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Contents
Modifications of the Floyd-Warshall algorithm with nearly quadratic
expected-time
Andrej Brodnik, Marko Grgurovič, Rok Požar . . . . . . . . . . . . . . . . 1
Z-oriented triangulations of surfaces
Adam Tyc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
The cubical matching complex revisited
Duško Jojić . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
From Italian domination in lexicographic product graphs to w-domination
in graphs
Abel Cabrera Martínez, Alejandro Estrada-Moreno,
Juan Alberto Rodríguez-Velázquez . . . . . . . . . . . . . . . . . . . . . . 57
Sum-list-colouring of θ-hypergraphs
Ewa Drgas-Burchardt, Agata Drzystek, Elżbieta Sidorowicz . . . . . . . . 83
Cospectrality of multipartite graphs
Alireza Abdollahi, Niloufar Zakeri . . . . . . . . . . . . . . . . . . . . . . 99
Oriented regