−1
4 and s = q. Set F
∗
q = ⟨θ⟩ and
√
−1 = θ
q−1
4 .
In GL(2, q), we set
u =
(
1 1
0 1
)
, u′ =
(
1 0
1 1
)
, l =
(
θ 0
0 θ−1
)
, t =
(
0 1
−1 0
)
,
t(x, y) =
(
x yθ
y x
)
, t′(x, y) =
√
−1
(
1 0
0 −1
)
t(x, y) =
√
−1
(
x −yθ
y −x
)
, x ̸= 0.
Then up to conjugacy, H may be chosen as
H = {t(x, y), t′(x, y)
∣∣ x2 − y2θ = 1}.
Consider the action of NG(⟨u⟩) = ⟨u⟩⋊⟨l⟩ on the set of ⟨u⟩-orbits (blocks) on [G : H].
Then [G : H] can be divided into two parts, say B and B′, where
B = {B1, B2, · · · , B q−1
4
}, B′ = {B′1, B′2, · · · , B′q−1
4
},
where Bi = {Huj li
∣∣ j ∈ Zq} and B′i = {Htuj li ∣∣ j ∈ Zq}, where 1 ≤ i ≤ q−14 .
Lemma 3.8. Suppose q ≡ 1(mod 4). Then for G acting on [G : H],
(1) there are q−32 suborbits of length
q+1
2 , while
q−1
4 of them {HlitH
∣∣ 1 ≤ i ≤
q−1
4 } are self-paired and
q−5
4 of them {HliH
∣∣ 1 ≤ i ≤ q−14 } are non-self-paired
suborbits;
(2) there are q−14 suborbits of length q + 1, with the form Hu
iH , where i2 ∈ S∗ ∩
(4θ +N). All of them are self-paired.
430 Ars Math. Contemp. 23 (2023) #P3.05 / 417–440
Proof. Since q+1 ≡ 3(mod 4), for any g ∈ G, H ∩Hg is either Z2 or 1, so every suborbit
is of length either q+12 or q + 1.
(1) |∆| = q+12
Let ∆ = HgH be a suborbit of length q+12 . Then H
g ∩ H ∼= Z2 and so αg is
an involution of H , where α = l
q−1
4 ∈ H . Then αg = αh for some h ∈ H , and so
gh−1 ∈ CG(α) = ⟨l, t⟩. Since HgH = Hgh−1H , we may choose h = 1 so that
g ∈ CG(α). Set g = li or lit for some i. Moreover, direct computations show that for any
two distinct elements g1, g2 ∈ CG(α) = ⟨l, t⟩, Hg1H = Hg2H if and only if g1 = g2α.
Therefore, we have q−12 suborbits of length
q+1
2 . In particular, HgH = Hg
−1H if and
only if either g2 = 1 or g−1 = gα, where the second case gives g ∈ H . So we get q−14 self-
paired suborbits HgH where g is non-central involution in CG(α), noting HgαH = HgH .
So the remaining q−54 suborbits of length
q+1
2 are non self-paired.
(2) |∆| = q + 1
Let first consider the suborbits D = HuiH where i ∈ Z∗q . From the arguments in (1),
we know that |∆| = q + 1. Since HuiH = HαuiαH = Hu−iH , ∆ is self-paired. Set
g = ui.
Suppose that Hg ∩H = Z2, that is
u−it′(x1, y1)ui ∈ H,
which implies 2x1 − iy1 = 0. Insetting it in x21 − y21θ = 1, we get
i2 = 4θ + 4x−21 ∈ S∗ ∩ (4θ + S∗).
Therefore, ∆ is of length q + 1 if and only if i2 ∈ S∗ ∩ (4θ + N). By Proposition 2.1,
|S∗ ∩ (4θ+N)| = q−14 . Check that HuiH = HujH if and only if i = ±j. Therefore, we
get q−14 suborbits of length q + 1.
Since 1+ q−32
q+1
2 +
q−1
4 (q+1) =
q(q−1)
2 = |[G : H]|, we already find all suborbits.
In what follows we deal with all cases of suborbits ∆ in Lemma 3.8, separately.
Lemma 3.9. Suppose that ∆ is a self-paired suborbit of length q+12 . Then X(G, ∆) is
hamiltonian.
Proof. Let X = X(G, ∆), where ∆ is self-paired and of length q+12 . From the last lemma,
∆ = HlktH for some k. Note q−14 = r is a prime, the two smallest values for q are 13
and 29. One may find a H-cycle by Magma for q = 13 and 29. So let q ̸= 13, 29. First we
give a remark.
Remark: Suppose we may get two facts: 1⃝ for any B′ ∈ B′, d(H, B′) = 0, 2 or 4;
2⃝ d(H, ∪B′∈B′B′) ≥ 5. Then H is adjacent to at least two blocks B′i, B′j in B such that
d(H, B′i) = 2 or 4. Let Y be the block graph. Then Y is a bipartite graph of order 2r,
where r = q−14 is a prime. Note that H ∈ Br. Since ⟨l⟩/⟨l
r⟩ acts regularly on both B and
B′, we may set B′di = B
′
j for some d ∈ ⟨l⟩/⟨l
r⟩. Then we get a H-cycle of Y :
B′i, Br, B
′d
i , B
d
r , B
′d2
i , · · · , Bd
r−1
r , B
′
i.
S. Du et al.: Hamilton cycles in primitive graphs of order 2rs 431
Then by Proposition 2.4, we may find a H-cycle for X(G, ∆).
Now we continue to prove the lemma. Clearly, the neighborhood of H is:
X1(H) = ∆ = HlktH = {Hlktt(x1, y1)
∣∣ x21 − y21θ = 1}.
The vertex Hlktt(x1, y1) is contained in B′i if and only if
Hlktt(x1, y1) = Htuj li, for some j,
if and only if
lktt(x1, y1)(tuj li)
−1 ∈ H,
if and only if one of the following two systems of equations with unknowns j, i, x1 and y1
has solutions corresponding to (ε, η) = (1, −1) and (−1, 1):
y1jθ
2k = x1(θ
2k+2i − ε),
y1(θ
2i+2 + ηθ2k) = x1θj,
x21 − y21θ = 1.
(3.12)
Every such system has the same solutions with
y21 =
θ2k+1
ηθ4k+θ2ε
θ2i − εθ
ηθ4k+θ2ε
, (i)
y21 = θ
−1x21 − θ−1, (ii)
j = (θ2i − εθ−2k)x1y1 . (iii)
(3.13)
From (iii), we know that given a solution for x21, y
2
1 and i, we have two values of j, that
is ±j. Then the possible values for d(H, B′i) is 0, 2 or 4, noting we have two choices for
(ε, η), showing fact 1⃝.
Set b = −θ−1, a1 = θ
2k+1
ηθ4k+θ2ε
and a2 = − εθηθ4k+θ2ε . Then a1, a2 ̸= 0 and a2 ̸= b.
From (i) and (ii), we get that either
y21 ∈ S∗∩(S∗+a2)∩(N+b) if a1 ∈ S∗ or y21 ∈ S∗∩(N+a2)∩(N+b) if a1 ∈ N.
By using Lemma 3.3, we get that the number of solutions for y21 is at least ⌊ 18 (q − 11 −
2
√
q)⌋, which implies that the number of solutions for j, i, x1, y1 is at least 2⌊ 18 (q− 11−
2
√
q)⌋, for given (ε, η). In other words, d(H, ∪B′∈B′B′) is at least 2⌊ 18 (q − 11− 2
√
q)⌋.
Moreover, 2⌊ 18 (q − 11− 2
√
q)⌋ ≥ 5, showing fact 2⃝.
Lemma 3.10. Suppose that ∆ is a non self-paired suborbit of length q+12 . Then X(G,
∆ ∪∆∗) is hamiltonian.
Proof. Let X = X(G, ∆ ∪ ∆∗), where ∆ is non self-paired and of length q+12 . From
Lemma 3.8, ∆ = HlkH and ∆∗ = Hl−kH for some integer k. Note q−14 = r is a prime,
the three smallest values for q are 13, 29 and 53. One may find a H-cycle by Magma for
q = 13, 29 and 53. So let q ̸= 13, 29, 53.
From the remark in last lemma, it suffices to show two facts: (i) for any B′ ∈ B′,
d(H, B′) = 0, 2, 4, 6 or 8; (ii) d(H, ∪B′∈B′B′) ≥ 9.
432 Ars Math. Contemp. 23 (2023) #P3.05 / 417–440
Check that the neighborhood of H is:
X1(H) = ∆ ∪∆∗ = {Hlkt(x1, y1), Hl−kt(x1, y1)
∣∣ x21 − y21θ = 1}.
The vertex Hlkt(x1, y1) and Hl−kt(x1, y1) are contained in B′i if and only if either
Hlkt(x1, y1) = Htuj li, orHl−kt(x1, y1) = Htuj li, for some j
if and only if either
lkt(x1, y1)(tuj li)
−1 ∈ H, or l−kt(x1, y1)(tuj li)−1 ∈ H
if and only if one of the following four systems of equations with unknowns j, i, x1 and y1
has solutions corresponding to (ε, η) = (1, −1), (1, 1), (−1, −1) or (−1, 1):
y1(θ
i+ϵk+1 − ηθ−i−εk) = x1jθεk−i,
y1jθ
−εk−i+1 = x1(θ
i−ϵk+1 − ηθ−i+εk),
x21 − y21θ = 1.
(3.14)
Every such system has the same solutions with
y21 =
θ2iθ
ηθθ2εk−ηθθ−2εk −
ηθ2εk
ηθθ2εk−ηθθ−2εk , (i)
y21 = θ
−1x21 − θ−1, (ii)
j = θ
i+εk+1−ηθ−i−εk
θεk−i
y1
x1
. (iii)
(3.15)
From (iii), we know that given a solution for x21, y
2
1 and i, we have two values of j, that
is ±j. Then the possible values for d(H, B′i) is 0, 2, 4, 6 or 8, noting we have four choices
for (ε, η), showing fact (i).
Set b = −θ−1, a1 = ηθθ2εk − ηθθ−2εk and a2 = − ηθ
2εk
ηθθ2εk−ηθθ−2εk . Then a1, a2 ̸= 0
and a2 ̸= b. From (i) and (ii), we get that either
y21 ∈ S∗∩(N+a2)∩(N+b) if a1 ∈ S∗ or y21 ∈ S∗∩(S∗+a2)∩(N+b) if a1 ∈ N.
By using Lemma 3.3, we get that the number of solutions for y21 is at least ⌊ 18 (q − 11 −
2
√
q)⌋, which implies that the number of solutions for j, i, x1, y1 is at least 2⌊ 18 (q− 11−
2
√
q)⌋, for given (ε, η). In other words, d(H, ∪B′∈B′B′) is at least 2⌊ 18 (q − 11− 2
√
q)⌋.
Moreover, 2⌊ 18 (q − 11− 2
√
q)⌋ ≥ 9, showing fact (ii).
Lemma 3.11. Suppose that ∆ is a self-paired suborbit of length q + 1. Then X(G, ∆) is
hamiltonian.
Proof. Let X = X(G, ∆), where ∆ is self-paired and of length q + 1. From Lemma 3.8,
∆ = HukH for some integer k. Note q−14 = r is a prime.
If we may get two facts: (i) for any B′ ∈ B′, d(H, B′) = 0, 2 or 4; (ii) for any
B′ ∈ B′, d(H, B′) = 0, 2 or 4, then every vertex in block graph has the valency at least
(q+1)−2
4 =
q−1
4 =
1
2
q−1
2 . So Y contains a H-cycle. Since d(Bi, B
′
j) is even, this cycle
can lift a H-cycle for X(G, ∆) by Proposition 2.4.
In fact, check that the neighborhood of H is:
X1(H) = ∆ = HukH = {Hukt(x1, y1), Hukt′(x1, y1)
∣∣ x21 − y21θ = 1}.
S. Du et al.: Hamilton cycles in primitive graphs of order 2rs 433
By observing the neighbor, one can see these neighbors contained in B q−1
4
are just Huk
and Hu−k, which implies d(B q−1
4
) = 2. The vertex Hukt(x1, y1) and Hukt′(x1, y1) are
contained in Bi if and only if either:
Hukt(x1, y1) = Huj li, or Hukt′(x1, y1) = Huj li
if and only if either:
ukt(x1, y1)(uj li)
−1 ∈ H, or ukt′(x1, y1)(uj li)−1 ∈ H
if and only if one of the following systems of equations with unknowns j, i, x1 and y1
has solutions corresponding to (ϵ, η, γ, δ) = (−1, 1, −1, 1), (1, −1, 1, −1), (−1, −1,
−1, −1) or (1, 1, 1, 1) :
ϵθ−iy1j = (x1 + ky1)θ
−i − ηx1θi,
γ(x1 + ky1)θ
−ij = y1θ
−iθ − δθi(y1θ + kx1),
x21 − y21θ = 1.
(3.16)
This system has the same solutions with
j =
(θ−i − ηθi)x1
εθ−iy1
+ kε−1 where δεθ2i = γ(k2y21 + 2kx1y1 + 1).
Calculating the equation δεθ2i = γ(k2y21 + 2kx1y1 + 1) we could get
(4k2θ − k4)u2 + (2k2 + 2δεγθ2ik2)u− (δεθ2i − γ)2 = 0,
where u = y21 . Since the product of the two solutions is
−(δεθ2i−γ)2
4k2θ−k4 , a non-square (as
4θ − k2 ∈ N ), there exists at most one solution for u = y21 . It is easy to see that there
are two solutions for j. Since there are just two different equations for δεθ2i = γ(k2y21 +
2kx1y1 +1), there are at most 4 solutions for j, that is d(H, Bi) = 0, 2 or 4, showing fact
(i).
The vertex Hukt(x1, y1) and Hukt′(x1, y1) are contained in B′i if and only if either
Hukt(x1, y1) = Htuj li or Hukt′(x1, y1) = Htuj li
if and only if either
ukt(x1, y1)(tuj li)
−1 ∈ H or ukt′(x1, y1)(tuj li)−1 ∈ H
if and only if one of the following systems of equations with unknowns j, i, x1 and y1 has
solutions corresponding to (ϵ, η, γ, δ) = (1, −1, 1, −1), (1, 1, 1, 1), (−1, −1, −1, −1)
or (−1, 1, −1, 1):
−(x1 + ky1)θ−ij = ηy1θ−i − ϵ(y1θ + kx1)θi−1
−y1θ−iθj = δ(x1 + ky1)θ−i − γx1θi−1θ
x21 − y21θ = 1.
(3.17)
434 Ars Math. Contemp. 23 (2023) #P3.05 / 417–440
This system has the same solutions with
j =
δθ−i − γθiθ
−θ−iθ
x1
y1
− kδ
θ
where γθ2iθ = δ(k2y21 + 2kx1y1 + 1).
Calculating the equation γθ2iθ = δ(k2y21 + 2kx1y1 + 1) we could get
(4k2θ − k4)u2 + (2k2 + 2δγk2θ2iθ)u− (−γθ2iθ + δ)2 = 0,
where u = y21 . Since the product of the two solutions is
−(−γθ2iθ+δ)2
4k2θ−k4 , a non-square (as
4θ − k2 ∈ N ), there exists at most one solution for u = y21 and it is easy to see there
are two solutions for j. Since there are just two different equations for γθ2i = δ(k2y21 +
2kx1y1 + 1), there are at most 4 solutions for j, that is d(B q−1
4
, B′i) = 0, 2 or 4, showing
fact (ii).
3.3 Groups in Table 2
In this subsection, we shall deal with the groups in Table 2, separately.
Lemma 3.12. Let G be one of groups in rows 1 and 2 of Table 2. Then every orbital graph
of G contains a Hamilton cycle.
Proof. Let T = PSL(m, q) where m = 4 or 5. It suffices to consider the group T . We
shall deal with two cases: m = 4 and m = 5, separately.
Case 1: m = 4.
Let Ω be the set of 2-dim. subspaces of a space V of dimension 4. Then n =
(q4−1)(q3−1)
(q−1)(q2−1) = (q
2 + q + 1)(q2 + 1), where s = q2 + q + 1 and r = q
2+1
2 are two
primes. Consider a subspace W0 of dimension d(W0) = 2. Then T has two nontrivial
suborbits relative to W0:
∆1 = {W ∈ Ω
∣∣ d(W ∩W0) = 1} and ∆2 = {W ∈ Ω ∣∣ d(W ∩W0) = 0},
where r1 := |∆1| = q
4−q
q2−q =
q3−1
q−1 and r2 := |∆2| = n − 1 − r1. Since r2 ≥
n
2 , the
corresponding orbital graph Γ(T, ∆2) has a H-cycle.
Now we are considering X(T, ∆1). Take a projective point ⟨α⟩ and extend it into a
base α, α1, α2, α3 of V . Let Σ(α) be the set of all 2-dim. subspaces containing α. Then
|Σ(α)| = q2 + q + 1. Since ⟨α1, α2, α3⟩ contains exactly q2 + q + 1 points and for any
two distinct points β, β′ in ⟨α1, α2, α3⟩, ⟨α, β⟩ ≠ ⟨α, β′⟩, one may see
Σ(α) = {⟨α, β⟩
∣∣ β ∈ ⟨α1, α2, α3⟩}.
Let ⟨h⟩ be the Singer subgroup of PSL(3, q) and β ∈ ⟨α1, α2, α3⟩. Since s = q2 + q + 1
is a prime, ⟨β⟩, ⟨βh⟩, ⟨βh2⟩, · · · , ⟨βhs−1⟩ are all the projective points of ⟨α1, α2, α3⟩.
Denote βh
i
= βi. Since the subgraph induced by Σ(α) is a complete graph, we may
consider a H-cycle of the subgraph, say
⟨α, β0⟩, ⟨α, β1⟩, ⟨α, β2⟩, · · · , ⟨α, βs−2⟩, ⟨α, βs−1⟩, ⟨α, β0⟩,
S. Du et al.: Hamilton cycles in primitive graphs of order 2rs 435
⟨α, β0⟩ ⟨α, β1⟩ ⟨α, β2⟩ ⟨α, β3⟩ ⟨α, βr−1⟩ ⟨α, β0⟩
⟨β0, β1⟩ ⟨β1, β2⟩ ⟨β2, β3⟩ ⟨β3, β4⟩ ⟨βr−1, β0⟩ ⟨β0, β1⟩
H1 H2 H3 H0
Figure 1:
where βi ∈ ⟨α1, α2, α3⟩ and s = q2 + q + 1.
Set
A = {⟨βi, βi+1⟩, ⟨βs−1, β0⟩|i = 0, 1, · · · , s− 2} = {⟨β, βh⟩h
i
|i = 0, 1, · · · , s− 1},
Xi = Σ(βi) \ (
i−1⋃
j=1
Σ(βj)
⋃
A), i ∈ {1, 2, · · · , s− 1},
X0 = Σ(β0) \ (
s−1⋃
j=1
Σ(βj)
⋃
A).
Since every 2-subspace ⟨η, γ⟩ can be expressed as ⟨η, γ− b0a0 η⟩, where η = a0α+ a1α1 +
a2α2 + a3α3 and γ = b0α + b1α1 + b2α2 + b3α3, every 2-subspace of V is contained in
(
⋃s−1
i=0 Xi)
⋃
A. Moreover, from the definition, we know that X0, X1, · · · , Xs−1, A are
mutually disjoint.
Now we are ready to find a H-cycle for X(T, ∆1). For i = 0, 1, · · · , r − 2 , con-
sider a H-path Hi+1 in the subgraph induced by Xi+1
⋃
{⟨βi, βi+1⟩} with the starting
vertex ⟨βi, βi+1⟩ and the ending vertex ⟨α, βi+1⟩. Consider H0 in the subgraph induced
by X0
⋃
{⟨βs−1, β0⟩} with the starting vertex ⟨βs−1, β0⟩ and the ending vertex ⟨α, β0⟩.
Then by replacing every arc (⟨α, βi⟩, ⟨α, βi+1⟩) by the path (⟨α, βi⟩, Hi+1) and the arc
(⟨α, βs−1⟩, ⟨α, β0⟩) by the path (⟨α, βs−1⟩, H0), we get a cycle:
⟨α, β0⟩, H1, H2, · · · , Hs−1, H0,
which is clearly a H-cycle of X(T, ∆1), as shown in Figure 1.
Case 2: m = 5.
Let Ω be the set of 2-dim. subspaces of V . Then
n = |Ω| = (q
5 − 1)(q4 − 1)
(q − 1)(q2 − 1)
= (q4 + · · ·+ 1)(q2 + 1) = 2rs.
Then s = q4 + · · · + 1 is a prime and r = q
2+1
2 are two prime. Let S = ⟨h⟩ be a Singer
subgroup of PSL(5, q), where |S| = s. Take a projective point α. Then α, αh, · · · , αhs−1
are all the projective points. Set Wi = ⟨α, αh
i⟩ where i = 1, 2, · · · , s − 1. Then G has
two nontrivial suborbits relative to W1:
∆1 = {W ∈ Ω|d(W ∩W1) = 1} and ∆2 = {W ∈ Ω|d(W ∩W1) = 0},
436 Ars Math. Contemp. 23 (2023) #P3.05 / 417–440
where
r1 := |∆1| = ( q
4
q−1 − 1)(q + 1) = q(q + 1)(q
2 + q + 1),
r2 := |∆2| = (q
5−q2)(q5−q3)
(q2−1)(q2−q) = q
4(q2 + q + 1).
Since r2 ≥ n2 , the corresponding orbital graph X(T, ∆2) has a H-cycle.
Now we are considering X(T, ∆1). Let Si be the path
Wi, W
hi
i , W
h2i
i , W
h3i
i , · · · , Wh
(s−1)i
i .
Since ⟨hi⟩ acts nontrivially on Wi and it is of order a prime s, ⟨hi⟩ moves Wi. Since every
2-subspace must be contained in some clique and either |Si ∩ Sj | = 0 or Si = Sj for
any two distinct cliques Si and Sj , we could pick up q2 + 1 distinct cliques which cover
all 2-dim. subspaces, denoted by Wµ1 , Wµ2 , · · · , Wµq2+1 . Then we can get a H-cycle
of X(T, ∆1) : Wµ1 , W
hµ1
µ1 , W
h2µ1
µ1 , · · · ,W
h(s−1)µ1
µ1 , Wµ2 , W
hµ2
µ2 , W
h2µ2
µ2 , W
h3µ2
µ2 , · · · ,
Wh
(s−1)µ
q2+1
µq2+1
,Wµ1 .
Lemma 3.13. Every orbital graph of G = PΩ−(2m, q) in row 3 of Table 2 is hamiltonian.
Proof. Let G = PΩ−(2m, q) act on n totally singular (t.s.) 1-spaces, where n =
(qm+1)(qm−1−1)
q−1 = 2rs and m = 2
2l . Then m − 1 is a prime. Since m − 1 = (22l−1 −
1)(22
l−1
+1), we get 22
l−1 − 1 = 1, which implies l = 1 and then m = 4. Now r =: q
3−1
q−1
is a prime. Let Ω be the set of all t.s.1-spaces. Recall that SO−(8, q) ≤ GL(8, q) and
|GL(8, q)| = q28Π8i=1(qi − 1). To describe SO
−(8, q), take a symmetric bilinear form,
given by the following matrix:
J =
 0 E3 0E3 0 0
0 0 J2
 , J2 = (1 00 −t
)
, t ∈ N.
Let ⟨A⟩ be a Singer subgroup of GL(3, q), C = Aq−1 and D = (C−1)′, where C ′
denotes the transpose of C. Set B = C
⊕
(C ′)−1
⊕
E2, the block diagonal matrix.
Then we have BJB′ = J , which means B ∈ SO−(8, q). Since B is of prime order,
B ∈ (PSO−(8, q))′ = PΩ−(8, q). Set S = ⟨B⟩ and α = (1, 0, · · · , 0). Then there are
two nontrivial suborbits for the action of G⟨α⟩ relative to ⟨α⟩, see [22]:
∆1 = {⟨β⟩ ∈ Ω \ {⟨a⟩}
∣∣ (α, β) = 0} and ∆2 = {⟨β⟩ ∈ Ω \ {⟨a⟩} ∣∣ (α, β) ̸= 0},
where |∆1| = q5+ q4+ q2+ q and |∆2| = q6. Since |∆2| ≥ 12n, we only need to consider
X(G, ∆1).
Noting that S acts semiregularly on Ω, we consider the block graph X induced by S-
orbits, where V (X) = q4 + 1. For any γ = (γ1, γ2, γ3) ∈ Ω, where γ1 = (c1, c2, c3),
γ2 = (c4, c5, c6) and γ3 = (c7, c8), we have γB
i
Jα′ = 0 if and only if γ2Diα′ = 0,
that is γ2Di = (0, c′5, c
′
6) for some c
′
5, c
′
6. Since ⟨C⟩ (and so ⟨D⟩) is regular on nonzero
1-spaces, we know that α has q + 1 (resp. q2 + q) neighbors in the block γS if γ ̸∈ αS
(resp. γ ∈ αS). From ((q5 + q4 + q2 + q)− (q2 + q))/(q + 1) = q4 we know that X is a
complete graph. By Propsosition 2.4, X(G, ∆1) is hamiltonian.
Lemma 3.14. Every orbital graph of G = PΩ+(2m, q) in row 4 of Table 2 is hamiltonian.
S. Du et al.: Hamilton cycles in primitive graphs of order 2rs 437
Proof. Let G = PΩ+(2m, q) act on n totally singular 1-spaces, where the degree n =
(qm−1)(qm−1+1)
q−1 = 2rs, m = 2
2l + 1, and s = q
m−1
q−1 and r =
qm−1+1
2 are primes. Let
Ω be the set of all totally singular 1-spaces. Recall that SO+(2m, q) ≤ GL(2m, q). To
describe SO+(2m, q), take a symmetric bilinear form, given by the following matrix:
J =
(
0 Em
Em 0
)
.
Let ⟨A⟩ be a Singer subgroup of GL(m, q), C = Aq−1 and D = (C−1)′, where C ′
denotes the transpose of C. Set B = C
⊕
(C ′)−1. Then we have BJB′ = J , which means
B ∈ SO+(2m, q). Since B is of prime order, B ∈ (PSO+(m, q))′ = PΩ+(m, q). Set
S = ⟨B⟩ and α = (1, 0, · · · , 0). Then there are two nontrivial suborbits for the action of
G⟨α⟩ relative to ⟨α⟩, see By [22]:
∆1 = {⟨β⟩ ∈ Ω \ {⟨a⟩}
∣∣ (α, β) = 0} and ∆2 = {⟨β⟩ ∈ Ω \ {⟨a⟩} ∣∣ (α, β) ̸= 0},
where |∆1| = (q
m−1+q)(qm−1−1)
q−1 and |∆2| = q
2m−2. Since |∆2| ≥ 12n, we only need to
consider X(G, ∆1).
Noting that S acts semiregularly on Ω, we consider the block graph X induced by S-
orbits, where V (X) = qm−1 + 1. For any γ = (γ1, γ2) ∈ Ω, we have γsiJα′ = 0 if
and only if γ2Diα′ = 0, which implies that the first coordinate of γ2Di is 0. Since ⟨C⟩
(and so ⟨D⟩) is regular on nonzero 1-spaces, we know that α has q
m−1−1
q−1 (resp.
qm−1
q−1 −
1) neighbors in the block γS if γ ̸∈ αS (resp. γ ∈ αS). From ( (q
m−1+q)(qm−1−1)
q−1 −
( q
m−1
q−1 − 1))/(
qm−1−1
q−1 ) = q
m−1 we know that X is a complete graph. By Proposition 2.4,
X(G, ∆1) is hamiltonian.
Lemma 3.15. Vertex-transitive graphs arising from the action of Ac on 2-subsets given in
row 6 of Table 2 are hamiltonian.
Proof. Let Ω = {α1, α2, · · · , αc}, where c ≥ 5. Then we only have the following two
orbital graphs:
(1) Two subsets are adjacent if and only if they intersect at a single point. In this case,
the orbital graph is the Johnson graph J(c, 2). Then we may get a H-cycle as the
following way:
first pick up a cycle of c vertices, say {α1, α2}, {α2, α3}, {α3, α4}, · · · ,
{αc−1, αc}, {αc, α1}, {α1, α2}; then
replace the edge {α1, α2}, {α2, α3} by any H-path of all 2-subsets containing α2,
with the starting vertex {α1, α2} and the ending vertex {α2, α3}; then
replace the edge {α2, α3}, {α3, α4} by any H-path of all 2-subsets containing α3,
with the starting vertex {α2, α3} and the ending vertex {α3, α4}; then for 5 ≤ i ≤ c,
replace the edge {αi−2, αi−1}, {αi−1, αi} by any H-path of all 2-subsets con-
taining αi−1 but removing {{α2, αi−1}, {α3, αi−1}, · · · , {αi−3, αi−1}}, with the
starting vertex {αi−2, αi−1} and the ending vertex {αi−1, αi}.
438 Ars Math. Contemp. 23 (2023) #P3.05 / 417–440
(2) Two subsets are adjacent if and only if they have no intersecting point. Then the
orbital graph is the Kneser graph K(c, 2). If c ≥ 8, then the degree of the graph is
more than n2 and so it is hamiltonian, where n is the order of the graph. For the cases
when c ≤ 7, we do it just by Magma.
Lemma 3.16. Let G be one of the groups listed in row 5, 7 − 10 of Table 2. Then every
orbital graph of G is hamiltonian.
Proof. Using Magma, we compute the suborbits for these groups and show that every
corresponding orbital graph is hamiltonian.
(1) The action of PSL(3, 5).2 on the flags has three nontrivial suborbits, with the respec-
tive length 10, 50 and 125;
(2) The action of M11 on the cosets of a subgroup isomorphic to S5 has three nontrivial
suborbits, with the respective length 15, 20 and 30;
(3) The action of M12 on the cosets of a subgroup isomorphic to M10 : 2 has two
nontrivial suborbits, with the respective length 20 and 45;
(4) The action of M23 on the cosets of a subgroup isomorphic to A8 has three nontrivial
suborbits, with the respective length 15, 210 and 280;
(5) The action of J1 on the cosets of a subgroup isomorphic to PSL(2, 11) has four
nontrivial suborbits, with the respective length 11, 12, 110 and 132.
ORCID iDs
Shaofei Du https://orcid.org/0000-0001-6725-9293
Yao Tian https://orcid.org/0000-0001-5391-6870
Hao Yu https://orcid.org/0000-0001-5271-576X
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P3.06 / 441–454
https://doi.org/10.26493/1855-3974.2340.a61
(Also available at http://amc-journal.eu)
Bootstrap percolation via automated
conjecturing*
Neal Bushaw †, Blake Conka, Vinay Gupta, Aidan Kierans, Hudson
Lafayette, Craig Larson, Kevin McCall, Andriy Mulyar, Christine
Sullivan, Scott Taylor, Evan Wainright, Evan Wilson, Guanyu Wu
Department of Mathematics and Applied Mathematics,
Virginia Commonwealth University, Richmond VA, USA
Sarah Loeb
Department of Mathematics and Computer Science, Hampden-Sydney College,
Hampden-Sydney VA, USA
Received 6 June 2020, accepted 28 February 2022, published online 24 January 2023
Abstract
Bootstrap percolation is a simple monotone cellular automaton with a long history in
physics, computer science, and discrete mathematics. In k-neighbor bootstrap percolation,
a collection of vertices are initially infected. Vertices with at least k infected neighbors sub-
sequently become infected; the process continues until no new vertices become infected. In
this paper, we hunt for graphs which can become entirely infected from initial sets which
are as small as possible. We use automated conjecture-generating software and a large
group lab-based model as a fundamental part of our exploration.
Keywords: Bootstrap percolation, automated conjecturing, graph theory, percolation, cellular au-
tomata.
Math. Subj. Class. (2020): 05C35,68R05
*We thank the anonymous referees for their detailed and useful comments, suggestions, and references.
†Corresponding author.
E-mail addresses: nobushaw@vcu.edu (Neal Bushaw), conkaba@vcu.edu (Blake Conka),
guptavp@vcu.edu (Vinay Gupta), kieransaf@vcu.edu (Aidan Kierans), lafayettehl@vcu.edu (Hudson
Lafayette), clarson@vcu.edu (Craig Larson), mccallkj@vcu.edu (Kevin McCall), andriy.mulyar@gmail.com
(Andriy Mulyar), sullivanca2@vcu.edu (Christine Sullivan), taylorsm9@vcu.edu (Scott Taylor),
wainrightep@vcu.edu (Evan Wainright), wilsonea@vcu.edu (Evan Wilson), wug2@vcu.edu (Guanyu Wu),
sloeb@hsc.edu (Sarah Loeb)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
442 Ars Math. Contemp. 23 (2023) #P3.06 / 441–454
1 Introduction
1.1 History
Bootstrap percolation can be thought of as a graph process in which an arbitrary initial
configuration of infected vertices is selected from a graph; remaining uninfected vertices
with many infected neighbors are successively added to the infected set until the system
stabilizes. Bootstrap percolation serves as a model of nucleation and growth [12] and has
been applied in the study of crack formation [1], magnetic alloys [13], hydrogen mixtures
[2], and computer storage arrays [20]. More generally, it provides an important stepping
stone towards understanding other cellular automaton models with applications in physics,
biology, information technology, epidemiology, and more.
The k-bootstrap model has a long and interesting history. First introduced by Chalupa,
Leath, and Reich [13] in 1979 as a way to model magnetic materials, it is perhaps the
simplest example of a monotone cellular automata (which were introduced by von Neu-
mann [26], based on a suggestion of Ulam [25]). Most of the work related to bootstrap
percolation has focused on finding thresholds for growing families of graphs, in which the
initially infected sets are chosen at random. For the interested reader, this fascinating direc-
tion can be explored through, e.g., [3, 4, 5, 10, 11, 19]). These bootstrap models have been
generalized significantly in recent years, with the advent of graph bootstrap percolation [8].
Here, however, we go in a somewhat different direction. Rather than selecting the
initially infected vertices at random, we allow them to be chosen very carefully. How small
could such an initial set be, given that it eventually infects the entire graph?
It is clear that if the infection can only spread to vertices with at least k infected neigh-
bors, then such an initial infected set can contain no fewer than k vertices (otherwise, not
even a single uninfected vertex can become infected). In this note, we search for those
graphs which can be infected from a set exactly k vertices – that is, those graphs which can
be infected as easily as possible.
This work is inspired by earlier results of Dairyko, Ferrara, Lidický, Martin, Pfender
and Uzzell [15], and Freund, Poloczek, and Reichman [17]. These groups gave degree
conditions for graphs to be infectable from a small set, in the case that k = 2. Similar
results were later proven by Gunderson [18] and Wesolek [27] for the k ≥ 3 cases. We stay
with the k = 2 case, but provide non-degree based conditions for percolation from small
sets.
1.2 Conjecturing
The conjectures reported below are the product of the property-relations version of the
CONJECTURING program of Larson and Van Cleemput [21, 22]. While the program is
described in these papers it is worth mentioning that produced conjectures are produced if
they are both true for all input (graph) examples and significant—here this means that the
produced conjecture was an improvement of either temporarily stored potential conjectures
or user-supplied theoretical knowledge (theorems). The CONJECTURING program is open-
source, and written to work with Sage; the code, examples, and set-up instructions are
available at: nvcleemp.github.io/conjecturing/. A substantial effort has also
been made to code graph-theoretic knowledge; this is available at: math1um.github.
io/objects-invariants-properties/.
CONJECTURING is used a tool in this research; while we don’t mean to add anything
to the papers that describe how the program works, we will add some context for inter-
N. Bushaw et al.: Bootstrap percolation via automated conjecturing 443
ested readers. In this paper we investigate both sufficient and necessary conditions for a
graph to be 2-bootstrap-good. Sufficient conditions are themselves properties, often them-
selves boolean functions of more basic properties. The CONJECTURING program allows
the user to input any list of pre-coded properties to use as “ingredients” for these sufficient
conditions. These input (or basic, or pre-coded) properties have minimum complexity—or
“complexity-1”. A unary boolean operator, such as negation, applied to a complexity-1
property yields a complexity-2 property. The CONJECTURING will systematically build
every possible property-expression from these input properties and (built-in) boolean op-
erators. The CONJECTURING program also allows the user to input a list of graphs. A
property P will be considered to be a possible sufficient condition for a graph to be 2-
bootstrap-good if every input graph G that has property P is also 2-bootstrap-good. A
possible sufficient condition P will only be added as a (potential) conjecture if it is true
for some input graph G which is false for every other currently stored sufficient condition
conjecture.
The user of the CONJECTURING can improve the quality of the produced conjectures
by adding more pre-coded properties, and by adding as input graphs any graphs that have
been found to be counterexamples to previous conjectures. The CONJECTURING program
simply systematizes and automates what a human mathematician already does: a human
mathematician’s sufficient condition condition conjectures for a graph to be 2-bootstrap-
good are necessarily properties “built” from graph properties she already knows and should
be true at least for the specific graphs she has tested them on. In a precise sense, a human
cannot make a “better” conjecture for a graph to be 2-bootstrap-good than the conjectures
the CONJECTURING program makes (from the same inputs). Maybe the most important
feature of the program is its ability to systematically consider every property up to some
complexity—no human can do this.
A last feature we will mention of the CONJECTURING program is its ability to use
theorems or theoretical knowledge. Suppose it is known that property P is a sufficient
condition for a graph to be 2-bootstrap-good. This can be added as an input to the program:
any conjectured sufficient condition property Q must be true for some input graph G which
does not have property P . This feature of the program can be useful to “grow” a theory.
In fact, some simple theorems may later be superseded. There is utility still in simple
theorems: Dirac’s Theorem, for instance, is still of interest—even though it is now implied
by less-simple, more comprehensive, theorems.
1.3 Definitions
Here, we define precisely the k-bootstrap percolation model. Wherever possible, we use
standard graph theoretic notation (see, e.g., [7]).
Let k be a natural number, G a graph, and let I ⊆ V (G) be a set of vertices which
we think of as being initially infected. We then grow the infected set as follows: if an
uninfected vertex v has at least k neighbors which are infected, then we add it to I. That
is, whenever we have a vertex v ∈ V (G) \ I with |N(v) ∩ I| ≥ k, then we move v to
I.1 Eventually, this process stabilizes – either every remaining uninfected vertex has fewer
1There is some abiguity here – we have described this process as happening a single vertex at a time. That
is, each vertex of the graph in sequence checks its number of infected neighbors, and becomes infected if this is
large. It is more standard to think of this infection as occurring in ‘rounds’, where every vertex with lots of infected
neighbors is infected simultaneously. Because the process is monotone (infected vertices never uninfect), both
versions reach the same final percolating set. We won’t be concerned with things like the time to percolate, which
444 Ars Math. Contemp. 23 (2023) #P3.06 / 441–454
than k neighbors in I, or every vertex has joined I. We denote this final infected set by ⟨I⟩.
When ⟨I⟩ = V (G), we say that G k-percolates from I. When G is clear from context,
we will say that I k-percolates; when k is also clear from context, we simply say that I
percolates.
For a graph with more than k vertices, any set I which k-percolates must have |I| ≥ k;
otherwise, there are not enough vertices in total for any uninfected vertex to join I. With
this minimum size in mind, we call a graph k-bootstrap-good if there is a set of size exactly
k which k-percolates2. A graph which is not k-bootstrap-good is k-bootstrap-bad.
We define m(G, k) to be the minimum size of a set I such that G k-percolates from I.
As such, our k-bootstrap-good graphs are those which have m(G, k) = k. In the rest of
this paper, we focus on finding conditions related to 2-bootstrap-good graphs3.
2 Lemmata (useful lemmas)
In this section, we note a few very simple results which we shall use frequently in the
remainder of the paper. To be explicit, since we’re only interested in graphs which might
be 2-bootstrap-good, all theorems and conjectures following should be assumed to have the
following extra conditions:
1. We focus exclusively on graphs with at least 3 vertices, since it requires two neigh-
bors to become infected.
2. All graphs are connected. (The only disconnected graph which is 2-bootstrap-good
is the graph with two isolated vertices)
3. All graphs have at most two blocks (as discussed in the following paragraph.).
Recall that a block in a graph G is a maximal connected subgraph with no cut vertex.
We enforce the third condition above due to the following lemma.
Lemma 2.1. If a graph is 2-bootstrap-good, then it has at most two blocks.
Proof. Assume G is a connected graph with three blocks B1, B2, B3. Since G is con-
nected, the blocks are nontrivial (that is, the blocks are either K2 or 2-connected graph). If
both infected vertices are in a single block (say B1), then at most one vertex vertex of B2
will be infected – the cut vertex separating B1 and B2, if such a vertex exists. Thus B2 will
not be infected, and so the set cannot percolate.
If, instead, the infected vertices are in different blocks, then either no infection will
spread, or if the two vertices are adjacent to a common cut vertex, they will infect first only
that common cut vertex. This can then spread to the two blocks, but as before it will not
move to the remaining block (since it cannot spread beyond the cut vertex).
As a consequence of this, we note that in particular if G contains a cut edge between
two bad subgraphs, then G is itself 2-bootstrap-bad.
We shall frequently make use the following two lemmas, which help us to decompose
graphs which are k-bootstrap-good.
is itself a fascinating area of research, and so we shall use either the ‘vertex-by-vertex’ or ‘rounds’ perspective as
we see fit.
2It is worth noting here that we only require the existence of a single small percolating set – not that every set
of k vertices percolates.
3Thus wherever it is not stated, the reader should assume that we are discussing 2-bootstrap percolation and
that a graph declared ‘good’ is in fact ‘2-bootstrap-good’.
N. Bushaw et al.: Bootstrap percolation via automated conjecturing 445
Lemma 2.2. If G is k-bootstrap-good and H is formed by adding a vertex v with at least
k neighbors inside G, then H is also good.
By infecting the initial percolating set I of size k in G, all of G will become infected,
including (at least) k neighbors of v, and so v will also become infected. And so, our initial
set I inside G actually percolates to all of H .
Lemma 2.3. If G is an n vertex graph which is k-bootstrap-good, then it can be con-
structed from an n−1 vertex k-bootstrap-good graph G′ and adding a new vertex adjacent
to at least k vertices of G′.
This is immediate – consider a minimum size infecting set, and let v be the very last
vertex which becomes infected.
To be explicit, as we are only interested in graphs which might be 2-bootstrap-good, in
order to avoid very long theorem statements, we really wish all of our theorems to have the
following additional conditions:
1. As we stated in the introduction, we focus exclusively on graphs with at least 3
vertices.
2. All graphs are connected. (The only disconnected graph which is 2-bootstrap-good
is the graph with two isolated vertices)
3. All graphs have at most two blocks (as discussed in the Lemmata).
We collect together this set of ‘potentially bootstrap good’ graphs in the definition be-
low. This will allow us to simplify our theorems tremendously; rather than, e.g., “every
connected chordal graph with at least three vertices and at most two blocks is 2-bootstrap-
good”, we can simply say “A graph in G which is chordal is 2-bootstrap-good”.
Definition 2.4. We let G denote the set of all connected graphs of order at least three which
have at most two blocks. We emphasize that the all large graphs which are 2-bootstrap-good
are in G.
2.1 Which graphs are bad?
In this section, we collect some easy properties of 2-bootstrap-bad graphs. While none of
these results are new (and several seem to be folklore), we give their very short proofs here
for completeness. The first two of these rely on the simple observation that pendant vertices
must be initially infected in any percolating set.
Proposition 2.5. A graph with at least two leaves with distinct parents is 2-bootstrap-bad.
Again, this is straightforward – leaves can never become infected if they are not ini-
tially infected; thus any leaves must be initially infected. So, both leaves must be initially
infected, and since these have distinct parents the infection does not spread.
Proposition 2.6. Any graph with at least three leaves is 2-bootstrap-bad.
As above, leaves must be initially infected, and there are simply too many to infect.
Proposition 2.7. The path graph Pk of order k ≥ 4 is 2-bootstrap-bad.
446 Ars Math. Contemp. 23 (2023) #P3.06 / 441–454
Initially infected vertices x and y are either adjacent (and no spread happens), or not
(in which case they infect exactly the vertex in between them if d(x, y) = 2 and no vertices
otherwise).
Proposition 2.8. The cycle Ck of order k ≥ 4 is 2-bootstrap-bad.
Consider a cycle x1x2 . . . xk, with initially infected vertices xi, xj with i < j. If
j − i ∈ 1, 3, 4, . . . , k − 1, then no new vertices are infected; otherwise, xi+1 is infected
and the spread stops.
For the next results, we denote by d(G) the average degree of G, we denote the maxi-
mum average degree by mad(G) := maxG′⊆G d(G′); this is a well known graph parameter
arising in chromatic theory. We will prove the following using a simple counting technique
due to Riedl (who also uses wasted and used edges similar to our usable edges above) [23].
Theorem 2.9. Let ε > 0. Then there is some N = N(ε) such that every graph with
mad(G) < 4− ε and |G| > N is 2-bootstrap-bad.
It is worth noting that this theorem is sharp, as is seen by the square of the cycle C2n for
each n; such graphs have mad(G) = 4 and are 2-bootstrap-good for each n. In fact, we
will prove the corresponding result for the more general k-bootstrap model; this is again
shown to be sharp by the kth power of the cycle.
Theorem 2.10. Let ε > 0. Then there is some N = N(ε) such that every graph with
mad(G) < 2k − ε and |G| > N is k-bootstrap-bad.
Proof. Assume G is k-bootstrap-good, with vertices infected one at a time; let Ht be the
graph induced by those vertices which are infected within the first t steps. Then since we
initially infected k vertices, followed by one vertex at each time step, we have |Ht| = t+k.
Further, each vertex was infected because it had at least k edges to the preceding infected
vertices and so ∥Ht∥ ≥ kt. Thus d(G) ≥ ktt+k , and for t sufficiently large this is larger than
2k − ε; this contradicts the maximum average degree condition.
2.2 What is required to be good?
As is common in such problems, we provide only a few necessary conditions for a graph
to be 2-bootstrap-good. The first of these is immediate from Lemma 2.3.
Proposition 2.11. If G is good, then ∥G∥ ≥ 2(|G| − 2).
The next result will be of considerable use to us later. Recall that the girth of a graph is
the minimum of the cycle lengths present.
Proposition 2.12. If G is 2-bootstrap-good and not P3, then it has girth less than five.
Proof. Consider two initially infected vertices u and v which percolate. Since we’re assum-
ing our graphs have at least 3 vertices, there is some vertex w which is becomes infected
next – it is adjacent to both u and v. If uv ∈ E(G), then we already have a triangle. Oth-
erwise, if G is not K1,2, then there is a fourth vertex which becomes infected; say x. Then
x must be adjacent to two of {u, v, w} – if it is adjacent to both v, w, we form a triangle; if
it is adjacent to u,w or u, v we form a C4.
Note that this result shows that the Petersen graph is not 2-bootstrap-good.
N. Bushaw et al.: Bootstrap percolation via automated conjecturing 447
2.3 What will guarantee goodness?
In this section, we provide a number of theorems giving sufficient conditions for a graph
to be 2-bootstrap-good. The first of these require little to prove; however, they were the
first conjectures provided by the CONJECTURING program, so we record them here for
completeness.
Proposition 2.13. Complete graphs are 2-bootstrap-good.
Proposition 2.14. Complete bipartite graphs are 2-bootstrap-good.
Proof. Since |G| > 2, one of the bipartition classes class has at least two vertices; assume
that G has bipartition (X,Y ) with |X| > 1. Initially infect two vertices of X . Since the
graph is complete bipartite, every vertex of Y is infected immediately. Then, the remaining
vertices of X become infected in the next step.
Indeed, this remains true for the similar class of split graphs – those graphs whose
vertex set can be partitioned into a clique and an independent set.
Theorem 2.15. If G is a split graph with at most two blocks, then G is 2-bootstrap-good.
Proof. First, notice that if the complete side has only one vertex, then the graph is a star
(and thus either K2 or K1,2, since it has at most two blocks, and thus good.) The graph
can have at most one pendant, v, which must lie in the independent set. Choosing v and
any vertex of the complete graph which is not the parent of v will infect the entire graph,
since the complete graph will become immediately infected and each non-pendant in the
independent set must be adjacent to at least two vertices of the complete graph. If there is
no pendant, then infecting any two vertices of the complete graph will suffice.
The above classes of graphs percolate very quickly (in at most 3 steps). Next, we see a
class of graphs which percolates, but not necessarily in a fixed number of steps. Recall that
a graph is locally connected if the open neighborhood of every vertex is a connected graph.
Theorem 2.16. If a graph G ∈ G is locally connected, then it is 2-bootstrap-good.
Proof. First, note that a locally connected graph has no pendants – otherwise, the neighbor-
hood of the pendant vertex’s parent contains an isolated vertex. Hence let G be a locally-
connected graph, v be any vertex and w be any neighbor of v. We initially infect {v, w}.
Recall that ⟨{v, w}⟩ is then the set of vertices eventually infected from {v, w}.
As the (open) neighborhood N(v) is connected there is a path w = x1...xk = u from w
to any other vertex u in the graph H = G[N(v)] induced by N(v). Note that each vertex xi
in this path is necessarily a neighbor of v. Since v and w are infected and x2 is a neighbor
of both, x2 is also infected. Similarly x3, ..., xk = u must all be infected. So N(v) is a
subset of ⟨{v, w}⟩. By a symmetric argument N(w) is also a subset of ⟨{v, w}⟩.
Suppose ⟨{v, w}⟩ does not equal V . Let x be any vertex in V \⟨{v, w}⟩ that is adjacent
to some vertex y ∈ ⟨{v, w}⟩. Since y ∈ ⟨{v, w}⟩ and our graph is connected, there must
also be a neighbor z of y in ⟨{v, w}⟩. By the reasoning above it follows that N(y) must be
a subset of ⟨{v, w}⟩. But then x must be in ⟨{v, w}⟩.
It is worth noting that the above proof in fact shows that if G is locally connected and
pendant-free, then G 2-percolates from any set of two adjacent vertices.
448 Ars Math. Contemp. 23 (2023) #P3.06 / 441–454
The CONJECTURING program made several conjectures of the form that a known suffi-
cient condition for graph Hamiltonicity is a sufficient condition for 2-bootstrap-goodness.
It is a well-known result that Dirac graphs are Hamiltonian; indeed Freund, Poloczek, and
Reichmann [16] proved that they are also 2-bootstrap-good. As we will use similar tech-
niques later, we provide a short proof here.
Theorem 2.17. If a graph in G is Dirac then it is 2-bootstrap-good.
Proof. It is easy to check that graphs with order three with the Dirac property are 2-
bootstrap-good. Assume that Dirac graphs with fewer than n vertices are 2-bootstrap-good.
Let G be a Dirac graph with n vertices; so every vertex in G has degree at least n2 .
Let H be a 2-bootstrap-good subgraph of H with a maximum number of vertices. Note
that no vertex in V \ H has more than one neighbor in H , otherwise H would not be a
maximum 2-bootstrap-good subgraph of G. So every vertex in V \ H has at least n2 − 1
neighbors in V \H . So V \H induces a Dirac subgraph of G. By our inductive assumption
the graph G[V \H] is 2-bootstrap-good. Since every vertex in G[V \H] has degree at least
n
2 − 1 and the order of G[V \H] is no more than the order of h, it follows that both H and
V \H have order n2 .
So G has the structure of two complete order n2 complete sugraphs with a matching
from H to V \H . Let v be a vertex in H , v′ be the vertex it is matched to in V \H and
w be any other vertex in V \ H . It is easy to see that {v, w} percolates G and thus G is
2-bootstrap-good.
As a consequence, we obtain the following easy corollary.
Corollary 2.18. If a graph in G is 2-bootstrap-good then it is either not cubic or it is Dirac.
Proof. A graph which is both Dirac and cubic has order at most six (and no cubic graph has
order seven). Hence it suffices to prove that if G is cubic with at least eight vertices, then
it is not 2-bootstrap-good. We’ll call an edge ‘usable’ at any particular step in the infection
process if it has one infected endpoint and one uninfected endpoint. If an infected graph
has less than two usable edges, then the infection cannot spread any further. Consider an
initial set of two infected vertices in G. There are at most six usable edges leaving this set,
since G is cubic. Any new infected vertex will make two edges unusable, and add at most
one usable edges; thus the total number of usable edges drops by at least one with each
newly infected vertex. Hence, the final number of infected vertices can be at most five,
since at this point there will be at most one usable edge remaining.
A graph with order n is Ore if every pair of non-adjacent vertices have degree sum at
least n: being Ore is also a sufficient condition for being Hamiltonian. The CONJECTURING
program also conjectured that Ore graphs are 2-bootstrap-good – this is strictly weaker than
the result proven in [15], who prove that in fact degree sum at least n− 2 is enough.
Recall that a graph is chordal if it has no induced cycle of length longer than three.
In order to prove that all chordal graphs are 2-bootstrap-good, we will need the following
lemma.
Lemma 2.19. Let G ∈ G be a 2-connected chordal graph and S ⊊ V (G) such that |S| ≥ 2
and G[S] is connected. Then there is some x ∈ V (G) \S such that x is adjacent to at least
two vertices v, w ∈ S.
N. Bushaw et al.: Bootstrap percolation via automated conjecturing 449
Proof. Consider building an auxiliary graph G′ by adding a new vertex y adjacent to ev-
erything in S. By its construction, G′ is 2-connected. Pick x ∈ V \ S to minimize the
total length of two internally disjoint paths from x to p; call these paths P1 = xv1 . . . vny
and P2 = xw1 . . . wmy. By minimality, both vn and wm are both in S. Since G[S] is
connected, there is a path from vn to wm in S; let P = vnp1 . . . pkwm be a minimum
length such path. By taking the union of P1vn, P2vm and P , we find a cycle, which by
minimality must be induced. But since G is chordal, this means the cycle is a triangle, and
it means that x is the vertex we wanted.
From this lemma, we easily deduce that all chordal graphs are 2-bootstrap-good.
Theorem 2.20. If G ∈ G is chordal, then it is 2-bootstrap-good.
Proof. If G contains a single block, infect any two adjacent vertices. Otherwise, infect one
vertex of each block (both of which are adjacent to the cut vertex). Thus in each block we
have two infected adjacent vertices, and so by repeatedly applying Lemma 2.19 we infect
a new vertex as long as there is some uninfected vertex.
In fact, something somewhat stronger is true – every chordal graph is a strangulated
graph; this somewhat less common class of graphs consists of those graphs in which delet-
ing the edges of any induced cycle of length greater than three would disconnect the re-
maining graph.
Theorem 2.21. If G ∈ G is a strangulated graph, then it is 2-bootstrap-good.
Proof. A strangulated graph can be constructed from chordal graphs and maximal planar
graphs by gluing along cliques [24]. If such a graph is a block, then all the gluings occur
along cliques of size at least two. We argue that any two adjacent vertices will infect the
graph since both chordal graphs and maximal planar graphs can be infected from any initial
adjacent pair (this is argued above for chordal graphs).
Indeed, let H be a maximal 2-bootstrap-good subgraph of a maximal planar graph G
(such maximal planar graphs are well known to be triangulations). If H ̸= G, then there
must be some vertex v ∈ G −H with a neighbor w ∈ H ∩ N(v). We orient the vertices
around w clockwise as x1, x2, . . . , xk. Note that v = xi for some i ∈ [k], and there is
at least one xj from H (possibly with i = j). Hence at some point there must be a pair
xℓ, xℓ+1 (we think cyclically, allowing xk, x1) with exactly of the pair in H and the other
in G − H . But then {w, xℓ, xℓ+1} lie on a common face, which must be a triangle. As
such, any infection which percolates on H also spreads to all of w, xℓ, xℓ+1 contradicting
maximality. Thus it must be that H = G, and so each triangulation can be percolated from
any adjacent pair. Note that if a strangulated graph has two blocks, then there it has only
one gluing that is along a single vertex; infecting a single neighbor from each adjacent
block will infect the entire graph via the infection processes described above inside each
block.
A different superclass of chordal graphs is that of dually chordal graphs (so named
because they are the clique graphs of chordal graphs, and thus dual in nature to chordal
graphs). An alternate characterization is that a graph is dually chordal if and only if the
hypergraph of its maximal cliques is the dual is a hypertree [9] (we give a more technical
version of this somewhat non-standard term inside the proof). These graphs, like chordal
graphs, are always 2-bootstrap-good.
450 Ars Math. Contemp. 23 (2023) #P3.06 / 441–454
Theorem 2.22. If G ∈ G is dually chordal, then it is 2-bootstrap-good.
Proof. We first make use of an alternate characterization. If a graph is dually chordal if the
auxiliary hypergraph formed with V (H) = V (G) and E(H) = {X : G[X] is a maximal
clique} is a hypertree (that is, it is connected and has no cycles). If G contains a single
block, then each pair of cliques intersects in at least two vertices; thus infecting any adjacent
vertices will infect the entire block (as it percolates through the cliques of the instersection
hypertree). For a graph with two blocks, we once again infect a single vertex of each block,
with both adjacent to the cut vertex.
Next, recall that a graph is called a cograph (short for complement reducible) when it
contains no induced copy of the path P4. The CONJECTURING program conjectured that
such graphs are 2-bootstrap-good.
Theorem 2.23. If G ∈ G is a cograph, then it is 2-bootstrap-good.
Proof. Cographs can be constructed by taking disjoint unions and joins of cographs, start-
ing from single vertices [14]. We proceed by strong induction on order of our cograph; the
base cases are trivial. Consider next a cograph which is a single block. Since G ∈ G, we
know that G is connected and thus it arises from taking the join of two cographs G1 and
G2. Consider infecting two vertices from G1; this will infect all of G2 in the next step, and
these will infect the remainder of G2 in the second step. Note that this shows something
slightly stronger – we can infect any two vertices in either part of their block. Therefore, if
G is constructed from two blocks G1 and G2 sharing a cut-vertex, and each Gi was con-
structed by taking the join of Hi,1 and Hi,2 with at least two vertices each, then we simply
select a vertex in H1,k and H2,j which are adjacent to the cut vertex; this will infect the cut
vertex, and then spread to the blocks by the preceding argument.
3 Which Kneser graphs are good?
Finally, we make a somewhat different attack; rather than proving a general condition is
sufficient, we explore a particular class of graphs and characterize those which are good. In
particular, recall that the Kneser graph KG(t)s is a graph whose vertices are the t element
subsets of [s], with two vertices adjacent when their corresponding subsets are disjoint.
Trivially, this graph is an independent set whenever s < 2t. But when is it 2-bootstrap-
good?
KG
(1)
1 and KG
(1)
2 are both trivially 2-bootstrap-good, and these are the only interesting
Kneser graphs with s ≤ 2. Further, we note that for k ≥ 2, the graph KG(t)2t is a collection
of disjoint edges; this is clearly not 2-bootstrap-good, so we may assume that s ≥ 2t + 1.
All remaining possibilities are covered by the following theorem.
Theorem 3.1. Assume s ≥ 3. A Kneser graph KG(t)s ∈ G is 2-bootstrap-good if and only
if s ≥ min{3t, 2t+ 3}.
Proof of Theorem 3.1. Necessity: Assume that s < 3t, that s ≤ 2t + 2, and let v, w be
vertices of KG(t)s (that is, v and w are size t subsets of [s]) with which our infection begins.
Note that since v and w are t element sets, we have |v∩w| ∈ [0, t−1]. Let A := v∪w ⊆ [s],
and let B := [s] \ A. Note that if a vertex x is adjacent to both v and w (that is, x can be
infected by {v, w}), then x must be disjoint from A – and thus x ⊆ B.
N. Bushaw et al.: Bootstrap percolation via automated conjecturing 451
Since s ≥ 3, we have at least three vertices in KG(t)s , and so if |B| < t, then there
are no vertices disjoint from |A| and so v and w cannot infect any vertices. So, if our
infection is to percolate we must choose v and w in order to guarantee |B| ≥ t, and so
|A| = s− |B| ≤ t+ 2. Now we need only consider two cases – either |A| = t+ 1 (and so
v and w share t − 1 common elements) or |A| = t + 2 (and so v, w share t − 2 common
elements). Most of our work will lie in proving the first case; the second will fall shortly
after.
Assume |A| = t + 1; then |B| = t or |B| = t + 1. If |B| = t, then there is only a
single vertex x, which will become infected by v and w. Since s < 3t, there are no vertices
adjacent to both x and v or to both x and w. Thus the infection stops at precisely three
vertices, and since
(
s
t
)
≥ 4 for all s, t satisfying our conditions, this is not the entire graph.
If |B| = t + 1, then similarly v and w can infect t + 1 new vertices. At the next
stage, any two of these vertices will infect all vertices disjoint from B – these are precisely
X = {y : |y| = t and y ⊆ A}. In these last two steps, we’ve built a complete bipartite
infected Kt+1,t+1. We will show that the infection can spread no further.
Let X be as above, and let Y be the corresponding vertices from B. Since s < 3t, there
is again no vertex adjacent to both a vertex from X and a vertex from Y . Further, any two
vertices a, b ∈ X (or both in Y ) contain t − 1 common elements, so between them they
both contain all t + 1 elements of A (or of B). Then, the only vertices adjacent to both
a, b are those vertices in the other half of our bipartite graph – which are already infected.
Again, the infection process must stop. Since vertices with some elements from A and
some elements from B are not yet infected, the initial set has not percolated.
Finally, assume |A| = t + 2. Then, |B| = t and there is only one vertex infected by
v and w. As above, the infection cannot spread any further, and since there are more than
three vertices in KG(t)s we cannot infect the whole graph.
Sufficiency: Suppose we have s ≥ 3, along with s ≥ 3t or s ≥ 2t + 3. We note that
there are only two Kneser graphs for which s ≥ 3t but s < 2t + 3 – these are KG(1)3
and KG(2)6 . Since KG
(1)
3
∼= K3, this is trivially 2-bootstrap-good. Further, one can easily
check that {{1, 2}, {2, 3}} percolates in KG(2)6 (as does any other pair of vertices sharing
a common element). For all other values of our parameters, we may assume s ≥ 2t + 3
(since s ≥ 3t will guarantee this).
As in KG(2)6 we choose two vertices v and w with |v ∩ w| = t − 1. Then, letting
A := v ∪ w ⊆ [s] and B := [s] \ A as before, we have |A| = t + 1 and |B| ≥ t + 2.
Now, we partition the vertices x of KG(t)s according to the size of |A ∩ x|, noting that
|B ∩ x| = t− |A ∩ x|. We denote these sets A0, A1, . . . , At, where |Ai| = i.
Initially, we infect vertices v and w. In the second round, v and w infect all vertices
disjoint from A; that is, all those vertices in A0. These vertices in A0 then infect all those
vertices in At (which are disjoint from B). Since |B| ≥ t+ 2, we can choose two vertices
b1 and b2 in A0 which overlap which share t− 1 elements, so that |b1 ∪ b2| = t+ 1. Then,
there will be at least one element of y ∈ B \ (b1 ∪ b2), so b1 and b2 can infect the vertices
in At−1. Finally, we show that our infection percolates from this point.
Claim 3.2. If all vertices in At, At−1, and A0 are infected, the entire graph will become
infected.
Proof of Claim 3.2. For any choice of k ∈ [1, t] we can choose two vertices of Ak such
that we can from them infect a vertex in At−k and in At−k+1. Choose two vertices v and
452 Ars Math. Contemp. 23 (2023) #P3.06 / 441–454
w from Ak for which |v ∩A| = |w ∩A| and |(v ∩B)∩ (w ∩B)| = t− k− 1. Now, since
|A| = t + 1, we have t + 1 − k elements of A at our disposal, and since |B| ≥ t + 2 we
have at least t + 2 − (t − k + 1) = k + 1 elements available from B. Since a vertex of
At−k+1 requires t− k + 1 elements of A and t− (t− k + 1) = k − 1 elements of B, and
so such a vertex exists. Further, since we have every element of Ak already infected, we
can choose v ∩A and w ∩A such that any t+ 1− k elements are available, and since our
choice of v ∩ B and w ∩ B is independent of our choice these, we can choose v ∩ B and
w ∩ B so that any k + 1 entries are available from B; this allows us to infect our desired
vertex in At−k+1.
To infect a vertex in At−k, choose v and w such that |(v ∩A) ∩ (w ∩A)| = k − 1 and
such that v ∩B = w ∩B. Then we have again t+ 1− (k+ 1) = t− k elements available
from A, and at least t+ 2− (t− k) = k + 2 elements of B. Thus we can find a vertex in
At−k adjacent to both v and w, and which will thus become infected. As before, we can
infect any vertex of At−k this way.
End of Proof of Theorem 3.1.
4 Conclusion and further work
This is an introductory exploration to the area of very small percolating sets. Building on
the work of Dairkyo et al. [15], and others, we used the automated conjecturing frame-
work to begin a systematic search for classes of graphs which are 2-bootstrap-good (or
2-bootstrap-bad). From this starting point, we’ve given a number of not-so-hard-to-prove
but quite-hard-to-discover conditions (both necessary and sufficient) for a graph to be 2-
bootstrap-good. It remains an intriguing open question to find a full characterization of
such graphs (however, at this early stage we do not even have a conjecture of what such a
characterization might look like).
Further, there are many natural generalizations of these results to explore. In particular,
what properties will guarantee that a graph has a k-element percolating set in k-bootstrap
percolation? This paper explores the k = 2 case, but k = 3 and higher are as interesting. In
addition, bootstrap percolation is just one of many monotone cellular automata which one
can define on a graph (as a group, these are all examples of graph bootstrap percolation
defined in the 1960s by Bollobás under the name weak saturation [6]). What graphs have
the smallest possible percolating sets in these more general models?
Finally we report a conjecture that attracted our interest but which we did not resolve.
The diameter of a (connected) graph is the maximum distance between any pair of its
vertices. Notice that a graph with diameter no more than two has at most two blocks. A
graph is perfect if the chromatic number and clique number of every subgraph is equal.
This class of graphs includes, for instance, bipartite graphs and chordal graphs. As such,
there is relation between this conjecture and Theorem 2.20.
Conjecture 4.1. If a graph in G is perfect and its diameter is no more than two then the
graph is 2-bootstrap-good.
This conjecture was produced by the CONJECTURING program and, like the proved
conjectures reported above, only guaranteed to be true for the input graphs used when the
program was run. It is a surprising fact that many conjectures of the program are in fact
true.
N. Bushaw et al.: Bootstrap percolation via automated conjecturing 453
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P3.07 / 455–465
https://doi.org/10.26493/1855-3974.2573.90d
(Also available at http://amc-journal.eu)
On the existence of zero-sum perfect matchings
of complete graphs
Teeradej Kittipassorn , Panon Sinsap *
Department of Mathematics and Computer Science, Faculty of Science,
Chulalongkorn University, Bangkok, Thailand
Received 28 February 2021, accepted 28 August 2022, published online 24 January 2023
Abstract
In this paper, we prove that given a 2-edge-coloured complete graph K4n that has the
same number of edges of each colour, we can always find a perfect matching with an equal
number of edges of each colour. This solves a problem posed by Caro, Hansberg, Lauri,
and Zarb. The problem is also independently solved by Ehard, Mohr, and Rautenbach.
Keywords: Graphs, zero-sum perfect matching.
Math. Subj. Class. (2020): 05C15
1 Introduction
Note that in this paper, we will use the word ‘matching’ when in fact we mean ‘perfect
matching’.
For an edge-colouring function f : E(G) → S of a graph G where S ⊆ Z and a
subgraph H of G, if
∑
e∈E(H) f(e) = 0 then H is called a zero-sum subgraph of G.
The research in zero-sum problems can be traced back to the three theorems that
give them the algebraic foundation. These are the Erdős-Ginzberg-Ziv Theorem [13], the
Cauchy-Davenport Theorem [11], and Chevalley’s Theorem [10]. Early zero-sum results
concern with the sum taken in additive group Zk, the area is called Zero-sum Ramsey
Theory. This theory studies the zero-sum Ramsey number R(G,Zk) which is the smallest
number n such that in every Zk-edge-colouring of r-uniform hypergraph on n vertices K(r)n
there exists a zero-sum modulo k copy of G. It also studies the zero-sum bipartite Ramsey
number B(G,Zk) which is the smallest number n such that in every Zk-edge-colouring of
Kn,n there exists a zero-sum modulo k copy of G. For more complete developments of the
topic consult [1, 3].
*Corresponding author.
E-mail addresses: teeradej.k@chula.ac.th (Teeradej Kittipassorn), panon.sinsap@gmail.com (Panon Sinsap)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
456 Ars Math. Contemp. 23 (2023) #P3.07 / 455–465
In [2], Caro gave the complete characterization of the zero-sum modulo 2 Ramsey
number R(G,Z2). In [7], Caro and Yuster gave the characterization of zero-sum modulo
2 bipartite Ramsey numbers. Along with [8] and [14], these four papers completely solved
the zero-sum Ramsey theory over Z2. Caro and Yuster [9] were the first to consider zero-
sum problems over Z. Recently, several variants of the zero-sum problems have been
studied (see, e.g. [5, 6]).
In [4], Caro, Hansberg, Lauri, and Zarb had studied zero-sum subgraphs where S =
{−1, 1} from various host graphs and various kinds of subgraphs. They then proved what
they call the ‘Master Theorem’ which covers many results of this kind. However there is
a remarkable variation of zero-sum subgraph problem that has not been decided by their
work in that paper. So they posed it at the end of the paper and the problem is the following:
Problem 1.1. Suppose f : E(K4n) → {−1, 1} is such that it is a zero-sum graph. Does a
zero-sum matching always exist?
Observe that this problem essentially wants us to find a matching that has an equal
number of edges that were assigned with −1 and 1 out of a complete graph of degree 4n
that had been assigned an equal number of −1 and 1 to their edges. This allows us to
discard the arithmetic meanings of −1 and 1 and replace them with general colour names.
In this paper, we choose to use black and red.
Our main result is the following theorem whose proof is equivalent to the solution of
Problem 1.1.
Theorem 1.2. For any 2-edge-colouring of K4n with an equal number of edges of each
colour, there exists a matching with an equal number of edges of each colour.
Recently, Problem 1.1 had been independently resolved by Ehard, Mohr, and Rauten-
bach [12].
2 Terminology
To facilitate the language of our proof, we introduce the following notations and terminolo-
gies.
For a graph G, we define M(G) to denote the set of all matchings in G.
In K4n, n ∈ N, we define the operation S : M(G)× V (G)4 → M(G) by
S(M,u, v, x, y) =
{
(V (M), E(M) ∪ {ux, vy} − {uv, xy}) if uv ∈ M and xy ∈ M,
M otherwise.
This operation will be called a swapping (see Figure 1).
If M is a matching extracted from 2-edge-coloured K4n, VB(M) will denote the set of
all vertices of M incident to a black edge in M , while VR(M) will denote the same thing
for red.
For any disjoint subsets S, T ⊆ V (G), E(S, T ) denotes the set of all edges with one
endpoint in S and one endpoint in T .
Let b(M) and r(M) denote the number of black edges and red edges in M respectively.
Lastly, sometimes we will shorten the phrase ‘the difference between the number of
edges of each colour’ to merely ‘the difference’. As there will be only one kind of differ-
ence in this work, this should cause no ambiguity.
Now we have all the terminologies needed for our proof.
T. Kittipassorn and P. Sinsap: On the existence of zero-sum perfect matchings of complete graphs 457
u v
x y
M
u v
x y
S(M,u, v, x, y)
Figure 1: The result of a swapping. All other edges in the matching stays the same.
3 Proof of the Theorem
We first state an important observation as a lemma.
Lemma 3.1. For M ∈ M(G), if there are more red edges than black edges in E(VR(M),
VB(M)), then we can make a swapping that will increase the number of red edges and
decrease the number of black edges in M by 1 each. In particular, if there are more black
edges than red edges in M and there are more red edges than black edges in E(VR(M),
VB(M)), then we can make a swapping that will reduce the difference by 2.
Proof. Consider two edges of M , one black and one red, uv and xy respectively. The edges
joining between the vertices of these two edges will be among the following six varieties
u v
x y
u v
x y
u v
x y
Figure 2: The first three varieties.
u v
x y
u v
x y
u v
x y
Figure 3: The first three varieties.
Observe that in the first three varieties the number of black edges are no fewer than the
numbers of red edges and that the resulting matchings from S(M,u, v, x, y) and S(M,u, v,
y, x) will never reduce the difference between the numbers of black edges and red edges
that we had from M .
458 Ars Math. Contemp. 23 (2023) #P3.07 / 455–465
u v
x y
u v
x y
u v
x y
u v
x y
S(M,u, v, x, y)
u v
x y
u v
x y
u v
x y
S(M,u, v, y, x)
u v
x y
u v
x y
u v
x y
While in the last three varieties the numbers of red edges are no fewer than the numbers
of black edges and that at least one of the resulting matchings from S(M,u, v, x, y) or
S(M,u, v, y, x) reduces the difference between the numbers of black edges and red edges
that we had from M .
u v
x y
u v
x y
u v
x y
u v
x y
S(M,u, v, x, y)
u v
x y
u v
x y
u v
x y
S(M,u, v, y, x)
u v
x y
u v
x y
u v
x y
So if there are more red edges than black edges joining between VB(M) and VR(M),
we can guarantee the existence of a pair of edges, one black and one red, in M such that
the edges joining between them form one of the latter three (in fact two) varieties.
T. Kittipassorn and P. Sinsap: On the existence of zero-sum perfect matchings of complete graphs 459
Using this pair of edges and appropriate order of vertices, we can make a swapping that
will increase the number of red edges and decrease the number of black edges that we had
from M by 1 each.
Note that in any cases, we have replaced one black edge and one red edge with two red
edges, so that the difference between the number of black edges and red edges that we had
from M will change by 2 in the resulting matching. If initially there are more black edges
than red edges in M , the difference will reduce by 2. But if initially there are more red
edges than black edges in M , the difference will increase by 2.
Remark 3.2. If we are to read the proof of this lemma with the colour red and black in
place of each other, the same thing will happen to the colour black when there are more
black edges than red edges joining between VB(M) and VR(M).
We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2. In our attempt to prove this statement, we will first take an arbitrary
matching of K4n, then gradually reduce the difference between the numbers of edges of
each colour.
To achieve that, we will construct a finite sequence of matchings in M(G) which as our
sequence progress |b(M)− r(M)|, the difference between the numbers of edges of each
colour, will gradually and strictly decrease until it reaches zero.
To start the proof, we first pick an arbitrary matching M of G.
We take this M as M0, the zeroth term of our sequence.
Next, we proceed to obtain next terms of our sequence by the following method.
For nonnegative integer i, if Mi is a term in our sequence, it is without loss of generality
to assume that b(Mi) ⩾ r(Mi).
Now we have three cases to consider.
Case 1: b(Mi) = r(Mi)
In this case, we end our sequence and take Mi as the matching we have been looking
for.
Case 2: b(Mi)− r(Mi) > 2
Case 2.1: G[VB(Mi)] is monochromatic.
We claim that there are more red edges than black edges between VB(Mi) and VR(Mi).
Since b(Mi) > r(Mi), |VB(Mi)| > |VR(Mi)| so there are more edges in G[VB(Mi)]
than in G[VR(Mi)].
Recall that in our graph, the number of red edges and the number of black edges are
equal.
Since G[VB(Mi)] is monochromatic(black) and e(G[VB(Mi)]) > e(G[VR(Mi)]), there
must be more red edges between VB(Mi) and VR(Mi) than black edges.
By applying the lemma to Mi, we can make a swapping that will reduce the difference
by 2.
We take the resulting matching of this swapping as Mi+1 in our sequence.
Case 2.2: G[VB(Mi)] is not monochromatic.
Since G[VB(Mi)] is not monochromatic, there is a red edge, ux, in G[VB(Mi)].
Since u, x ∈ VB(Mi) and ux is red, there must be v, y ∈ VB(Mi) such that uv, xy ∈
Mi.
We take S(Mi, u, v, x, y) to be Mi+1 in our sequence.
460 Ars Math. Contemp. 23 (2023) #P3.07 / 455–465
This resulting matching will reduce the difference between the number of black edges
and red edges by 2 or 4 compared to that of Mi, depending on the colour of vy (see Fig-
ure 4).
u x
v y
vy is black
u x
v y
vy is red
u x
v y
If vy is black, the difference reduces by 2. If vy is red, the difference reduces by 4.
Figure 4: Two possibilities of swapping.
Case 3: b(Mi)− r(Mi) = 2
Case 3.1: G[VB(Mi)] is monochromatic.
The reasoning and execution of this case are the same as the Case 2.1.
Case 3.2: G[VB(Mi)] is not monochromatic.
Claim A. If there are more red edges than black edges between VB(Mi) and VR(Mi), then
we are done.
Proof. By applying the lemma to Mi, we can make a swapping that will reduce the differ-
ence by 2.
We take the resulting matching of this swapping as Mi+1 in our sequence.
Thus we may assume that there are not more red edges than black edges between
VB(Mi) and VR(Mi).
Claim B. If there are u, v, x, y ∈ VB(Mi) such that uv, xy ∈ E(Mi), ux is red and vy is
black in G, then we are done.
Proof. This claim is the same as Case 2.2 where vy is black. We take S(Mi, u, v, x, y) to
be Mi+1.
This reduce the difference by 2, so that it becomes 0.
T. Kittipassorn and P. Sinsap: On the existence of zero-sum perfect matchings of complete graphs 461
Thus we may assume that for all u, v, x, y ∈ VB(Mi) such that uv, xy ∈ E(Mi), if ux
is red then vy is also red in G
We observe that , as a consequence of this assumption, red edges always appear in pairs
and each red edge only involve two edges of Mi, namely those that share a vertex with it.
So if we count the number of red edges in G[VB(Mi)], it must be an even number.
As it will become important in the rest of the proof, let us make explicit that , as a
consequence of this assumption, there must be n + 1 black edges and n − 1 red edges in
Mi.
So |VB(Mi)| = 2n+2, |VR(Mi)| = 2n−2, e(G[VB(Mi)]) =
(
2n+2
2
)
= 2n2+3n+1,
e(G[VR(Mi)]) =
(
2n−2
2
)
= 2n2 − 5n + 3, the number of edges between VB(Mi) and
VR(Mi) is (2n + 2)(2n − 2) = 4n2 − 4, and finally the total number of edges of each
colour is 12
(
4n
2
)
= 4n2 − n.
Claim C. If there is an equal number of red edges and black edges between VB(Mi) and
VR(Mi), then we are done.
Proof. We claim that there is an odd number of black edges in G[VR(Mi)]. Note that there
is an even number of red edges in G[VB(Mi)] and an even number of edges of each colour
joining VB(Mi) and VR(Mi).
Since the argument is just a simple parity analysis, to avoid a verbose and confusing
argument, we present the following table as our argument.
n total black edges e(G[VB(Mi)]) black edges in G[VR(Mi)]
even even odd odd
odd odd even odd
So there is an odd number of black edges in G[VR(Mi)].
Thus there are p, q, r, s ∈ VR(Mi) such that pq, rs ∈ E(Mi), pr is red and qs is black
in G.
Let M ′i = S(Mi, p, q, r, s). Now M
′
i has n+ 2 black edges and n− 2 red edges.
Observe that after the latest swapping occurs, those vertices and edges originally in
G[VB(Mi)] are all contained in G[VB(M ′i)].
As a premise of this Case 3.2 states that there is a red edge in G[VB(Mi)], there must
be u, v, x, y ∈ VB(Mi) ⊂ VB(M ′i) such that uv, xy ∈ E(Mi) and ux, vy are red in G.
Since uv, xy ∈ E(M ′i), we take Mi+1 to be S(M ′i , u, v, x, y).
This reduce the difference by 2, so that it becomes 0.
Thus we may assume that there are more black edges than red edges between VB(Mi)
and VR(Mi).
For the sake of clarity of how we divide our next cases, we will consider the question
”What is the least number of red edges that have to be in G[VB(Mi)]?”.
So we will have to maximize the number of red edges outside of G[VB(Mi)].
Thus all the edges in G[VR(Mi)] and 4n
2−4
2 − 1 = 2n
2 − 3 edges between VB(Mi)
and VR(Mi) have to be red. Since there are 4n2 − n red edges in total, there are at least
(4n2 − n)− (2n2 − 5n+ 3)− (2n2 − 3) = 4n red edges in G[VB(Mi)]
Claim D. If there are more than 4n red edges in G[VB(Mi)], then we are done.
462 Ars Math. Contemp. 23 (2023) #P3.07 / 455–465
Proof. Since there are more black edges than red edges joining VB(Mi) and VR(Mi),
from our lemma, there must be a black edge uv and a red edge xy of Mi such that
S(Mi, u, v, x, y) will increase the number of black edges from that of Mi by one.
But before we make the swapping, we consider that there are 4n edges adjacent to uv
in G[VB(Mi)]. So that there is a red edge not adjacent to uv.
As a consequence of the assumption right after Claim B, a red edge implies an existence
of another red edge, there are p, q, r, s ∈ VB(Mi) − {u, v} such that pq, rs ∈ E(Mi) and
pr, qs are red in G.
Let M ′i = S(Mi, u, v, x, y) so that there are n+ 2 black edges and n− 2 red edges.
We take S(M ′i , p, q, r, s) to be our Mi+1.
The first swap increase the difference by 2 to be 4, then the second swap reduce the
difference by 4 to 0.
Thus we may assume that there are exactly 4n red edges in G[VB(Mi)]
As a consequence of this assumption, there are 4n
2−4
2 −1 = 2n
2−3 red edges between
VB(Mi) and VR(Mi) and G[VR(Mi)] is monochromatic (red).
Claim E. If there is a pair of edges in Mi of different colours such that the edges that lie
between them are of the latter three varieties shown in the proof of our lemma, then we are
done.
Proof. In this case we are guaranteed a swapping that will reduce the difference between
the numbers of black edges and red edges by 2 to 0. We take the resulting matching of that
swapping to be Mi+1
Thus we may assume that each pair of edges of different colours in Mi have edges of
the first three varieties between them.
Since there are 2n2−1 black edges and 2n2−3 red edges between VB(Mi) and VR(Mi)
and the edges are from just the first three varieties, there is only one possibility.
That is there is a black uv and a red xy connected to each other by edges of type two,
while other pairs of edges in Mi are connected by edges of type three in G.
Without loss of generality let ux be red in G.
Claim F. There is a red edge not adjacent to uv in G[VB(Mi)]
Proof. As in Claim D, there must be p, q, r, s ∈ VB(Mi)−{u, v} such that pq, rs ∈ E(Mi)
and pq, rs are red in G.
As in Claim D, we take S(S(Mi, u, v, y, x), p, q, r, s) as our Mi+1.
This effectively reduce the difference from 2 to 0.
Thus we may assume that all 4n red edges are adjacent to uv in G[VB(Mi)]
In G[VB(Mi)], u and v each connecting to 2n vertices apart from each other.
Thus they collectively involve 4n edges, so that all of those edges are red.
Claim G. There is a member of VB(Mi)− {u, v} that is joined with y by a black edge.
Proof. Let this member of VB(Mi)− {u, v} be called z.
Since z ∈ VB(Mi), there is a w ∈ VB(Mi) such that zw ∈ E(Mi).
Now vw is red and vy, yz, zw are black in G.
We take S(S(Mi, u, v, x, y), v, y, w, z) as our Mi+1 (see Figure 5).
First swapping does not change the difference, while the second one reduce it by 2
to 0.
T. Kittipassorn and P. Sinsap: On the existence of zero-sum perfect matchings of complete graphs 463
u v w z
x y
S(Mi, u, v, x, y)
u v w z
x y
S(S(Mi, u, v, x, y), v, y, w, z)
u v w z
x y
Figure 5
v u p q
x y
S(Mi, u, v, y, x)
v u p q
x y
S(S(Mi, u, v, y, x), u, y, p, q)
v u p q
x y
Figure 6
Thus we may assume that all members of VB(Mi) − {u, v} are joined with y by red
edges.
Observe that from our premises every red edge in G[VB(Mi)] has to have either u or v
as an endpoint and that all the edges joining u or v with any member in VB(Mi) − {u, v}
are red.
Now choose any p, q ∈ VB(Mi) − {u, v}. The edge pq is black while up and yq are
red.
We take S(S(Mi, u, v, y, x), u, y, p, q) as our Mi+1 (see Figure 6).
First swapping increase the difference by 2 to be 4, then the second one reduce it by 4
to 0.
Now, every case, except for Case 1 which is the terminal case, strictly reduces the
difference between the numbers of black edges and red edges as we are creating new terms
for our sequence.
Note that in no case the difference was reduced by more than its value.
Thus as our sequence progresses the difference strictly decreases and when it termi-
nates, the last term of the sequence has the difference of 0.
That is by following this method, we can always guarantee a matching which has the
same number of black edges and red edges.
This proves our theorem.
4 Concluding Remarks
We have proved Theorem 1.2 which settles the problem posed in [4]. Now we know that
if we are given a 2-edge-coloured complete graph of order 4n with the same number of
edges of each colour, we can extract a matching that has the same number of edges of each
colour.
We would like to pose two problems that are related to the result that we have proven.
464 Ars Math. Contemp. 23 (2023) #P3.07 / 455–465
The first problem is a generalization of our result. In our theorem, we study only the case
where the complete graph K4n is 2-edge-coloured and those two colours colour an equal
number of edges. An obvious generalization of this problem is to consider the similar
situation when there are more than two colours involved. Now we pose the following
question which is a generalization of our result.
Problem 4.1. For any k-edge-colouring of K2kn such that there are an equal number of
edges of each colour. Does there exist a matching such that there are an equal number of
edges of each colour?
The second problem comes from the fact that when we take a matching with an equal
number of edges of each colour out of our original complete graph, we are left with a graph
that has the same number of edges of each colour. One question that comes up is ‘can
we take another such matching?’. If we can, can we continue until all edges are gone?
If we cannot exhaust the edges with an arbitrary order of taking matchings out, is there
any sequence of taking matchings out that would use all edges? This leads us to pose the
following problem.
Problem 4.2. Given a 2-edge-coloured K4n with an equal number of edges of each colour.
Can the graph be decomposed into perfect matchings such that each matching has the same
number of edges of each colour?
ORCID iDs
Teeradej Kittipassorn https://orcid.org/0000-0001-8039-393X
Panon Sinsap https://orcid.org/0000-0001-8671-4773
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P3.08 / 467–480
https://doi.org/10.26493/1855-3974.2694.56a
(Also available at http://amc-journal.eu)
The A-Möbius function of a finite group
Francesca Dalla Volta
Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca,
Via Cozzi 55, 20126 Milano, Italy
Andrea Lucchini *
Università degli Studi di Padova, Dipartimento di Matematica “Tullio Levi-Civita”,
Via Trieste 63, 35121 Padova, Italy
Received 11 September 2021, accepted 30 September 2022, published online 27 January 2023
Abstract
The Möbius function of the subgroup lattice of a finite group G has been introduced
by Hall and applied to investigate several different questions. We propose the following
generalization. Let A be a subgroup of the automorphism group Aut(G) of a finite group
G and denote by CA(G) the set of A-conjugacy classes of subgroups of G. For H ≤ G
let [H]A = { Ha | a ∈ A} be the element of CA(G) containing H. We may define an
ordering in CA(G) in the following way: [H]A ≤ [K]A if Ha ≤ K for some a ∈ A. We
consider the Möbius function µA of the corresponding poset and analyse its properties and
possible applications.
Keywords: Groups, subgroup lattice, Möbius function.
Math. Subj. Class. (2020): 20D30, 05E16
1 Introduction
The Möbius function of a finite partially ordered set (poset) P is the map µP : P ×P → Z
satisfying µP (x, y) = 0 unless x ≤ y, in which case it is defined inductively by the
equations µP (x, x) = 1 and
∑
x≤z≤y µP (x, z) = 0 for x < y.
In a celebrated paper [5], P. Hall used for the first time the Möbius function µ of the
subgroup lattice of a finite group G to investigate some properties of G, in particular to
compute the number of generating t-tuples of G. A detailed investigation of the properties
of the function µ associated to a finite group G is given by T. Hawkes, I. M. Isaacs and
*Corresponding author.
E-mail addresses: francesca.dallavolta@unimib.it (Francesca Dalla Volta), lucchini@math.unipd.it (Andrea
Lucchini)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
468 Ars Math. Contemp. 23 (2023) #P3.08 / 467–480
M. Özaydin in [6]. In that paper, the authors also consider the Möbius function λ of the
poset of conjugacy classes of subgroups ofG,where [H] ≤ [K] ifH ≤ Kg for some g ∈ G
(see [6, Section 7]). In particular, they propose the interesting and intriguing question of
comparing the values of µ and λ.
In this paper we aim to generalize the definitions and main properties of the func-
tions µ and λ to a more general context. Let G and A be a finite group and a subgroup
of the automorphism group Aut(G) of G, respectively. Denote by CA(G) the set of A-
conjugacy classes of subgroups of G. For H ≤ G let [H]A = { Ha | a ∈ A} be
the element of CA(G) containing H. We may define an ordering in CA(G) in the following
way: [H]A ≤ [K]A if Ha ≤ K for some a ∈ A; we consider the Möbius function µA
of the corresponding poset. We will write µA(H,K) in place of µA([H]A, [K]A). When
A = Inn(G), we write C(G) and [H], in place of CInn(G)(G) and [H]Inn(G). When A = 1,
µA = µ is the Möbius function in the subgroup lattice of G, introduced by P. Hall. In
the case when A = Inn(G) is the group of the inner automorphism, µInn(G) coincides the
Möbius function λ of the poset of conjugacy classes of subgroups of G, defined above.
Note that for any subgroup A of Aut(G), we get [G]A = {G}.
In Section 2, we prove some general properties of µA. In particular we prove the
following result:
Proposition 1.1. LetG be a finite solvable group. IfG′ ≤ K ≤ G andA is the subgroup of
Inn(G) obtained by considering the conjugation with the elements ofK, then µA(H,G) =
λ(H,G) for every H ≤ G.
To illustrate the meaning of the previous proposition, consider the following example.
LetG = A4 be the alternating group of degree 4 andA the subgroup of Inn(G) induced by
conjugation with the elements of G′ ∼= C2×C2. The posets C(G) and CA(G) are different.
For example there are three subgroups of G of order 2, which are conjugated in G, but not
A-conjugated. However λ(H,G) = µA(H,G) for any H ≤ G.
In Section 3, we generalize some result given by Hall in [5], about the cardinality
ϕ(G, t) of the set Φ(G, t) of t-tuples (g1, . . . , gt) of group elements gi such that G =
⟨g1, . . . , gt⟩. As observed by P. Hall, using the Möbius inversion formula, it can be proved
that
ϕ(G, t) =
∑
H≤G
µ(H,G)|H|t. (1.1)
We generalize this formula, showing that ϕ(G, t) can be computed with a formula involving
µA for any possible choice of A.
Theorem 1.2. For any finite group G and any subgroup A of Aut(G),
ϕ(G, t) =
∑
[H]A∈CA(G)
µA(H,G)| ∪a∈A (Ha)t|.
If G is not cyclic, then ϕ(G, 1) = 0, so we obtain the following equality, involving the
values of µA.
Corollary 1.3. If G is not cyclic, then
0 =
∑
[H]A∈CA(G)
µA(H,G)| ∪a∈A Ha|.
F. Dalla Volta and A. Lucchini: The A-Möbius function of a finite group 469
Further generalizations are given in Section 4, where we consider the function ϕ∗(G, t),
which is an analogue of ϕ(G, t): actually, ϕ∗(G, t) denotes the cardinality of the set of of
t-tuples (H1, . . . ,Ht) of subgroups of G such that G = ⟨H1, . . . ,Ht⟩. As a corollary of
our formula for computing ϕ∗(G, t), we obtain we following unexpected result.
Proposition 1.4. Let σ(X) denote the number of subgroups of a finite group X. For any
finite group G, the following equality holds:
1 =
∑
H≤G
µ(H,G)σ(H).
Finally, in Section 5, we consider one question originated from a result given by Hawkes,
Isaacs and Özaydin in [6]: they proved that the equality
µ(1, G) = |G′|λ(1, G)
holds for any finite solvable group G; later Pahlings [7] generalized the result proving that
µ(H,G) = |NG′(H) : G′ ∩H| · λ(H,G) (1.2)
holds for any H ≤ G whenever G is finite and solvable. Following [3], we say that G
satisfies the (µ, λ)-property if (1.2) holds for any H ≤ G. Several classes of non-solvable
groups satisfy the (µ, λ)-property, for example all the minimal non-solvable groups (see
[3]). However it is known that the (µ, λ)-property does not hold for every finite group. For
instance, it does not hold for the following finite almost simple groups: A9, S9, A10, S10,
A11, S11, A12, S12, A13, S13, J2, PSU(3, 3), PSU(4, 3), PSU(5, 2), M12, M23, M24,
PSL(3, 11), HS, Aut(HS), He Aut(H), McL, PSL(5, 2), G2(4), Co3, PΩ−(8, 2),
PΩ+(8, 2). It is somehow intriguing to notice that although the (µ, λ)-property fails for
the sporadic groups M12, J2, McL, it holds for their automorphism groups.
We prove the following generalization of Pahlings’s result.
Theorem 1.5. Let N be a solvable normal subgroup of a finite group G. If G/N satisfies
the (µ, λ)-property, then G also satisfies the (µ, λ)-property.
An almost immediate consequence of the previous theorem is the following.
Corollary 1.6. PSU(3, 3) is the smallest group which does not satisfy the (µ, λ) property.
In the last part of Section 5, we use Theorem 1.2 to deduce some consequences of the
(µ, λ)-property. In particular we prove the following theorem.
Theorem 1.7. Suppose that a finite group G satisfies the (µ, λ)-property. Then, for every
positive integer t, the following equality is satisfied:
∑
[H]∈C(G)
λ(H,G)
(
|H|t−1|G||G′H|
|G′NG(H)|
− | ∪a∈A (Ha)t|
)
= 0.
Some open questions are proposed along the paper.
470 Ars Math. Contemp. 23 (2023) #P3.08 / 467–480
2 Applying some general properties of the Möbius function
Given a poset P , a closure on P is a function¯ : P → P satisfying the following three
conditions:
(a) x ≤ x̄ for all x ∈ P ;
(b) if x, y ∈ P with x ≤ y, then x̄ ≤ ȳ;
(c) x̄ = x̄ for all x ∈ P .
If ¯ is a closure map on P, then P = {x ∈ P | x̄ = x} is a poset with order induced by the
order on P . We have:
Theorem 2.1 (The closure theorem of Crapo [2]). Let P be a finite poset and let ¯: P → P
be a closure map. Fix x, y ∈ P such that y ∈ P . Then
∑
x≤z≤y,z̄=y
µP (x, z) =
{
µP̄ (x, y) if x = x̄
0 otherwise.
In [5], P. Hall proved that if H < G, then µ(H,G) ̸= 0 only if H is an intersection
of maximal subgroups of G. Using the previous theorem, the following more general
statement can be obtained.
Proposition 2.2. If H < G and µA(H,G) ̸= 0, then H can be obtained as intersection of
maximal subgroups of G.
Proof. Let H be a proper subgroup of G and let H be the intersection of the maximal
subgroups of G containing H . Moreover let G = G. The map [H]A 7→ [H]A is a well
defined closure map on CA(G). Apply Theorem 2.1, with x = [H]A and y = [G]A. Since
K = G if and only if K = G, we have that µA(H,G) = 0 if H ̸= H.
An element a of a poset P is called conjunctive if the pair {a, x} has a least upper
bound, written a ∨ x, for each x ∈ P.
Lemma 2.3 ([6, Lemma 2.7]). Let P be a poset with a least element 0, and let a > 0 be a
conjunctive element of P. Then, for each b > a, we have∑
a∨x=b
µP(0, x) = 0.
From the above 2.3, the following Lemma 2.4 follows easily. Together with Lemma 2.5
and Lemma 2.7, this allows us to prove Proposition 1.1.
Lemma 2.4. Let N be an A-invariant normal subgroup of G and H ≤ G. If H < HN <
G, then
µA(H,G) = −
∑
[Y ]A∈SA(H,N)
µA(H,Y ),
with SA(H,N) = {[Y ]A ∈ CA(G) | [H]A ≤ [Y ]A < [G]A and Y N = G}.
Proof. Let P be the interval {[K]A ∈ CG(A) | [H]A ≤ [K]A ≤ [G]A]}. Notice that
[HN ]A is a conjunctive element of P. Indeed [HN ]A ∨ [K]A = [KN ]A for every [K]A ∈
P. So the conclusion follows immediately from Lemma 2.3.
F. Dalla Volta and A. Lucchini: The A-Möbius function of a finite group 471
Lemma 2.5. Let K and A be a subgroup of G and the subgroup of Inn(G) induced by
the conjugation with the elements of K, respectively. Assume that N is an abelian minimal
normal subgroup of G contained in K and H < HN ≤ G. Then
µA(H,G) = −µA(HN,G)γA(N,H),
where γA(N,H) is the number of A-conjugacy classes of complements of N in G contain-
ing H.
Proof. If HN = G, then H is a maximal subgroup of G, hence µA(H,G) = −1, while
µA(HN,G) = µA(G,G) = 1 and γA(N,H) = 1, so the statement is true. So we may
assume HN < G and apply Lemma 2.4. Suppose [Y ]A ∈ SA(H,N). Notice that, since
Y N = G andN is abelian, Y ∩N is normal inG. MoreoverN ̸≤ G, since Y < G = Y N.
By the minimality of N as normal subgroup, we conclude Y ∩N = 1. Let
C = {J ≤ G | H ≤ J ≤ Y }, D = {L ≤ G | HN ≤ L}
CA = {[J ]A ∈ CA(G) | [H]A ≤ [J ]A ≤ [Y ]A},DA = {[L]A ∈ CA(G) | [HN ]A ≤ [L]A}.
The map η : C → D sending J to JN is an order preserving bijection. Clearly, if J2 = Jx1
for some x ∈ K, then η(J2) = NJ2 = NJx1 = (NJ1)x = (η(J1))x. Conversely assume
η(J2) = (η(J1))
x with x ∈ K. Since Y N = G, x = yn with n ∈ N and y ∈ Y ∩ K.
Thus J2N = (J1N)x = (J1N)y and consequently, applying the Dedekind law, J2 =
J2(Y ∩ N) = J2N ∩ Y = (J1N)y ∩ Y = (J1N ∩ Y )y = Jy1 . It follows that η induces
an order preserving bijection from CA to DA, but then µA(H,Y ) = µA(HN,Y N) =
µA(HN,G).
The statement of the previous lemma leads to the following open question.
Question 2.6. LetG be a finite group,A ≤ Aut(G) andN anA-invariant normal subgroup
of G. Does µA(HN,G) divide µA(H,G) for every H ≤ G?
The following lemma is straightforward.
Lemma 2.7. Let A be a subgroup of Aut(G) and N an A-invariant normal subgroup of
G. Every a ∈ A induces an automorphism a of G/N . Let A = {a | a ∈ A}. Then, for any
H ≤ G, µA(HN,G) = µA(HN/N,G/N).
Proof of Proposition 1.1. We work by induction on |G| · |G : H|. The statement is true if
G is abelian. Assume thatG′ contains a minimal normal subgroup, sayN, ofG. IfN ≤ H,
then, by Lemma 2.7
λ(H,G) = λ(H/N,G/N) = µA(H/N,G/N) = µA(H,G).
So we may assume N ̸≤ H. If H is not an intersection of maximal subgroups of G, then
λ(H,G) = µA(H,G) = 0. Suppose H =M1 ∩ · · · ∩Mt where M1, . . . ,Mt are maximal
subgroups of G. In particular N is not contained in Mi for some i, so Mi is a complement
of N in G containing H and N ∩H = 1. By Lemma 2.5, we have
λ(H,G) = −λ(HN,G)γ(N,H), µA(H,G) = −µA(HN,G)γA(N,H),
where γ(N,H) is the number of conjugacy classes of complements of N in G containing
H and γA(N,H) is the number of A-conjugacy classes of these complements. Suppose
that K1,K2 are two conjugated complements of N in G containing H. Then K2 = Kn1 for
some n ∈ NN (H). Since N ≤ G′ ≤ K, it follows γ(N,H) = γA(N,H). Moreover, by
induction, λ(HN,G) = µA(HN,G), hence we conclude λ(H,G) = µA(H,G).
472 Ars Math. Contemp. 23 (2023) #P3.08 / 467–480
3 Generalizing a formula of Philip Hall
We begin with introducing the functions ΨA(H, t) and ψA(H, t), analogue of Φ(H, t) and
ϕ(H, t) in the general case of any possible subgroup A of Aut(G).
For any H ∈ CA(G) and any positive integer t, let
1. ΩA(H, t) =
⋃
a∈A(H
a)t;
2. ωA(H, t) = |ΩA(H, t)|;
3. ΨA(H, t) = {(g1, . . . , gt) ∈ Gt | ⟨g1, . . . , gt⟩ = Ha for some a ∈ A};
4. ψA(H, t) = |ΨA(H, t)|.
If (x1, . . . , xt) ∈ ΩA(H, t), then ⟨x1, . . . , xt⟩ ≤ Ha for some a ∈ A, hence ⟨x1, . . . , xt⟩ =
K for some K ≤ G with [K]A ≤ [H]A. Thus∑
[K]≤A[H]
ψA(K, t) = ωA(H, t)
and therefore, by the Möbius inversion formula,∑
[H]∈CA(G)
µA(H,G)ωA(H, t) = ψA(G, t).
On the other hand ψA(G, t) = ϕ(G, t) so we have proved the following formula.
Theorem 3.1. For any finite group G and any subgroup A of Aut(G),
ϕ(G, t) =
∑
[H]∈CA(G)
µA(H,G)ωA(H, t).
Notice that if A = 1, then ωA(H, t) = |Ht|, so that the result by Hall given in (1.1) is
a particular case of the previous theorem.
Corollary 3.2. If G is not cyclic, then
0 = ϕ(G, 1) =
∑
[H]∈CA(G)
µA(H,G)ωA(H, 1).
Taking A = Inn(G), we deduce in particular that if G is not cyclic, then∑
H∈C(H)
λ(H,G)ωInn(G)(H, 1) =
∑
H∈C(H)
λ(H,G)| ∪g Hg| = 0.
For example, ifG = S4, then the values of λ(H,G) and |∪gHg| are as in the following
table and 24− 12− 16− 15 + 4 + 9 + 7− 1 = 0.
F. Dalla Volta and A. Lucchini: The A-Möbius function of a finite group 473
λ(H,G) | ∪g Hg|
S4 1 24
A4 -1 12
D4 -1 16
S3 -1 15
K 1 4
⟨(1, 2, 3, 4)⟩ 0 10
⟨(1, 2, 3)⟩ 1 9
⟨(1, 2⟩ 1 7
⟨(1, 2)(3, 4)⟩ 0 4
1 -1 1
If G = A5, then the values of λ(H,G), ωInn(G)(H, 1) = | ∪g Hg|, ωInn(G)(H, 2) =
| ∪g (Hg)2| (taking only the subgroups H with λ(H,G) ̸= 0) are as in the following table
and 60− 36− 36− 40 + 21 + 32− 1 = 0.
λ(H,G) | ∪g Hg| | ∪g (Hg)2|
A5 1 60 3600
A4 -1 36 636
S3 -1 36 306
D5 -1 40 550
⟨(1, 2, 3)⟩ 1 21 81
⟨(1, 2)(3, 4)⟩ 2 16 46
1 -1 1 1
Moreover
3600− 636− 306− 550 + 81 + 2 · 46− 1 = 2280 = 19
30
· 3600 = ϕ(A5, 2).
If G = Dp = ⟨a, b | ap = 1, b2 = 1, ab = a−1⟩ and p is an odd prime, then the
behaviour of the subgroups in C(G) is described by the following table.
λ(H,G) | ∪g Hg|
Dp 1 2p
⟨a⟩ -1 p
⟨b⟩ -1 p+ 1
1 -1 1
Another interesting example is given by considering G = Cnp and A = Aut(G).
Let H ∼= Cn−1p be a maximal subgroup of G. Then, for K ≤ G, µA(K,G) ̸= 0
if and only if either [K]A = [G]A or [K]A = [H]A. Clearly ∪α∈Aut(G)Hα = G so
µA(G,G)ωA(G, 1)− µA(H,G)ωA(H, 1) = |G| − |G| = 0. More generally, ΩA(H, t) is
the set of t-tuples (x1, . . . , xt) such that (x1, . . . , xt) ∈ Kt for some maximal subgroup
K of G, so µA(G,G)ωA(G, t)− µA(H,G)ωA(H, t) = |G|t − ωA(H, t) is the number of
generating t-tuples of G.
474 Ars Math. Contemp. 23 (2023) #P3.08 / 467–480
Another generalization of (1.1), essentially due to Gaschütz, has been described by
Brown in [1, Section 2.2]. Let N be a normal subgroup of G and suppose that G/N admits
t generators for some integer t. Let y = (y1, . . . , yt) be a generating t-tuple of G/N and
denote by P (G,N, t) the probability that a random lift of y to a t-tuple of G generates
G. Then P (G,N, t) = ϕ(G,N, t)/|N |t, where ϕ(G,N, t) is the number of generating t-
tuples ofG lying over y. As is showed in [1, Section 2.2], using again the Möbius inversion
formula it can be proved:
ϕ(G,N, t) =
∑
H≤G,HN=G
µ(H,G)|H ∩N |t. (3.1)
This formula can be generalized in our contest in the following way:
Theorem 3.3. Let N be an A-invariant normal subgroup of G and fix g1, . . . , gt ∈ G with
the property that G = ⟨g1, . . . , gt⟩N. Define
• ΩA(H,N, t) = {(n1, . . . , nt) | ⟨g1n1, . . . , gtnt⟩ ≤ Ha for some a ∈ A};
• ωA(H,N) = |ΩA(H,N, t)|
and let CA(G,N) = {[H]A ∈ CA(G) | HN = G}. Then
ϕ(G,N, t) =
∑
[H]A∈CA(G,N)
µA(H,G)ωA(H,N, t).
Proof. Fix g1, . . . , gt ∈ G with the property that G = ⟨g1, . . . , gt⟩N. Then ϕ(G,N, t) is
the cardinality of the set
Φ(G,N, g1, . . . , gt) = {(n1, . . . , nt) ∈ N t | ⟨g1n1, . . . , gtnt⟩ = G}.
Set:
ΨA(H,N, g1, . . . , gt) = {(n1, . . . , nt)∈N t | ⟨g1n1, . . . , gtnt⟩=Ha for some a ∈ A};
ψA(H,N, t) = |ΨA(H,N, g1, . . . , gt)|.
Notice that ωA(H,N, t) ̸= 0 if and only if [H]A ∈ CA(G,N). If (n1, . . . , nt) ∈
ΩA(H,N, t), then ⟨g1n1, . . . , gtnt⟩ ≤ Ha for some a ∈ A, and ⟨g1n1, . . . , gtnt⟩ = K
for some K ≤ G with [K]A ≤ [H]A. Thus∑
[K]A≤[H]A
ψA(K,N, t) = ωA(H,N, t)
and therefore, by the Möbius inversion formula∑
[H]∈CA(G,N)
µA(H,G)ωA(H,N, t) = ψA(G,N, t) = ϕ(G,N, t)
F. Dalla Volta and A. Lucchini: The A-Möbius function of a finite group 475
4 Another application of Möbius inversion formula
Denote by Φ∗(G, t) the set of t-tuples (H1, . . . ,Ht) of subgroups of G such that G =
⟨H1, . . . ,Ht⟩ and by ϕ∗(G, t) the cardinality of this set. For any H ∈ CA(G) and any
positive integer t, let
1. ΣA(H, t) = {(H1, . . . ,Ht) | ⟨H1, . . . ,Ht⟩ ≤ Ha for some a ∈ A};
2. σA(H, t) = |ΣA(H, t)|;
3. ΓA(H, t) = {(H1, . . . ,Ht) | ⟨H1, . . . ,Ht⟩ = Ha for some a ∈ A};
4. γA(H, t) = |ΓA(H, t)|.
Theorem 4.1.
ϕ∗(G, t) =
∑
[H]∈CA(G)
µA(H,G)σA(H, t).
Proof. If (H1, . . . ,Ht) ∈ ΣA(H, t), then ⟨H1, . . . ,Ht⟩ = K for some K ≤ G with
[K]A ≤ [H]A. Thus ∑
[K]≤A[H]
γA(K, t) = σA(H, t)
and therefore, by the Möbius inversion formula,∑
[H]∈CA(G)
µA(H,G)σA(H, t) = γA(G, t) = ϕ
∗(G, t).
In the particular case whenA = 1, σA(H, t) = σ(H)t, denoting with σ(H) the number
of subgroups of H. So we obtain the following corollary:
Corollary 4.2.
ϕ∗(G, t) =
∑
H≤G
µ(H,G)σ(H)t.
Clearly Σ∗(G, t) = {G}, so ϕ∗(G, 1) = 1 and therefore it follows:
Corollary 4.3.
1 =
∑
H∈HA
µA(H,G)σA(H, 1).
In particular:
Corollary 4.4.
1 =
∑
H≤G
µ(H,G)σ(H).
For example, if G = A5 then the subgroups of G with µ(H,G) ̸= 0 are listed in the
following table (where κ(H,G) denote the numbers of conjugate of H in G).
476 Ars Math. Contemp. 23 (2023) #P3.08 / 467–480
µ(H,G) κ(H,G) σ(H)
A5 1 1 59
A4 -1 5 10
S3 -1 10 6
D5 -1 6 8
⟨(1, 2, 3)⟩ 2 10 2
⟨(1, 2)(3, 4)⟩ 4 15 2
1 -60 1 1
According with Corollary 4.4, 1 = 59− 5 · 10− 10 · 6− 6 · 8 + 2 · 10 · 2 + 4 · 15 · 2− 60.
For a finite group G, denote by P (G, t) and P ∗(G, t) the probability of generating
G with, respectively, t elements or t subgroups. It can be easily seen that P (G, t) =
P (G/Frat(G), t), but in general P ∗(G, t) ̸= P ∗(G/Frat(G), t). For example, if G ∼=
Cpa , then G and H ∼= Cpa−1 are the unique subgroups of G with non trivial Möbius
number and therefore
P (G, t) =
|G|t − |H|t
|G|t
= 1− 1
pt
,
P ∗(G, t) =
σ(G)t − σ(H)t
σ(G)t
= 1− a
t
(a+ 1)t
.
So P (G, t) is independent of a, while P ∗(G, t) tends to 0 when a tends to infinity.
5 The (µ, λ)-property
Proof of Thereom 1.5. Working by induction on the order of G, it suffices to prove the
statement in the particular case when N is an abelian minimal normal subgroup of G. Let
H be a subgroup of G. If N ≤ H, then
µ(H,G) = µ(H/N,G/N) = λ(H/N,G/N)|NG′N/N (H/N) : H/N ∩G′N/N |
= λ(H,G)|NG′N (H) : H ∩G′N | = λ(H,G)|NNG′(H) : N(H ∩G′)|
= λ(H,G)
|NG′(H) : H ∩G′|
|N ∩NG′(H) : N ∩H ∩G′|
= λ(H,G)
|NG′(H) : H ∩G′|
|N ∩G′ : N ∩G′|
= λ(H,G)|NG′(H) : H ∩G′|.
So we may assume N ̸≤ H. If H is not an intersection of maximal subgroups of G, then
µ(G,H) = λ(G,H) = 0. So we may assume H = M1 ∩ · · · ∩Mt where M1, . . . ,Mt
are maximal subgroups of G. Since N is not contained in H, then N is not contained in
Mi for some i, but then Mi is a complement of N in G containing H and N ∩H = 1. If
g ∈ NG(HN), then g = xn with x ∈ Mi and n ∈ N. In particular Hx ≤ HN ∩Mi =
H(N ∩Mi) = H , so NG(HN) = NG(H)N. By Lemma 2.5, we have
µ(H,G)
λ(H,G)
=
µ(HN,G)
λ(HN,G)
κ
δ
= |NG′N (HN) : HN∩G′N |
κ
δ
= |NNG′(H) : HN∩G′N |
κ
δ
where k is the number of complements of N in G containing H and δ is the number of
conjugacy classes of these complements. First assume that N ≤ Z(G). Then κ = δ,
F. Dalla Volta and A. Lucchini: The A-Möbius function of a finite group 477
G′ =M ′i ≤Mi, N ∩G′ = 1 and
µ(H,G)
λ(H,G)
= |NNG′(H) : HN ∩G′N |
κ
δ
= |NNG′(H) : HN ∩G′N |
= |NNG′(H) : N(H ∩G′)| = |NG′(H) : H ∩G′|.
Finally assume N ̸≤ Z(G). Then N ≤ G′, κ/δ = |NN (H)| and
µ(H,G)
λ(H,G)
= |NNG′(H) : HN ∩G′N |
κ
δ
= |NNG′(H) : N(H ∩G′)||NN (H)|
=
|N ||NG′(H)|
|NN (H)|
|NN (H)|
|N ||H ∩G′|
= |NG′(H) : H ∩G′|.
Proof of Corollary 1.6. Suppose thatG has minimal order with respect to the property that
G does not satisfy the (µ, λ) property. By the previous proposition, G contains no abelian
minimal normal subgroup and therefore soc(G) = S1 × · · · × St is a direct product of
nonabelian finite simple groups. If |G| ≤ |PSU(3, 3)| = 6048, then either t = 1 or
G = soc(G) = A5 × A5. So it suffices to check that A5 × A5 and any almost simple
group of order at most 6048 satisfies the (µ, λ) property. Recall that the table of marks
of a finite group G is a matrix whose rows and columns are labelled by the conjugacy
classes of subgroups of G and where for two subgroups A and B the (A,B)-entry is the
number of fixed points of B in the transitive action of G on the cosets of A in G. Since, for
every H ≤ G, λ(H,G) and µ(H,G) can be computed from the table of marks of G (see
[7, Proposition 1]), our proof can be easily completed using the library of table of marks
available in GAP [4].
We may use Theorem 3.1 to deduce some consequences of the (µ, λ)-property.
Theorem 5.1. Suppose that a finite group G satisfies the (µ, λ)-property. Then
∑
[H]∈C(G)
λ(H,G)
(
|H|t−1|G||G′H|
|G′NG(H)|
− ω(H, t)
)
= 0. (5.1)
Proof. By Theorem 3.1,∑
H∈C(G)
λ(H,G)ω(H, t) = ϕ(G, t) =
∑
H≤G
µ(H,G)|H|t
=
∑
H∈C(G)
µ(H,G)|G : NG(H)||Ht|
=
∑
H∈C(G)
λ(H,G)|NG′(H) : G′ ∩H||G : NG(H)||Ht|
=
∑
H∈C(G)
λ(H,G)
|H|t|G||NG′(H)|
|G′ ∩H||NG(H)|
=
∑
H∈C(G)
λ(H,G)
|H|t−1|G||G′H|
|G′NG(H)|
.
478 Ars Math. Contemp. 23 (2023) #P3.08 / 467–480
A natural question is whether (5.1) is also a sufficient condition for the (µ, λ)-property.
For any H ≤ G, set µ∗(H,G) = |NG′(H) : G′ ∩ H|λ(H,G). The validity of (5.1) is
equivalent to∑
H∈C(G)
λ(H,G)ω(H, t)−
∑
H∈C(G)
µ∗(H,G)|H|t|G : NG(H)| = 0.
In any case we must have∑
H∈C(G)
λ(H,G)ω(H, t)−
∑
H∈C(G)
µ(H,G)|H|t|G : NG(H)| = 0.
So (5.1) is equivalent to
∑
H∈C(G)
(µ(H,G)− µ∗(H,G))|H|t
|NG(H)|
= 0.
Let T = {[H] ∈ C(G) | µ(H,G) ̸= µ∗(H,G)}. Then (5.1) is true if and only if
∑
[H]∈T
(µ(H,G)− µ∗(H,G))|H|t
|NG(H)|
= 0. (5.2)
For example, if G = PSU(3, 3), then T consists of four conjugacy classes of sub-
groups and the corresponding values are given by the following table:
µ(H,G) µ∗(H,G) |H| |NG(H)|
-48 0 2 96
3 0 6 18
0 -4 8 32
1 2 24 24
In this case (5.2) is equivalent to
2t−1 − 6t−1 − 8t−1 + 24t−1 = 0
which is true only if t = 1.
For any positive integer n let
τ(n) =
∑
H∈T ,|H|=n
µ(H, g)− µ∗(H,G)
|NG(H)|
.
Proposition 5.2. A finite group G satisfies (5.1) for every positive integer t if and only if
τ(n) = 0 for any ∈ N.
Question 5.3. Does τ(n) = 0 for all n ∈ N imply µ∗(H,G) = µ(H,G) for all H ≤ G?
F. Dalla Volta and A. Lucchini: The A-Möbius function of a finite group 479
For any H ≤ G, consider
α(H, t) =
|H|t−1|G||G′H|
|G′NG(H)|
, β(H, t) = α(H, t)− ω(H, t).
Let C∗(G) = {[H] ∈ C(H) | [H] < [G] and λ(H,G) ̸= 0}. If G satisfies the (λ, µ)-
property, then for any t ∈ N, the vector
βt(G) = (β(H, t))[H]∈C∗(G)
is an integer solution of the linear equation∑
[H]∈C∗(G)
λ(H,G)xH = 0. (5.3)
One could investigate about the dimension of the vector space generated by the vectors
βt(G), t ∈ N. For example, if G = A5, then we may order the elements of C∗(G) so that
H1 = A4, H2 = S3, H3 = D5, H4 = ⟨(1, 2, 3)⟩, H5 = ⟨(1, 2)(3, 4)⟩, H6 = 1. Then (5.3)
can be written in the form∑
[H]∈C∗(G)
λ(H,G)xH = −xH1 − xH2 − xH3 + xH4 + 2xH5 − xH6
and
β1(G) = (24, 24, 20, 39, 44, 59),
β2(G) = (84, 54, 50, 99, 74, 59),
β3(G) = (264, 114, 110, 279, 134, 59),
β4(G) = (804, 234, 230, 819, 254, 59),
β5(G) = (2424, 474, 470, 2439, 494, 59),
β6(G) = (7284, 954, 950, 7299, 974, 59).
The first three vectors β1(G), β2(G), β3(G) are linearly independent, while β4(G), β5(G)
and β6(G) can be obtained as linear combinations of β1(G), β2(G), β3(G).
The situation is completely different when G = S3. We may order the elements of
C∗(G) so that H1 = ⟨(1, 2, 3)⟩, H2 = ⟨(1, 2)⟩, H3 = 1. The equation (5.3) has in this
case the form xH1 + xH2 − xH3 = 0 and βt(G) = (0, 2, 2) independently on the choice
of t.
Some properties of the vectors βt(G) are described in the following propositions.
Proposition 5.4. If H ∈ C∗(G), then β(H, t) ≥ 0 with equality if and only if G′ ≤ H. In
particular βt(G) is a non-negative vector and βt(G) = 0 if and only if G is nilpotent.
Proof. Notice that ω(H, t) ≤ |G : NG(H)|(|H|t − 1) + 1. So
β(H, t) ≥ |H|
t−1|G||G′H|
|G′NG(H)|
− |G : NG(H)|(|H|t − 1)− 1
= |H|t|G : NG(H)|
|G′ ∩NG(H)|
|G′ ∩H|
− |G : NG(H)|(|H|t − 1)− 1 ≥ 0
with equality if and only if H ≥ G′.
480 Ars Math. Contemp. 23 (2023) #P3.08 / 467–480
Proposition 5.5. The vector βt(G) is independent on the choice of t if and only if G is a
nilpotent group or a primitive Frobenius group, with cyclic Frobenius complement.
Proof. By the previous proposition, if G is nilpotent then βt(G) is the zero vector for any
t ∈ N, so we may assume that G is not nilpotent. Assume that βt(G) is independent on the
choice of t. Let H be a maximal non-normal subgroup of G. Then α(H, t) = |H|t · u with
u = |G : H|. Let H1, . . . ,Hu be the conjugates of H in G. For any J ⊆ {1, . . . , u}, let
αJ = | ∩j∈J Hj |. Then
β(H, t) =
∑
J ̸={1,...,u}
(−1)|J|+1|αJ |t.
We must have αJ = 1 for every choice of J, otherwise limt→∞ β(H, t) = ∞. Hence H
is a Frobenius complement and, since H is a maximal subgroup, the Frobenius kernel V
is an irreducible H-module. Since β(V, t) = |V |t(|H ′| − 1) does not depends on t, H
must be abelian, and consequently cyclic. So if βt(G) is independent of the choice of t,
then G is a primitive Frobenius group with a cyclic Frobenius complement. Conversely
assume G = V ⋊H, where H is cyclic and V and irreducible H-module. If X ∈ C∗(G),
then λ(X,G) ̸= 0, so X is an intersection of maximal subgroups of G and therefore either
V = G′ ≤ X, or X is conjugate to a subgroup of H. In the first case β(H, t) = 0.
Assume X = Kv for some K ≤ H and v ∈ V. Then β(H, t) = |K|t|V | − ω(K, t) =
|K|t|V | − (|V |(|K|t − 1) + 1) = |V | − 1.
ORCID iDs
Francesca Dalla Volta https://orcid.org/0000-0001-7368-4050
Andrea Lucchini https://orcid.org/0000-0002-2134-4991
References
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P3.09 / 481–500
https://doi.org/10.26493/1855-3974.2902.f01
(Also available at http://amc-journal.eu)
On adjacency and Laplacian cospectral
switching non-isomorphic signed graphs*
Tahir Shamsher † , Shariefuddin Pirzada ‡
Department of Mathematics, University of Kashmir, Srinagar, Kashmir, India
Mushtaq A. Bhat
Department of Mathematics, National Institute of Technology, Srinagar, India
Received 6 June 2022, accepted 2 January 2023, published online 30 January 2023
Abstract
Let Γ = (G, σ) be a signed graph, where σ is the sign function on the edges of G. In
this paper, we use the operation of partial transpose to obtain switching non-isomorphic
Laplacian cospectral signed graphs. We will introduce a new operation on signed graphs.
This operation will establish a relationship between the adjacency spectrum of one signed
graph with the Laplacian spectrum of another signed graph. As an application, this new
operation will be utilized to construct several pairs of switching non-isomorphic cospectral
signed graphs. Finally, we construct integral signed graphs.
Keywords: Signed graph, partial transpose, cospectral signed graphs, Laplacian cospectral signed
graphs, equienergetic signed graphs, integral signed graph.
Math. Subj. Class. (2020): 05C22, 05C50
1 Introduction
LetG = (V (G), E(G)) be a simple connected graph with vertex set V (G) = {v1, v2, . . . ,
vn} and edge set E(G) = {e1, e2, . . . , em}. A signed graph is defined to be a pair Γ =
(G, σ), with G = (V (G), E(G)) as the underlying graph and σ : E(G) → {−1, 1} as the
signing function. In this manuscript, bold lines denote positive edges, and dashed lines
*The authors are grateful to the referee for the useful comments which improved the presentation of the paper.
†The research of Tahir Shamsher is supported by SRF financial assistance by Council of Scientific and Indus-
trial Research (CSIR), New Delhi, India.
‡Corresponding author. The research of S. Pirzada is supported by SERB-DST research project number
CRG/2020/000109.
E-mail addresses: tahir.maths.uok@gmail.com (Tahir Shamsher), pirzadasd@kashmiruniversity.ac.in
(Shariefuddin Pirzada), mushtaqab@nitsri.net (Mushtaq A. Bhat)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
482 Ars Math. Contemp. 23 (2023) #P3.09 / 481–500
denote negative edges. Signed graphs are a generalization of graphs, since they are signed
graphs with each edge positive. The sign of a cycle in a signed graph is defined to be the
product of the signs of its edges. A signed cycle is said to be positive (resp. negative) if its
sign is positive (resp. negative). A signed graph is said to be balanced if none of its cycles
is negative, otherwise unbalanced.
In a signed graph Γ = (G, σ), the degree of a vertex v is the same as its degree in
the underlying graph G (denoted by dv(G)). For a signed graph Γ with vertex set V (G),
let X ⊂ V (G) be a nonempty set. Let ΓX denote the signed graph obtained from Γ
by reversing signs of edges between X and V (G) − X . Then, we say ΓX is switching
equivalent to Γ. Here, we note that the switching is an equivalence relation and preserves
the eigenvalues of the adjacency and the Laplacian matrix including their multiplicities. A
switching class is represented by a single signed graph.
The adjacency matrix of a signed graph Γ with vertex set {v1, v2, . . . , vn}, is the n×n
matrix A(Γ) = (aij), where
aij =
{
σ(vi, vj), if there is an edge from vi to vj ,
0, otherwise.
For a graph G, the Laplacian matrix is L(G) = D(G) − A(G) and signless Laplacian
matrix is Q(G) = D(G) + A(G), where A(G) and D(G) are respectively the adjacency
matrix and the diagonal matrix of vertex degrees of G. The Laplacian matrix of Γ is
L(Γ) = L(G, σ) = D(G) − A(Γ). Note that L(G,+) = L(G) and L(G,−) = Q(G).
The characteristic polynomial |xI − A(Γ)| and eigenvalues of the adjacency matrix A(Γ)
of Γ are denoted by ϕΓ(x) and λ1, λ2, . . . , λn, respectively. The characteristic polynomial
|xI − L(Γ)| and eigenvalues of the Laplacian matrix L(Γ) of Γ are denoted by ψΓ(x)
and µ1, µ2, . . . , µn, respectively. For a graph G (resp. signed graph Γ), eigenvalues of its
adjacency matrix and Laplacian matrix are called adjacency and Laplacian eigenvalues of
G (resp. Γ). Clearly, A(Γ) and L(Γ) are real symmetric and so all their eigenvalues are
real. Let the signed graph Γ of order n has distinct eigenvalues λ1, λ2, . . . , λk and let their
respective multiplicities be m1,m2, . . . ,mk. The adjacency spectrum of Γ is written as
Spec(Γ) = {λ(m1)1 , λ
(m2)
2 , . . . , λ
(mk)
k }. A signed graph is said to be an integral signed
graph if its adjacency spectrum consists of integers only.
Given a graph G, its subdivision graph S (G) is obtained from G by replacing each
of its edge by a path of length 2, or, equivalently, by inserting an additional vertex into
each edge of G. If two signed graphs have the same adjacency spectrum (resp. Lapla-
cian spectrum), they are said to be cospectral (resp. Laplacian cospectral); otherwise, they
are noncospectral (resp. Laplacian noncospectral). Any two switching isomorphic signed
graphs are cospectral (resp. Laplacian cospectral). A signed graph is said to be determined
by its adjacency spectrum if cospectral signed graphs are switching isomorphic. It is well-
known that in general the adjacency spectrum does not determine the signed graph and
this problem has attracted to identify, if any, switching non-isomorphic cospectral signed
graphs for a given class of signed graphs. For open problems in signed graphs, we refer to
[2].
The energy of a graph G is the sum of the absolute values of its adjacency eigenval-
ues. This concept was extended to signed graphs by Germina, Hameed and Zaslavsky
[9]. The energy of a signed graph Γ with eigenvalues x1, x2, . . . , xn is defined as E(Γ) =∑n
j=1 |xj |. Two signed graphs of same order are said to be equienergetic if they have the
same energy.
T. Shamsher et al.: On adjacency and Laplacian cospectral switching non-isomorphic . . . 483
Harary [12] pioneered the use of signed graphs in connection with the study of social
balance theory. Signed graphs have been intensively explored in a variety of fields such as
group theory, topological graph theory and classical root system. The reader is referred to
[17] for a complete bibliography on signed graphs.
The rest of the paper is organized as follows. In Section 2, we present some preliminary
results which will be used in the sequel. In Section 3, we define the concept of partial trans-
pose in signed graphs and use it to obtain switching non-isomorphic Laplacian cospectral
signed graphs. In Section 4, we introduce a new operation on signed graphs and this will
be utilized to construct switching non-isomorphic cospectral signed graphs, noncospectral
equienergetic signed graphs and integral signed graphs.
2 Preliminaries
In this section, we recall some previously established results which will be required in the
subsequent sections.
Definition 2.1 ([6]). Let P = (pij) ∈ Mm×n(R) and Q ∈ Mp×q(R). The Kronecker
product of P and Q, denoted by P ⊗Q, is defined as
P ⊗Q =

p11Q p12Q . . . p1nQ
p21Q p22Q . . . p2nQ
...
...
. . .
...
pm1Q pm2Q . . . pmnQ
 .
Lemma 2.2 ([6]). Let P , Q ∈ Mn(R) be two square matrices of order n. Let λ be an
eigenvalue of matrix P with corresponding eigenvector x and µ be an eigenvalue of matrix
Qwith corresponding eigenvector y. Then λµ is an eigenvalue of P⊗Qwith corresponding
eigenvector x⊗ y.
The Cartesian product (or sum) of two signed graphs Γ1 = (V (G1), E(G1), σ1) and
Γ2 = (V (G2), E(G2), σ2), denoted by Γ1×Γ2, is the signed graph (V (G1)×V (G2), E, σ),
where the edge set is that of the Cartesian product of underlying unsigned graphs and the
sign function is defined by
σ((ui, vj), (uk, vl)) =
{
σ1(ui, uk), if j = l,
σ2(vj , vl), if i = k.
The Kronecker product (or conjunction) of two signed graphs Γ1 = (V (G1), E(G1), σ1)
and Γ2 = (V (G2), E(G2), σ2), denoted by Γ1 ⊗ Γ2, is the signed graph (V (G1) ×
V (G2), E, σ), where the edge set is that of the Kronecker product of underlying unsigned
graphs and the sign function is defined by σ((ui, vj), (uk, vl)) = σ1(ui, uk)σ2(vj , vl).
Lemma 2.3 ([9]). Let Γ1 and Γ2 be two signed graphs with respective eigenvalues
x1, x2, . . . , xn1 and y1, y2, . . . , yn2 . Then
(i) the eigenvalues of Γ1 ×Γ2 are xi + yj , for all i = 1, 2, . . . , n1 and j = 1, 2, . . . , n2,
(ii) the eigenvalues of Γ1 ⊗ Γ2 are xiyj , for all i = 1, 2, . . . , n1 and j = 1, 2, . . . , n2.
484 Ars Math. Contemp. 23 (2023) #P3.09 / 481–500
Lemma 2.4 ([4]). Let Γ be an unbalanced signed graph with at least one edge, whose
spectrum is symmetric about the origin, having eigenvalues ξ1, ξ2, . . . , ξn. Then Γ×K2 and
Γ⊗K2, whereK2 is a complete signed graph on 2 vertices, are unbalanced, noncospectral
and equienergetic if and only if |ξj | ≥ 1, for all j = 1, 2, . . . , n.
Lemma 2.5 ([14]). Let P (•) be a given polynomial. If µ is an eigenvalue of A ∈ Mn,
while y is an associated eigenvector, then P (µ) is an eigenvalue of the matrix P (A) and y
is an eigenvector associated with P (µ).
Lemma 2.6 ([4]). Let Γ be a signed graph of order n. Then the following statements are
equivalent.
(i) The spectrum of Γ is symmetric about the origin,
(ii) ϕΓ(x) = xn +
∑⌊n2 ⌋
k=1(−1)kb2kxn−2k, where b2k are non negative integers for all
k = 1, 2, . . . , ⌊n2 ⌋,
(iii) Γ and −Γ are cospectral, where −Γ is the signed graph obtained by negating sign
of each edge of Γ.
Lemma 2.7 ([16]). For infinitely many n, there exists a family of 2k pairwise nonisomor-
phic Laplacian integral, Laplacian cospectral graphs on n vertices, where k > n(2log2(n)) .
3 Constructing Laplacian cospectral non-isomorphic signed graphs
Dutta [8] constructed large families of non-isomorphic signless Laplacian cospectral graphs
using partial transpose on graphs. In this section, we define the partial transpose in signed
graphs. Let Γ = (G, σ) be a signed graph on 2n vertices with vertex set V (G) =
V1 ∪ V2, such that V1 ∩ V2 = ∅, and V1 = {u1, u2, . . . un} , V2 = {v1, v2, . . . vn} .
We denote by ⟨V1⟩Γ and ⟨V2⟩Γ as the induced signed subgraphs of Γ formed by V1 and
V2 respectively. The spanning signed subgraph of Γ consisting of the signed edge set
{(ui, vj) ∈ E(Γ) : ui ∈ V1, vj ∈ V2} is denoted by ⟨V1, V2⟩Γ. Consider the set of
signed edges ̂E (⟨V1, V2⟩Γ) = {(uj , vi) : (ui, vj) ∈ E (⟨V1, V2⟩Γ)}. The definition of
̂E (⟨V1, V2⟩Γ) suggests that given any signed edge (ui, vj) ∈ E (⟨V1, V2⟩Γ) there is a
unique signed edge (uj , vi) ∈ ̂E (⟨V1, V2⟩Γ) with the same sign as the sign of edge (ui, vj)
in ⟨V1, V2⟩Γ.
The partial transpose of a signed graph Γ, denoted by Γτ , is defined as Γτ = Γ −
E (⟨V1, V2⟩Γ) + ̂E (⟨V1, V2⟩Γ). Note that, subtracting E (⟨V1, V2⟩Γ) indicates to remove
all the existing signed edges in Γ of the form (ui, vj) ∈ E (⟨V1, V2⟩Γ). Then we include
the signed edges (uj , vi) ∈ ̂E (⟨V1, V2⟩Γ) to construct Γτ . If i = j, then the edge (ui, vi)
will be removed and added again, that is the edge (ui, vi) is unaltered under partial trans-
pose. Therefore, partial transpose of a signed graph Γ is an operation on the edge set which
replaces the signed edge (ui, vj) with the sign σ = ±1, with the corresponding signed edge
(uj , vi) with the same sign σ.
Example 3.1. Consider the signed graphs Γ1 and Γτ1 as shown in Figure 1. Here, we have
V1 = {u1, u2, u3}, V2 = {v1, v2, v3} and E
(
⟨V1, V2⟩Γ1
)
= {(u1, v1), (u1, v3)}. Thus,
̂E
(
⟨V1, V2⟩Γ1
)
= {(u1, v1), (u3, v1)}. Here, we replace the signed edge (u1, v3) with the
signed edge (u3, v1).
T. Shamsher et al.: On adjacency and Laplacian cospectral switching non-isomorphic . . . 485
Remark 3.2. The partial transpose of a signed graph is labelling dependent. Therefore,
switching isomorphic signed graphs may have switching non-isomorphic partial trans-
poses, depending on the labellings. The partial transpose keeps ⟨V1⟩ and ⟨V2⟩ unaltered.
The total number of vertices and edges remains the same.
A cycle Cσl (v1, v2, . . . , vl, v1) in a signed graph Γ = (G, σ) is a finite sequence of
distinct vertices such that (vi, vi+1) ∈ E(Γ) for all i = 1, 2, . . . , l−1 and (vl, v1) ∈ E(Γ).
We denote the negative edges in the signed cycle Cσl (v1, v2, . . . , vl, v1) by putting the bar
over the corresponding adjacent vertices. For example, the cycle Cσ4 (v1, v2, v3, v4, v1) on
four vertices such that the only edge (v1, v2) ∈ E(Γ) has negative sign will be denoted by
C−4 (v1, v2, v3, v4, v1). Similarly if only two consecutive edges (v1, v2), (v2, v3) ∈ E(Γ)
have negative signs, then the cycle Cσ4 (v1, v2, v3, v4, v1) will be denoted by C
+
4 (v1, v2, v3,
v4, v1). In a signed graph Γ, a signed TU -subgraph H is a signed subgraph whose com-
ponents are trees or unbalanced unicyclic graphs, namely the unique cycle containing
an odd number of negative edges. Thus, if H is a signed TU -subgraph, then H =
T1∪T2∪· · ·∪Tp∪U1∪U2∪· · ·∪Uq , where T ′is are trees and U ′is are unbalanced unicyclic
graphs. The weight of the signed TU -subgraph H is defined as w(H) = 4q
∏p
i=1 |Ti|,
where |Ti| is the number of vertices in the tree Ti. Note that we define
∏p
i=1 |Ti| = 1 when
p = 0. The relation between the coefficients of the Laplacian characteristic polynomial
with the TU -subgraphs of a signed graph can be seen in [[3],Theorem 3.9]. If Γ is a signed
graph with Laplacian characteristic polynomial ψ(Γ, x) = xn+a1xn−1+· · ·+an−1x+an,
then its coefficients are given by
ai = (−1)i
∑
H∈Hi(Γ)
w(H) (i = 1, 2, . . . , n), (3.1)
where Hi(Γ) denotes the set of signed TU -subgraphs of Γ containing i edges. Two sets of
signed TU -subgraphs Hi(Γ) and Hi (Γ′) are comparable if∑
H∈Hi(Γ)
w(H) =
∑
H∈Hi(Γ′)
w(H).
Now, Equation (3.1) suggests that Γ and Γ′ are Laplacian cospectral if and only if the sets of
their signed TU -subgraphs are comparable for all i = 1, 2, . . .m, where m is the number
of edges in the signed graph Γ. We say two signed graphs Γ1 and Γ2 are comparable if
Hi(Γ1) and Hi(Γ2) are comparable for all i. As an example, two signed paths with equal
number of vertices are comparable.
Example 3.3. Consider the signed graphs Γ1 and Γτ1 as shown in Figure 1. We observe that
Γ1 contains two cycles C+3 (u1, u2, u3, u1) and C
−
4 (u1, v1, v2, v3, u1). The partial trans-
pose Γτ1 of Γ1 is obtained by replacing the signed edge (u1, v3) with (u3, v1). Clearly, Γ
τ
1
contains three cycles C+3 (u1, u2, u3, u1), C
−
4 (u1, v1, u3, u2, u1) and C
−
3 (u1, v1, u3, u1).
The cycle C+3 (u1, u2, u3, u1) remains invariant under partial transpose on Γ1. If the cycle
has an odd number of negative edges, then it contributes an unbalanced unicyclic graph in
the formation of signed TU -subgraphs. Therefore, the balanced cycle C+3 (u1, u2, u3, u1)
does not contribute in ψΓ1(x) and ψΓτ1 (x).
The unbalanced cycleC−4 (u1, v1, v2, v3, u1) in Γ1 is replaced byC
−
4 (u1, v1, u3, u2, u1)
in Γτ1 . Therefore, signed TU -subgraphs whose components contain cycles C
−
4 (u1, v1, v2,
v3, u1) and C−4 (u1, v1, u3, u2, u1) in Γ1 and Γ
τ
1 , respectively, have equal contribution in
ψΓ1(x) and ψΓτ1 (x).
486 Ars Math. Contemp. 23 (2023) #P3.09 / 481–500
Figure 1: Signed graph Γ1 and its partial transpose Γτ1 .
The signed edges (u1, v1), (u1, v3) and (u1, u3) induce star signed graph K1,3 in Γ1.
It is replaced by an unbalanced unicyclic TU -subgraph C−3 (u1, v1, u3, u1) in Γ
τ
1 . The
signed graphs K1,3 and C−3 (u1, v1, u3, u1) have the same weights equal to 4 as a com-
ponents to the signed TU -subgraphs. Therefore, the signed TU -subgraphs formed by
K1,3 and C−3 (u1, v1, u3, u1) in Γ1 and Γ
τ
1 , respectively, have equal contribution in ψΓ1(x)
and ψΓτ1 (x). Moreover, the role of signed TU -subgraphs which contain the signed edges
(u1, v1), (u1, v3) and (u1, u3) in Γ1 are replaced by the signed TU -subgraphs which con-
tain the signed cycle C−3 (u1, v1, u3, u1) in Γ
τ
1 . Thus, all the signed TU -subgraphs of Γ1
and Γτ1 are comparable. Hence they have same Laplacian characteristic polynomials, which
can be calculated by Equation (3.1) and is given as
ψΓ1(x) = ψΓτ1 (x) = x
6 − 14x5 + 73x4 − 176x3 + 196x2 − 88x+ 12.
In a signed graph Γ = (G, σ) with wi, wj ∈ V (Γ), if the signed edge (wi, wj) is
added, then the resultant signed graph is denoted by Γ′ = Γ + {(wi, wj)}. Similarly,
Γ′ = Γ − {(wi, wj)} denotes the signed graph obtained by removing an edge (wi, wj).
Whether the added/removed edge (wi, wj) is positive or negative, we denote a negative
edge by (wi, wj), and a positive edge without a bar over the edge (wi, wj).
Theorem 3.4. Let the signed subgraphs ⟨V1⟩Γ and ⟨V2⟩Γ of the signed graph Γ be two
paths on n vertices with each edge being positive. If ⟨V1, V2⟩Γ is an empty signed graph,
then
(i) the signed graph Γ1 = Γ+
{
(u1, un) , (u1, v1), (u1, vn)
}
is switching non-isomorphic
and Laplacian cospectral to its partial transpose Γτ1 .
(ii) the signed graphs Γ2 = Γ1 −
{
(un−1, un), (u1, v1)
}
+
{
(un−1, un), (u1, v1)
}
and
Γ3 = Γ
τ
1 −{(un−1, un)}+
{
(un−1, un)
}
are switching non-isomorphic and Lapla-
cian cospectral.
Proof. (i) The cycles in Γ1 generated by additional three edges and their incidence with
existing edges of Γ are C+n (u1, u2, u3, . . . , un, u1) and C
−
n+1 (u1, v1, v2, . . . , vn, u1). The
signed spanning subgraph ⟨V1, V2⟩Γ contains only two signed edges which are (u1, v1)
and (u1, vn) . Partial transpose replaces (u1, vn) with (un, v1) . Clearly, Γτ1 contains three
cycles C+n (u1, u2, u3, . . . , un, u1), C
−
n+1 (u1, v1, un, . . . , u2, u1) and C
−
3 (u1, v1, un, u1).
T. Shamsher et al.: On adjacency and Laplacian cospectral switching non-isomorphic . . . 487
Figure 2: Signed graphs Γ1, Γτ1 , Γ2 and Γ3.
The cycle C+n (u1, u2, u3, . . . , un, u1) remains invariant under partial transpose on Γ1. Re-
moving an edge from a cycle generates a tree. If the cycle has an odd number of negative
edges, then it contributes an unbalanced unicyclic graph in the formation of signed TU -
subgraphs. Therefore, the balanced cycle C+n (u1, u2, u3, . . . , un, u1) does not contribute
in ψΓ1(x) and ψΓτ1 (x).
The unbalanced cycle C−n+1 (u1, v1, v2, . . . , vn, u1) in Γ1 is replaced by C
−
n+1(u1, v1,
un, . . . , u2, u1) in Γτ1 . Therefore, signed TU -subgraphs whose components contain cycles
C−n+1 (u1, v1, v2, . . . , vn, u1) andC
−
n+1 (u1, v1, un, . . . , u2, u1) in Γ1 and Γ
τ
1 , respectively,
have equal contribution in ψΓ1(x) and ψΓτ1 (x).
The signed edges (u1, v1), (u1, vn) and (u1, un) induce star signed graph K1,3 in Γ1.
It is replaced by an unbalanced unicyclic TU -subgraph C−3 (u1, v1, un, u1) in Γ
τ
1 . They
have equal contribution in ψΓ1(x) and ψΓτ1 (x) which is seen in Example 3.3. Moreover,
the role of signed TU -subgraphs which contain the signed edges (u1, v1), (u1, vn) and
(u1, un) in Γ1 are replaced by the signed TU -subgraphs which contain the signed cycle
C−3 (u1, v1, un, u1) in Γ
τ
1 . Therefore, all the signed TU -subgraphs of Γ1 and Γ
τ
1 are com-
parable. Thus, they have the same Laplacian characteristic polynomials, which proves the
result in this case.
(ii) If a signed graph is switching equivalent to a signed graph whose each edge is nega-
tive, then its Laplacian matrix coincides with the signless Laplacian matrix of its underlying
graph. If n is odd, then the result follows by Corollary 2 of [8]. For even n, we observe that
Γ2 contains two cycles C−n (u1, u2, . . . , un−1, un, u1) and C
+
n+1 (u1, v1, . . . , vn, u1). The
underlying graph of Γ3 is the partial transpose of the underlying graph of Γ2. The signed
graph Γ3 contains three cycles C−n (u1, u2, . . . , un−1, un, u1), C
+
n+1(u1, v1, un, un−1,
un−2, . . . , u2, u1) and C−3 (u1, v1, un, u1). The cycle C
−
n (u1, u2, . . . , un−1, un, u1) is
common in Γ2 and Γ3. Therefore, the signed TU−subgraphs formed by the unbalanced
cycle C−n (u1, u2, . . . , un−1, un, u1) in Γ2 and Γ3 contribute same in ψΓ2(x) and ψΓ3(x).
The balanced cycle C+n+1 (u1, v1, . . . , vn, u1) in Γ2 is replaced by a balanced cycle
C+n+1(u1, v1, un, un−1, un−2, . . . , u2, u1) in Γ3. Therefore, they do not contribute in
ψΓ2(x) and ψΓ3(x).
The signed edges (u1, v1), (u1, vn) and (u1, un) induce star signed graph K1,3 in Γ2.
It is replaced by an unbalanced unicyclic TU -subgraph C−3 (u1, v1, un, u1) in Γ3 and pro-
ceeding similarly as in (i), we get the result.
Example 3.5. Consider the signed graphs Γ1, Γτ1 , Γ2 and Γ3 as given in Figure 2. They
are constructed by using Theorem 3.4. Their Laplacian characteristic polynomials are re-
spectively given as.
ψΓ1(x) = ψΓτ1 (x) = x
8−18x7+131x6−498x5+1061x4−1256x3+764x2−200x+16
488 Ars Math. Contemp. 23 (2023) #P3.09 / 481–500
Figure 3: Signed graphs Γ1 , Γτ1 , Γ2, Γ3, Γ4, Γ5 and Γ
′.
and
ψΓ2(x) = ψΓ3(x) = x
8−18x7+131x6−498x5+1065x4−1284x3+824x2−240x+20.
Clearly, the signed graphs Γ1 and Γτ1 are switching non-isomorphic and Laplacian cospec-
tral. Also, Γ2 and Γ3 are switching non-isomorphic and Laplacian cospectral signed graphs.
The proof of the following result is similar to Theorem 1 of [8] but for the self contain-
ment of the paper, we include it here.
Theorem 3.6. Let the signed subgraphs ⟨V1⟩Γ and ⟨V2⟩Γ of Γ be two cycles on n ver-
tices with each edge being positive. Let ⟨V1, V2⟩Γ be an empty signed graph. Given
two non-adjacent vertices ui and uj with i < j, construct a new signed graph Γ1 =
Γ +
{
(ui, uj) , (ui, vi), (ui, vj)
}
. Then
(i) the signed graph Γ1 is switching non-isomorphic and Laplacian cospectral to its
partial transpose Γτ1 ,
(ii) the signed graphs Γ2 = Γ1 −
{
(un−1, un), (ui, vi), (uj−1, uj)
}
+{
(un−1, un), (u1, v1), (uj−1, uj)
}
and Γ3 = Γτ1 − {(un−1, un), (uj−1, uj)}
+
{
(un−1, un), (uj−1, uj)
}
are switching non-isomorphic and Laplacian cospec-
tral,
(iii) the signed graphs Γ4 = Γ1 −
{
(vn−1, vn), (ui, vi), (uj−1, uj)
}
+{
(vn−1, vn), (u1, v1), (uj−1, uj)
}
and Γ5 = Γτ1 − {(vn−1, vn), (uj−1, uj)} +{
(vn−1, vn), (uj−1, uj)
}
are switching non-isomorphic and Laplacian cospectral.
Proof. (i) The set of all cycles in Γ1 includes two cycles of Γ. They are denoted by γ1 and
γ2. The new cycles formed by additional three signed edges and their incidence with exist-
ing edges in Γ are γ3 = C+j−i+1 (ui, ui+1, ui+2, . . . , uj , ui), γ4 = C
+
n−(j−i)+1(u1, u2, . . . ,
ui, uj , uj+1, . . . , un, u1), γ5 = C−j−i+2 (ui, vi, vi+1, vi+2, . . . vj , ui), and γ6 =
T. Shamsher et al.: On adjacency and Laplacian cospectral switching non-isomorphic . . . 489
C−n−(j−i)+2(v1, v2, . . . , vi, ui, vj , vj+1, . . . , vn, v1). Note that, ⟨V1, V2⟩Γ1 contains only
two edges which are (ui, vi) and (ui, vj). Partial transpose replaces (ui, vj) with (uj , vi).
The cycles γ1, γ2, γ3 and γ4 remain invariant under partial transpose on Γ1. If the cycle has
an odd number of negative edges, then it contributes an unbalanced unicyclic graph in the
formation of signed TU -subgraphs. Therefore, signed TU -subgraphs formed by the cycles
γ1, γ2, γ3 and γ4 in Γ1 and Γτ1 have equal contribution in ψΓ1(x) and ψΓτ1 (x).
Now, the signed cycle γ5 in Γ1 is replaced by γ′5 = C
−
j−i+2(ui, vi, uj , uj−1, . . . ,
ui+1, ui) in Γτ1 . They have equal length and equal contribution in the characteristic co-
efficients. The cycle γ6 in Γ1 and its counterpart γ′6 = C
−
n−(j−i)+2(u1, u2, . . . , ui, vi,
uj , uj+1, . . . , uq, u1) in Γτ1 have equal lengths. If (vk, vk+1) ∈ γ6 ∩ ⟨V2⟩Γ1 in Γ1, then
(uk, uk+1) ∈ γ′6 ∩ ⟨V1⟩Γτ1 in Γ
τ
1 . A signed TU -subgraphs containing more than
n − (j − i) + 2 edges contains edges from γ1 in Γ1. The role of γ1 in Γ1 is replaced
by the edges of γ2 in Γτ1 . We have assumed that γ1 and γ2 have equal length. Therefore,
replacement of γ6 in Γτ1 does not make any difference in the characteristic coefficients.
The new edges (ui, uj) , (ui, vi) and (ui, vj) form a star K1,3 in Γ1. It is replaced by
a unicyclic signed TU -subgraph C−3 (ui, uj , vi, ui) in Γ
τ
1 . They have equal contribution in
ψΓ1(x) andψΓτ1 (x) which is seen in Example 3.3. Therefore, all the signed TU−subgraphs
of Γ1 and Γτ1 are comparable as well as they form equal characteristic polynomials. This
proves (i) The proof of (ii) and (iii) is similar to (i). Hence, the result follows.
Example 3.7. Consider the signed graphs Γ1, Γτ1 , Γ2, Γ3, Γ4 and Γ5 as shown in Figure 3.
Here n = 4, i = 1 and j = 3. The signed graph Γ1, which is constructed by Theorem 3.6 is
switching non-isomorphic and Laplacian cospectral to its partial transpose Γτ1 . We obtain
the signed graph Γ2 from Γ1 by replacing the positive edges (u2, u3) and (u3, u4) with
negative edges (u2, u3) and (u3, u4) and negative edge (u1, v1) with the positive edge
(u1, v1). Also, the signed graph Γ3 is obtained from Γτ1 by replacing the positive edges
(u2, u3) and (u3, u4) with negative edges (u2, u3) and (u3, u4). The signed graphs Γ2 and
Γ3 are switching non-isomorphic and Laplacian cospectral. Similarly the switching non-
isomorphic and Laplacian cospectral signed graphs Γ4 and Γ5 are obtained from Γ1 and
Γτ1 , respectively, as in Theorem 3.6.
Remark 3.8. In Example 3.7, we have seen that Γ1 and Γτ1 are Laplacian cospectral signed
graphs. Also, we have mentioned that Γτ1 is the partial transpose of Γ1. But, not all signed
graphs are Laplacian cospectral to their partial transpose, for instance, consider the signed
graphs Γ and Γτ as given in Figure 4. It is easy to calculate that the Laplacian characteristic
polynomials of Γ and Γτ are ψΓ(x) = x6 − 12x5 + 51x4 − 96x3 + 81x2 − 30x + 4 and
ψΓτ (x) = x
6 − 12x5 + 51x4 − 94x3 + 72x2 − 18x.
Let G be a graph and Γ = (G, σ) be a signed graph on G. Hou et al. [15] raised the
following two problems.
Problem 1. Let G be a graph, Γ1 = (G, σ1) and Γ2 = (G, σ2) be two signed graphs on
G, and det(L (Γ1)) = det(L (Γ2)). Are L (Γ1) and L (Γ2) cospectral?
Problem 2. Do there exist pairs Γ1 = (G1, σ1) and Γ2 = (G2, σ2) of signed graphs that
have either of the following properties (i) and (ii)?
(i) Γ1 and Γ2 are not balanced but Laplacian cospectral such that G1 and G2 are noni-
somorphic.
490 Ars Math. Contemp. 23 (2023) #P3.09 / 481–500
Figure 4: Signed graphs Γ and Γτ .
(ii) Γ1 and Γ2 are not balanced but Laplacian cospectral such that G1 and G2 are not
cospectral.
The statement of Problem 1 is not always true. To see this, let Γ1 be a signed graph as
shown in Figure 3. Let Γ′ (shown in Figure 3) be the signed graph obtained from Γ1 by
replacing the negative edge (u1, v1) with positive edge (u1, v1) and positive edge (v3, v4)
with negative edge (v3, v4). The Laplacian characteristic polynomials of Γ1 and Γ′ are re-
spectively given by
ψΓ1(x) = x
8−22x7+197x6−928x5+2476x4−3736x3+2976x2−1056x+128
and
ψΓ′(x) = x
8−22x7+197x6−928x5+2476x4−3748x3+3048x2−1152x+128.
The underlying graphs of Γ1 and Γ′ are isomorphic and det(L (Γ1)) = det(L (Γ′)). It
is clear that the signed graphs Γ1 and Γ′ are not Laplacian cospectral and this answers
Problem 1.
For Problem 2, consider the signed graph Γ1 and its partial transpose Γτ1 as given in
Figure 3. Clearly, the underlying graphs of Γ1 and Γτ1 are non-isomorphic. The unbalanced
signed graphs Γ1 and Γτ1 are Laplacian cospectral. Also, it is easy to see that the underlying
graph of Γ1 and Γτ1 are not cospectral and this answers Problem 2.
4 Constructing switching non-isomorphic cospectral signed graphs,
integral signed graphs and equienergetic signed graphs
The novel non-isomorphic cospectral graph constructions have implications for the com-
plexity of the graph isomorphism problem. This necessitates the creation of methods
for detecting and/or creating non-isomorphic cospectral graphs. Seidel switching, God-
sil–McKay (GM) switching, and others are well-known approaches for constructing cospec-
tral graphs. In 2019, Belardo et al. [1] used the Godsil-Mckay-type procedures developed
for graphs to construct the pairs of switching non-isomorphic cospectral signed graphs. In
this section, we will introduce a new operation on signed graphs. This operation establishes
the relationship of the adjacency spectrum of one signed graph with the Laplacian spectrum
of another signed graph. Furthermore, this operation will be utilized to construct the pairs
of switching non-isomorphic cospectral signed graphs and integral signed graphs. Before
that, we need the following motivation which can also be seen in [3].
T. Shamsher et al.: On adjacency and Laplacian cospectral switching non-isomorphic . . . 491
Figure 5: Bidirected edges in signed graphs.
The usual orientation of edges in digraphs differs from the orientation of signed graphs.
In fact in signed graphs, instead of one arrow, we can use two arrows assigned to edges,
which results in bidirected graphs. An orientated signed graph, more exactly, is an ordered
pair Γϑ = (Γ, ϑ), where
ϑ : V (G)× E(G) → {0, 1,−1} (4.1)
satisfying the following three conditions.
(a) ϑ(u, vw) = 0 whenever u ̸= v, w;u, v, w ∈ V (G) and vw ∈ E(G),
(b) ϑ(v, vw) = 1 ( or −1) if an arrow at v is going into (rep. out of) v. For illustration,
see Figure 5,
(c) ϑ(v, vw) ϑ(w, vw) = −σ(vw).
As a result, positive edges are oriented edges, whereas negative edges are unoriented (see
Figure 5). Therefore, every bidirected graph is also a signed graph. The converse is likewise
true, however, one arrow (at any end) can be taken at random, whereas the other arrow (in
light of (c) above) cannot. For an oriented signed graph Γϑ, its incidence matrix Bϑ =
(bij) is a matrix, whose rows correspond to vertices and columns to edges of G, with
bij = ϑ(vi, ej) (here vi ∈ V (G), ej ∈ E(G)). Usually, when only Γ is given, then we
use an arbitrary orientation. So each row of the incidence matrix corresponding to vertex
vi contains dvi non-zero entries, each equal to +1 or −1. On the other hand, each column
of the incidence matrix corresponding to edge ej contains two non-zero entries, each equal
to +1 or −1. Therefore, even in the case that multiple edges exist, we easily obtain
BϑB
T
ϑ = D(G)−A(Γϑ) = L(Γϑ), (4.2)
where D(G) is the diagonal matrix of vertex degrees of G. It is easy to observe that L(Γϑ)
is positive-semidefinite.
The subdivision signed graph S(Γϑ) is the signed graph whose underlying graph is
S(G) with vertex set V (G)∪E(G). It preserves the orientation ϑ and its adjacency matrix
can be represented in the block form as follows
A(S(Γϑ)) =
(
On Bϑ
BTϑ Om
)
,
where Or ∈ Mr(R). It is easy to see that the signature σ of the subdivision signed graph
is defined by σ(viej) = ϑij . An example of a subdivision signed graph of a signed graph
is shown in Figure 6.
492 Ars Math. Contemp. 23 (2023) #P3.09 / 481–500
Figure 6: A signed graph and the corresponding subdivision signed graph.
Remark 4.1. Any orientation (random) ϑ to the edges of Γ gives rise to the same ma-
trices A(Γϑ) = A(Γ) and L(Γϑ) = L(Γ), while the matrix A(S(Γϑ)) does depend
on ϑ. Let S be a ±1 diagonal matrix such that Bϑ′ = BϑS. Clearly, A(S(Γϑ′)) =
[In+̇S]A(S(Γϑ))[In+̇S], where +̇ denotes the direct sum of two matrices. From now on,
the subscript ϑ in Bϑ will be not specified anymore.
Lemma 4.2 ([3]). If B is the incidence matrix of a connected signed graph Γ = (G, σ)
having n vertices. Then
rank(B) =
{
n− 1 if Γ is balanced,
n if Γ is unbalanced.
Operation. Let Γ be a signed graph with vertex set V (G) = {v1, v2, . . . , vn} and edge
set E(G) = {e1, e2, . . . , em}. Let S(Γ) be a subdivision signed graph of a signed graph
Γ with vertex set V (G) ∪ E(G). In S(Γ), replace each vertex vi, i = 1, 2, . . . , n, by
k vertices and join every vertex to the neighbours of vi with the same sign as that of the
signed edge joining vi with corresponding neighbours in S(Γ). Then in the resulting signed
graph, replace each vertex ej , j = 1, 2, . . . ,m, by p vertices and join every vertex to the
neighbours of ej with the same sign as that of the signed edge joining ej with corresponding
neighbours in S(Γ). The resulting signed graph is denoted by Sk,p(Γ). That is, for a given
signed graph Γ with a compatible orientation ϑ, the signed graph Sk,p(Γ) has vertex set
V (Γ)×{1, 2, . . . , k} ∪E(Γ)×{1, 2, . . . , p}(k copies of V (Γ) and p copies of E(Γ)) and
the edges of Sk,p(Γ) are all between pairs of vertices (v, i) and (e, j), where e ∈ E(Γ) is
incident to v ∈ V (Γ) in Γ, and with sign given by ϑ(v, e).
If k = p = 1, then Sk,p(Γ) coincides with the subdivision signed graph S(Γ). For
convenience, if k = 1, then Sk,p(Γ) will be denoted by Sp(Γ). To illustrate the above
operation, S2(Γ) is shown in Figure 7 and S2,2(Γ) for k = p = 2 is shown in Figure 8.
Theorem 4.3. Let Γ be a signed graph with n vertices and m edges. Let µ1 ≥ µ2 ≥ · · · ≥
µn−1 > µn ≥ 0 be the Laplacian eigenvalues of the signed graph Γ. Then the adjacency
spectrum of Sp(Γ) is Spec(Sp(Γ)) ={
{0(pm−n+2),±√pµ1(1),±
√
pµ2
(1), . . . ,±√pµn−1(1)} if Γ is balanced,
{0(pm−n),±√pµ1(1),±
√
pµ2
(1), . . . ,±√pµn−1(1),±
√
pµn
(1)} if Γ is unbalanced.
Proof. We first label the vertices of Sp(Γ) as follows. Let V (Γ) = {v1, v2, . . . , vn} and
let {uj1, u
j
2, . . . , u
j
p}, j = 1, 2, . . . ,m, denote the vertex set replaced corresponding to the
vertex ej in S(Γ). Denote by
Vi =
{
u1i , u
2
i , . . . , u
m
i
}
, i = 1, 2, . . . , p.
T. Shamsher et al.: On adjacency and Laplacian cospectral switching non-isomorphic . . . 493
Figure 7: Signed graphs Γ, S(Γ) and S2(Γ).
Then V (Γ)∪V1∪V2∪· · ·∪Vp is a partition of V (Sp(Γ)). With this partition, the adjacency
matrix of Sp(Γ) can be written as
A(Sp(Γ)) =

O B B . . . B
BT O O . . . O
BT O O . . . O
...
...
. . .
...
BT O O . . . O
 .
Now, we have
A(Sp(Γ))
2 =

O B B . . . B
BT O O . . . O
BT O O . . . O
...
...
. . .
...
BT O O . . . O


O B B . . . B
BT O O . . . O
BT O O . . . O
...
...
. . .
...
BT O O . . . O

=

pBBT O O . . . O
O BTB BTB . . . BTB
O BTB BTB . . . BTB
...
...
. . .
...
O BTB BTB . . . BTB

=
(
pBBT O1×p
Op×1 Jp×p ⊗BTB
)
,
where Jp×p is a square matrix whose all entries are equal to 1. Therefore
Spec(A(Sp(Γ))
2) = Spec(pBBT ) ∪ Spec(Jp×p ⊗BTB).
As BTB is a real symmetric matrix of order m, so all its eigenvalues are real. Let
x1 ≥ x2 ≥ · · · ≥ xm be the eigenvalues of the matrix BTB. Note that rank(BBT ) =
rank(BTB) = rank(B). Therefore, by Lemma 4.2, we have
Spec(BTB) =
{
{0(m−n+1), x1, x2, . . . , xn−1} if Γ is balanced,
{0(m−n), x1, x2, · · · , xn−1, xn} if Γ is unbalanced,
494 Ars Math. Contemp. 23 (2023) #P3.09 / 481–500
and
Spec(BBT ) = Spec(L(Γ)) =
{
{0, µ1, µ2, . . . , µn−1} if Γ is balanced,
{µ1, µ2, . . . , µn−1, µn} if Γ is unbalanced,
where xn ̸= 0 and µn ̸= 0. As Spec(Jp×p) is {0p−1, p}, then by Lemma 2.2, we have
Spec(Jp×p ⊗BTB) =
{
{0(pm−n+1), px1, px2, · · · , pxn−1} if Γ is balanced,
{0(pm−n), px1, px2, . . . , pxn−1, pxn} if Γ is unbalanced.
We know that the underlying graph of a subdivision signed graph is always bipartite. Simi-
larly, the underlying graph of Sp(Γ) is always bipartite. Note that the eigenvalues of BTB
are given by the eigenvalues of BBT , together with 0 of multiplicity m−n. Therefore, by
Lemmas 2.5 and 2.6, we have Spec(Sp(Γ)) ={
{0(pm−n+2),±√pµ1(1),±
√
pµ2
(1), . . . ,±√pµn−1(1)} if Γ is balanced,
{0(pm−n),±√pµ1(1),±
√
pµ2
(1), . . . ,±√pµn−1(1),±
√
pµn
(1)} if Γ is unbalanced.
Hence, the result follows.
The following result can also be seen in Theorem 2.2 of [3].
Corollary 4.4. Let Γ be a signed graph with n vertices andm edges. Let µ1 ≥ µ2 ≥ · · · ≥
µn−1 > µn ≥ 0 be the Laplacian eigenvalues of the signed graph Γ. Then the adjacency
spectrum of S(Γ) is Spec(S(Γ)) ={
{0(m−n+2),±√µ1(1),±
√
µ2
(1), . . . ,±√µn−1(1)} if Γ is balanced,
{0(m−n),±√µ1(1),±
√
µ2
(1), . . . ,±√µn−1(1),±
√
µn
(1)} if Γ is unbalanced.
Theorem 4.5. Let Γ be a signed graph with n vertices and m edges. Let µ1 ≥ µ2 ≥ · · · ≥
µn−1 > µn ≥ 0 be the Laplacian eigenvalues of the signed graph Γ. Then the adjacency
spectrum of Sk,p(Γ), where p ∈ {k, k − 1}, is Spec(Sk,p(Γ)) ={
{0((k−2)n+pm+2),±
√
pkµ1
(1)
, . . . ,±
√
pkµn−1
(1)} if Γ is balanced,
{0((k−2)n+pm),±
√
pkµ1
(1)
, . . . ,±
√
pkµn−1
(1)
,±
√
pkµn
(1)} if Γ is unbalanced.
Proof. We first label the vertices of Sk,p(Γ) as follows. Let
{
vj1, v
j
2, . . . , v
j
k
}
, j =
1, 2, . . . , n, denote the vertex set replaced corresponding to the vertex vj and {uj1, u
j
2, . . . ,
ujp}, j = 1, 2, . . . ,m, denote the vertex set replaced corresponding to the vertex ej in S(Γ).
Denote by
V i =
{
v1i , v
2
i , . . . , v
n
i
}
, i = 1, 2, . . . , k,
and
Vi =
{
u1i , u
2
i , . . . , u
m
i
}
, i = 1, 2, . . . , p.
T. Shamsher et al.: On adjacency and Laplacian cospectral switching non-isomorphic . . . 495
Then V 1 ∪ V1 ∪ V 2 ∪ V2 ∪ · · · ∪ V k ∪ Vp is a partition of V (Sk,p(Γ)) when k = p.
With this partition, the adjacency matrix of Sk,p(Γ) can be written as
A(Sk,p(Γ)) =

O B O . . . O B
BT O BT . . . BT O
O B O . . . O B
...
...
. . .
...
O B O . . . O B
BT O BT . . . BT O

.
If p = k− 1, then V 1 ∪V1 ∪V 2 ∪V2 ∪ · · · ∪Vk−1 ∪V k is a partition of V (Sk,p(Γ)). With
this partition, the adjacency matrix of Sk,p(Γ) is given by
A(Sk,p(Γ)) =

O B O . . . B O
BT O BT . . . O BT
O B O . . . B O
...
...
. . .
...
BT O BT . . . O BT
O B O . . . B O

.
To prove the result, the following two cases arise.
Case 1. Let Γ be a balanced signed graph with n vertices and m edges. Let Z =
(
X
Y
)
∈
M(n+m)×1(R), where X ∈Mn×1(R) and Y ∈Mm×1(R), be an eigenvector correspond-
ing to the non-zero eigenvalue λi, 1 ≤ i ≤ 2n − 2, of S(Γ). Then A(S(Γ))Z = λiZ
implies that BY = λiX and BTX = λiY . To find the eigenvalues of Sk,p(Γ), consider
the following two subcases.
Subcase 1.1. If k = p, then let U =

X
Y
...
X
Y
 . Clearly, U ∈ M(kn+pm)×1(R) is a non-zero
column vector. We have
A(Sk,p(Γ))U =

O B O . . . O B
BT O BT . . . BT O
O B O . . . O B
...
...
. . .
...
O B O . . . O B
BT O BT . . . BT O


X
Y
...
X
Y
 =

pλiX
pλiY
...
pλiX
pλiY

= pλiU.
Therefore pλi is an eigenvalue of Sk,p(Γ) corresponding to an eigenvector U . As k = p,
thus pλi can be written as
√
kpλi. Hence the result follows by Corollary 4.4.
496 Ars Math. Contemp. 23 (2023) #P3.09 / 481–500
Figure 8: Signed graphs Γ, Γϑ, S(Γ), S2,1(Γ) and S2,2(Γ).
Subcase 1.2. If p = k − 1, then let U =

√
pX√
kY
...√
pX√
kY√
pX

. Clearly, U ∈ M(kn+pm)×1(R) is a
non-zero column vector. We have
A(Sk,p(Γ))U =

O B O . . . B O
BT O BT . . . O BT
O B O . . . B O
...
...
. . .
...
BT O BT . . . O BT
O B O . . . B O


√
pX√
kY
...√
pX√
kY√
pX

=

pλi
√
kX
kλi
√
pY
...
pλi
√
kX
kλi
√
pY
pλi
√
kX

=
√
kpλiU.
Therefore
√
kpλi is an eigenvalue of Sk,p(Γ) corresponding to an eigenvector U . Hence
the result follows by Corollary 4.4.
Case 2. When Γ is an unbalanced signed graph with n vertices and m edges, the proof is
similar to that of Case 1.
Various constructions for non-isomorphic cospectral regular graphs, non-isomorphic
Laplacian cospectral graphs and non-isomorphic signless Laplacian cospectral graphs can
be seen in [5, 7, 8, 10, 11, 13]. The following results show that these constructions including
the constructions obtained in the last section can be utilized to obtain infinite families of
switching non-isomorphic cospectral signed graphs.
Corollary 4.6. Let Γ1 and Γ2 be two switching non-isomorphic signed graphs which are
Laplacian cospectral. Then
(i) the signed graphs Sp(Γ1) and Sp(Γ2) are switching non-isomorphic and cospectral,
(ii) for p ∈ {k, k − 1}, the signed graphs Sk,p(Γ1) and Sk,p(Γ2) are switching non-
isomorphic and cospectral.
T. Shamsher et al.: On adjacency and Laplacian cospectral switching non-isomorphic . . . 497
Figure 9: Cospectral signed graphs S2(Γ1) and S2(Γ2).
Proof. Let Γ1 and Γ2 be two switching non-isomorphic signed graphs. Then, clearly
Sp(Γ1) and Sp(Γ2) are switching non-isomorphic signed graphs and Sk,p(Γ1) and Sk,p(Γ2)
are switching non-isomorphic signed graphs. Hence the result follows by Theorems 4.3
and 4.5.
Example 4.7. Consider the two non-isomorphic signed graphs Γ1 and Γ2 as shown in
Figure 9. Their Laplacian spectrum is respectively given by SpecL(Γ1) = {0, 2, 3(2), 3 +√
5, 3−
√
5} and SpecL(Γ2) = {0, 2, 3(2), 3 +
√
5, 3−
√
5}. So Γ1 and Γ2 are Laplacian
cospectral. It is easy to see that S2(Γ1) and S2(Γ2) are non-isomorphic signed graphs
which are cospectral as their adjacency spectrum are, respectively, given by Spec(S2(Γ1))
= {0(10),±2,±
√
6
(2)
,±(
√
6 +
√
20),±(
√
6−
√
20)} and Spec(S2(Γ2)) = {0(10),±2,
±
√
6
(2)
,±(
√
6 +
√
20),±(
√
6−
√
20)}.
Corollary 4.8. Let Γ1 and Γ2 be two switching non-isomorphic cospectral r−regular
signed graphs. Then
(i) the signed graphs Sp(Γ1) and Sp(Γ2) are switching non-isomorphic and cospectral,
(ii) for p ∈ {k, k − 1}, the signed graphs Sk,p(Γ1) and Sk,p(Γ2) are switching non-
isomorphic and cospectral.
Proof. If Γ1 and Γ2 are two switching non-isomorphic cospectral regular signed graphs,
then L(Γ1) = D(Γ1) − A(Γ1) and L(Γ2) = D(Γ2) − A(Γ2) are cospectral. Hence the
result follows by Corollary 4.6.
Corollary 4.9. Let Γ be a signed graph whose all Laplacian eigenvalues are perfect
squares. Then
(i) the signed graph Sp(Γ) is integral, if p is a perfect square,
(ii) for p ∈ {k, k − 1}, the signed graph Sk,p(Γ) is integral, if kp is a perfect square.
Example 4.10. Let Kn be a balanced complete signed graph on n vertices, where n = t2,
t ≥ 2 is a positive integer. Then
(i) the signed graph Sp(Kn) is integral, if p is a perfect square,
498 Ars Math. Contemp. 23 (2023) #P3.09 / 481–500
(ii) the signed graph Sk,p(Kn) is integral, if kp is a perfect square.
The following result is the graceful implication of Lemma 2.7 and Corollaries 4.6
and 4.8.
Theorem 4.11. For infinitely many n, there exists a family of 2k pairwise switching noni-
somorphic cospectral signed graphs on n vertices, where k > n(2log2(n)) .
The next result directly follows from Theorems 4.3 and 4.5.
Theorem 4.12. Let Γ be a signed graph with n vertices and m edges. Then
(i) E(Sp(Γ)) =
√
pE(S(Γ)),
(ii) E(Sk,p(Γ)) =
√
pkE(S(Γ)), where p ∈ {k, k − 1}.
Theorem 4.13. Let Γ be an unbalanced unicyclic signed graph with at least one edge and
having Laplacian eigenvalues µ1 ≥ µ2 ≥ · · · ≥ µn > 0. Then S(Γ)×K2 and S(Γ)⊗K2,
where K2 is a complete signed graph on 2 vertices, are noncospectral and equienergetic if
and only if µn ≥ 1.
Proof. Let Γ be an unbalanced unicyclic signed graph. Then, by Theorem 4.3, we have
Spec(S(Γ)) = {±√µ1(1),±
√
µ2
(1)
, . . . ,±√µn−1(1),±
√
µn
(1)}.
First, assume that µn ≥ 1. This implies that |
√
µj | ≥ 1, for all j = 1, 2, . . . , n. Also,
E(S(Γ)×K2) = 2
n∑
j=1
(|õj + 1|+ |
√
µj − 1|).
As |√µj | ≥ 1, for all j = 1, 2, . . . , n, we have
E(S(Γ)×K2) = 2
n∑
j=1
(|õj |+ 1 + |
√
µj | − 1)
= 2E(S(Γ))
= E(S(Γ))E(K2) = E(S(Γ)⊗K2).
Note that
√
µ1 + 1 ∈ Spec(S(Γ)×K2) but
√
µ1 + 1 /∈ Spec(S(Γ)⊗K2). Therefore
S(Γ)×K2 and S(Γ)⊗K2 are noncospectral. The converse is similar to that of the converse
in Lemma 2.4.
Example 4.14. Let C−3 = (C3,−) be an unbalanced unicyclic signed graph on 3 vertices.
Its Laplacian spectrum is given by SpecL(C−3 ) = {4, 1, 1}. Therefore C
−
3 meets the
requirement of Theorem 4.13. Hence S(C−3 ) × K2 and S(C
−
3 ) ⊗ K2 are noncospectral
and equienergetic.
The following corollary directly follows from Theorem 4.12.
Corollary 4.15. Let Γ1 and Γ2 be two signed graphs whose subdivision signed graphs are
noncospectral and equienergetic. Then
T. Shamsher et al.: On adjacency and Laplacian cospectral switching non-isomorphic . . . 499
Figure 10: Signed graphs Γ1, Γ2, Γ3 and Γ4.
(i) the signed graphs Sp(Γ1) and Sp(Γ2)are noncospectral and equienergetic,
(ii) for p ∈ {k, k − 1}, the signed graphs Sk,p(Γ1) and Sk,p(Γ2)are noncospectral and
equienergetic.
Example 4.16. Consider the signed graphs Γ1, Γ2, Γ3 and Γ4 as shown in Figure 10. The
adjacency spectrum of their subdivision signed graphs is respectively given by
Spec(S(Γ1)) = {±2,±1(2), 0}, Spec(S(Γ2)) = {±2,±1(2)}, Spec(S(Γ3)) = {±2(3),
±
√
2
(3)
,±
√
6, 0(6)} and Spec(S(Γ4)) = {±1(2),±2(2),±
√
2,±2
√
2,±
√
6, 0(7)}.
Clearly, the signed graphs S(Γ1) and S(Γ2) are noncospectral and equienergetic. Simi-
larly, the signed graphs S(Γ3) and S(Γ4) are noncospectral and equienergetic. Thus, by
Corollary 4.15, we have
(i) the signed graphs Sp(Γ1) and Sp(Γ2) are noncospectral and equienergetic,
(ii) the signed graphs Sp(Γ3) and Sp(Γ4) are noncospectral and equienergetic,
(iii) for p ∈ {k, k − 1}, the signed graphs Sk,p(Γ3) and Sk,p(Γ4) are noncospectral and
equienergetic.
Conclusion. In this paper, we generalized the construction of the subdivision graph S(Γ)
to Sk,p(Γ) of a signed graph Γ. The adjacency spectrum of S1,p(Γ)(Sp(Γ)), Sp,p−1(Γ)
and Sp,p(Γ) is completely determined by the Laplacian spectrum of Γ. Now, it remains a
problem to investigate the adjacency spectrum of Sk,p(Γ) for other values of k and p.
ORCID iDs
Tahir Shamsher https://orcid.org/0000-0002-0330-3395
Shariefuddin Pirzada https://orcid.org/0000-0002-1137-517X
Mushtaq A. Bhat https://orcid.org/0000-0001-8186-5302
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P3.10 / 501–508
https://doi.org/10.26493/1855-3974.2863.4b9
(Also available at http://amc-journal.eu)
The role of the Axiom of Choice in proper and
distinguishing colourings
Marcin Stawiski
AGH University, al. Mickiewicza 30, Kraków, Poland
Received 10 April 2022, accepted 30 November 2022, published online 2 February 2023
Abstract
Call a colouring of a graph distinguishing if the only automorphism which preserves it
is the identity. We investigate the role of the Axiom of Choice in the existence of certain
proper or distinguishing colourings in both vertex and edge variants with special emphasis
on locally finite connected graphs. We show that every locally finite connected graph has
a distinguishing colouring with at most countable number of colours or every locally finite
connected graph has a proper colouring with at most countable number of colours if and
only if Kőnig’s Lemma holds. This statement holds for both vertex and edge colourings.
Furthermore, we show that it is not provable in ZF that such colourings exist even for
every connected graph with maximum degree 3. We also formulate a few conditions about
distinguishing and proper colourings which are equivalent to the Axiom of Choice.
Keywords: Proper colourings, distinguishing colourings, asymmetric colourings, infinite graphs,
graph automorphisms, Axiom of Choice.
Math. Subj. Class. (2020): 05C15, 03E25, 05C25, 05C63
1 Introduction
Let c be a vertex or an edge colouring of a graph G. We say that an automorphism φ of G
preserves c if each vertex of G is mapped to a vertex of the same colour or each edge of
G is mapped to an edge of the same colour. Call a colouring c distinguishing if the only
automorphism which preserves c is the identity. If a colouring c is a mapping into ordinal
numbers (or any well-ordered set) we can think about the number of colours in c. The dis-
tinguishing number D(G) of a graph G is the least number of colours in a distinguishing
vertex colouring of G. Similarly, the distinguishing index D′(G) of a graph G is the least
number of colours in a distinguishing edge colouring of G. Distinguishing vertex colour-
ings were introduced by Babai [1] in 1977 under the name asymmetric colourings during
E-mail address: stawiski@agh.edu.pl (Marcin Stawiski)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
502 Ars Math. Contemp. 23 (2023) #P3.10 / 501–508
his study of the complexity of the graph isomorphism problem [2]. Distinguishing edge
colourings were introduced by Kalinowski and Pilśniak [9].
In this paper we study proper and distinguishing colourings in ZF, hence without as-
suming the Axiom of Choice. Proper vertex colourings in ZF were investigated by Galvin
and Komjáth [5]. They proved that the existence of the chromatic number of each graph is
equivalent to the Axiom of Choice. Distinguishing colourings and proper edge colourings
in ZF were not previously investigated.
In most of the papers about infinite graphs some version of the Axiom of Choice is
used though not always explicitly. The most popular methods often involve Zorn’s Lemma
or Kőnig’s Lemma. In particular, proofs of general bounds by a function of ∆(G) for
chromatic number [3], chromatic index [10], distinguishing number [11] and distinguishing
index [12] of connected infinite graphs all use Kőnig’s Lemma in the case of locally finite
graphs and the Axiom of Choice in the form of Hessenberg’s Theorem for general bounds.
We show that in all these cases the use of Kőnig’s Lemma or respectively the Axiom of
Choice is necessary.
Similar problems for graphs without the assumption of connectivity were previously
investigated for proper vertex colourings. The statement that every graph has a proper
vertex colouring using at most two colours if and only if each of its finite subgraphs has
such a colouring is equivalent to the Axiom of Choice for Pairs. If we replace two colours
with three colours, then we obtain the statement equivalent to the Prime Ideal Theorem.
See [6] p. 109–116 for details and further examples.
Arguably, most of the results related to proper or distinguishing colourings in graph
theory concern only locally finite connected graphs. From the results mentioned in the pre-
vious paragraph, it follows that one cannot prove in ZF that every locally finite connected
graph has a distinguishing or a proper colouring with at most countable number of colours.
We show that one cannot prove the existence of such colourings in ZF even in the simplest
case of connected graphs with maximum degree 3.
2 Preliminaries
By a cardinal number we mean an initial ordinal i.e. an ordinal which is not equinumerous
with any smaller ordinal. For every set there exists a cardinal number equinumerous with
it if and only if the Axiom of Choice holds.
Well-Ordering Theorem states that for every set X there exists a well-order on X .
Well-Ordering Theorem is equivalent to the Axiom of Choice.
We now present some weak choice principles. The axiom ACω⩽κ states that every
countable family of non-empty sets of cardinality at most κ has a choice function. The
axiom ACωfin states that every countable family of non-empty finite sets has a choice
function. The axiom ACω2 is the same as AC
ω
⩽2. The axiom AC
ω
fin is equivalent to
Kőnig’s Lemma stating that every locally finite infinite connected graph has a ray. The
axiom ACωfin is also equivalent to the statement that every countable union of finite sets is
countable. More about the Axiom of Choice, weak choice principles and their equivalent
forms may be found in the extensive monograph of Howard and Rubin [7].
Let G be a graph. Denote by ∆(G) the supremum over the degrees of all vertices of
G. If there exists a vertex v ∈ V (G) such that d(v) = ∆(G), then ∆(G) is called the
maximum degree of G. Graphs with maximum degree 3 are called subcubic. We say that a
graph is locally finite if each of its vertices has finite degree.
M. Stawiski: The role of the Axiom of Choice in proper and distinguishing colourings 503
Let Γ be a group acting on a set Ω and let A be a subset of Ω. The orbit of A is the
set {φ(a) : a ∈ A,φ ∈ Γ}. We say that A is fixed if every φ ∈ Γ acts trivially on A i.e.
if φ(a) = a for every a ∈ A. We say that A is stabilized if for every φ ∈ Γ, we have
φ(A) ⊆ A. In the definitions in this paragraph, if A = {a} is a singleton, then we often
refer to a instead of {a}. An automorphism of a graph G is a bijection φ : V (G) → V (G)
such that uv is an edge in G if and only if φ(u)φ(v) is an edge in G. They form a group
with composition as the operation. If not written explicitly, the meaning of Γ and Ω shall
follow from the context. In this paper Γ is usually a group of some automorphisms of a
graph G and Ω is a set of some vertices of G or some edges of G.
Colourings in this paper are not necessarily proper unless stated otherwise. For notions
which are not defined here, see [4] or [8].
3 The Axiom of Choice in proper and distinguishing colourings
Let κ be an arbitrary non-zero cardinal. Call a family A = {Ai : i ∈ ω} acceptable if A
is a countable family of pairwise disjoint non-empty sets. We say that a family A is almost
κ-acceptable if A is acceptable and every set in A has cardinality less than κ. We say that
A is κ-acceptable if it is almost κ+-acceptable. In other words, A is κ-acceptable if A is
acceptable and every set in A has cardinality at most κ.
Let A = {Ai : i ∈ ω} be an acceptable family and let Y =
⋃
A. Let Z = {zi : i ∈ ω}
and Z ′ = {z′i : i ∈ ω} be disjoint sets which are also disjoint from Y . We now define
graphs GA and HA by
V (GA) = V (HA) = Y ∪ Z ∪ Z ′,
E(GA) = {zizi+1 : i ∈ ω} ∪ {ziz′i : i ∈ ω} ∪ {aiz′i : i ∈ ω, ai ∈ Ai},
E(HA) = E(GA) ∪ {ab : a ̸= b, a, b ∈ Ai, i ∈ ω}.
From the definitions of GA and HA it follows that every vertex in Z \ {z0} has degree 3 in
both graphs, and vertex z0 has degree 2. For the rest of the paragraph, assume that for every
i ∈ ω the set Ai is well-orderable. The vertex z′i has degree |Ai|+1 for every i ∈ ω in both
GA and HA. Every vertex in Y has degree 1 in GA. However, every vertex a ∈ Ai has
degree |Ai| in HA since it has edges to every other vertex in Ai and to z′i. Summarizing,
we obtain ∆(GA) = ∆(HA) = max{3, sup{|Ai|+ 1 : i ∈ ω}}.
Claim 3.1. Let A = {Ai : i ∈ ω} be an acceptable family. Then for every natural
number i, the vertices in Ai form an orbit with respect to the groups of automorphisms of
GA and HA. The rest of the vertices in both graphs are fixed with respect to these groups.
Proof. First we show that z0 is fixed in both graphs. Suppose that z0 may be mapped into
zi for some i ̸= 0. Let Ri be the maximal induced ray with endvertex zi. Clearly R0 is a
tail of the ray R induced by Z. Let P be a maximal induced path with endvertex z0 which
is edge disjoint from R, and let Pi be a maximal induced path with endvertex zi which is
edge disjoint from Ri and contains z0. If Pi contains z0, then the length of Pi is larger than
the length of P . Notice that in this case Pi is the longest induced path with endvertex zi
which is edge disjoint from Ri. This leads to contradiction because Pi cannot be mapped
into P . Hence, z0 is fixed. The ray R is the only induced ray with endvertex z0. Therefore,
R is fixed.
504 Ars Math. Contemp. 23 (2023) #P3.10 / 501–508
Figure 1: Graphs GA and HA. Graph HA is obtained by adding the dashed edges. The ray
Ri is pink, and the remaining part of the ray R is blue.
Since every vertex of the fixed set Z has exactly one neighbour outside Z, all of these
neighbours are fixed. Hence, Z ′ is fixed, and Ai is stabilized for every i ∈ ω. The vertices
in Ai form an independent set in GA (or a clique in HA). Hence, they form an orbit
with respect to the group of automorphisms of GA, and also with respect to the group of
automorphisms of HA.
Notice that from Claim 3.1 it follows that the group of automorphisms of GA is the
same as the group of automorphism of HA. Now, we prove a lemma which allow us to
restrict part of the later considerations to the problem of the existence of the distinguishing
number for graphs of the form GA. With the lemma below we are able to simultaneously
obtain results about distinguishing colourings and proper colourings in both vertex and
edge versions.
Lemma 3.2. Let A be an acceptable family. Then the following conditions are equivalent.
(a) There exists the distinguishing number of GA.
(b) There exists the distinguishing index of GA.
(c) There exists the chromatic index of GA.
(d) There exists the chromatic number of HA.
Proof. By Claim 3.1 if c is a distinguishing vertex colouring of GA, then for every i ∈ ω
vertices in Ai have distinct colours. If for every i ∈ ω and each vertex v ∈ Ai we colour
the edge vz′i with colour c(v), and we colour the rest of the edges of GA arbitrarily, then
we obtain a distinguishing edge colouring of GA. Hence, condition (a) implies (b).
Now, let c be a distinguishing edge colouring of GA. Let c′ be a colouring in which
the edges incident to vertices in Y have the same colour as in c, the edges between Z and
Z ′ are coloured with the same new colour, and the edges between the vertices in Z are
coloured alternately with two new colours. The colouring c′ is a proper edge colouring.
Therefore, condition (b) implies condition (c).
Let c be a proper edge colouring of GA. We now define a proper vertex colouring c′
of HA. First, for every i ∈ ω, we colour each vertex v ∈ Ai with colour c′(v) = c(vz′i).
M. Stawiski: The role of the Axiom of Choice in proper and distinguishing colourings 505
Next, we colour the vertices in Z ′ with the same new colour, and we colour the vertices
in Z alternately with two new colours. Colouring c′ is a proper vertex colouring of HA.
Hence, condition (c) implies condition (d).
Implication between (d) and (a) follows directly from Claim 3.1 since every proper
vertex colouring of HA is a distinguishing vertex colouring of GA.
We can now proceed to the study of relations between the existence of certain colour-
ings and the Axiom of Choice. The first step is Lemma 3.3, which shows that the existence
of the distinguishing number of GA implies the existence of a choice function for A.
Lemma 3.3. Let A be an acceptable family and assume that there exists the distinguishing
number of GA. Then there exists a choice function for A.
Proof. Let c be a vertex colouring of GA with elements of some cardinal number κ. Then
f(A) = argmin{c(a) : a ∈ A} is a choice function for A.
We now prove the next lemma which in the case of non-zero natural number k and k-
acceptable family A allows us to construct a distinguishing colouring of GA using a choice
function for A.
Lemma 3.4. Let k be an arbitrary non-zero natural number and assume ACω⩽k. Then for
every k-acceptable family A graph GA has distinguishing number at most k.
Proof. The proof is by induction on k. Let A be a k-acceptable family. If k = 1, then G
has no non-trivial automorphism. Hence, its distinguishing number is equal to 1. Assume
that k ⩾ 2, and that the statement of the lemma holds for every l < k. Let f be a choice
function for A. From the inductive hypothesis GA − f(A) has a distinguishing vertex
colouring c′ using at most k − 1 colours. Colouring c which agrees with colouring c′ on
V (GA) \ f(A) and which assigns the rest of vertices of GA the same new colour is a
distinguishing colouring using at most k colours.
Lemmas 3.2 – 3.4 allows us to formulate the following corollary about the existence of
certain parameters for k-acceptable families in the case of finite k.
Theorem 3.5. Let k ⩾ 2 be an arbitrary natural number. Then the following conditions
are equivalent.
(a) ACω⩽k.
(b) For every k-acceptable family A the graph GA has the distinguishing number.
(c) For every k-acceptable family A the graph GA has the distinguishing index.
(d) For every k-acceptable family A the graph GA has the chromatic index.
(e) For every k-acceptable family A the graph HA has the chromatic number.
In particular for k = 2 condition (a) is the axiom ACω2 which is independent of ZF. It
follows that in ZF one cannot prove the existence of the above parameters even for every
connected subcubic graph.
Theorem 3.5 tells us about the existence of certain parameters for connected graphs
with finite maximal degree. Now, we establish the relations between Kőnig’s Lemma and
the existence of proper colourings and distinguishing colourings using at most countable
number of colours in the case of locally finite connected graphs.
506 Ars Math. Contemp. 23 (2023) #P3.10 / 501–508
Theorem 3.6. The following conditions are equivalent.
(KL) Kőnig’s Lemma.
(KL1) Every infinite locally finite connected graph has the distinguishing number.
(KL2) Every infinite locally finite connected graph has the distinguishing index.
(KL3) Every infinite locally finite connected graph has the chromatic index.
(KL4) Every infinite locally finite connected graph has the chromatic number.
(KL5) For every almost ℵ0-acceptable family A the graph GA has the distinguishing
number.
(KL6) For every almost ℵ0-acceptable family A the graph GA has the distinguishing
index.
(KL7) For every almost ℵ0-acceptable family A the graph GA has the distinguishing
index.
(KL8) For every almost ℵ0-acceptable family A the graph HA has the chromatic num-
ber.
Proof. First, we show that Kőnig’s Lemma implies conditions (KL1) – (KL4). Let G be
an infinite locally finite connected graph. Let v be some vertex of G. For a natural number
d denote B(v, d) = {x ∈ V (G) : d(v, x) = d}. By local finiteness of G each B(v, d) is
finite. By connectivity of G the vertex set of G may be represented as the countable union
of finite sets V (G) =
⋃
{B(v, d) : d < ω}. Recall that Kőnig’s Lemma is equivalent to the
statement that the sum of every countable family of finite sets is countable. Hence, V (G)
is countable. As E(G) ⊆ V (G) × V (G), then E(G) is also countable. Since both sets
V (G) and E(G) are countable, we can obtain the desired colourings by assigning to each
vertex (edge respectively) a unique natural number.
Implications (KL1) ⇒ (KL5), (KL2) ⇒ (KL6), (KL3) ⇒ (KL7) and (KL4) ⇒ (KL8)
are trivial. The equivalence of the conditions (KL5) – (KL8) follows from Lemma 3.2.
It remains to show that the condition (KL5) implies Kőnig’s Lemma. From (KL5) and
Lemma 3.3, we have that for every countable family of finite sets there exists its choice
function. This is the axiom ACωfin, which is equivalent to Kőnig’s Lemma.
As we have shown, the existence of the distinguishing number of GA for every almost
ℵ0-acceptable family A is equivalent to Kőnig’s Lemma and therefore to the Axiom of
Countable Choice for Finite Sets. One may think that the existence of the distinguishing
number of every graph of the form GA for some acceptable family A is equivalent to the
Axiom of Countable Choice. It turns out that this condition is much stronger and it implies
the full Axiom of Choice.
Theorem 3.7. If for every acceptable family A the graph GA has the distinguishing num-
ber, then the Axiom of Choice holds.
Proof. Let X be a non-empty set and let A be an acceptable family such that X ∈ A.
By the assumption there exists a distinguishing vertex colouring c of the graph GA using
colours from some cardinal κ. As the colouring c is distinguishing, the elements of X
have distinct colours in c. It follows that c|X is an injection from X to cardinal number κ.
Hence, the Well-Ordering Theorem holds and so does the Axiom of Choice.
M. Stawiski: The role of the Axiom of Choice in proper and distinguishing colourings 507
Theorem 3.7 allows to formulate a list of conditions equivalent to the Axiom of Choice.
The conditions (AC1) – (AC4) in the theorem below are equivalent to their restrictions to
connected graphs. Recall that the equivalence of the Axiom of Choice, the existence of
the chromatic number of every graph, and the existence of the chromatic number of every
connected graph was proved by Galvin and Komjáth [5].
Theorem 3.8. The following conditions are equivalent.
(AC) The Axiom of Choice.
(AC1) Every graph has the distinguishing number.
(AC2) Every graph without a component isomorphic to K1 or K2 has the distinguish-
ing index.
(AC3) Every graph has the chromatic index.
(AC4) Every graph has the chromatic number.
(AC5) For every acceptable family A the graph GA has the distinguishing number.
(AC6) For every acceptable family A the graph GA has the distinguishing index.
(AC7) For every acceptable family A the graph GA has the chromatic index.
(AC8) For every acceptable family A the graph HA has the chromatic number.
Proof. From the Well-Ordering Theorem we can well-order the set of vertices and the
set of edges of a given graph and then colour each vertex (edge respectively) of the said
graph with a unique colour. This means that the Axiom of Choice implies conditions
(AC1) – (AC4). Each of the condition (AC1) – (AC4) implies its restriction to connected
graphs and also the corresponding condition (AC5) – (AC8). By Lemma 3.2 conditions
(AC5) – (AC8) are equivalent. The implication between condition (AC5) and the Axiom
of Choice is Theorem 3.7.
ORCID iDs
Marcin Stawiski https://orcid.org/0000-0003-2554-1754
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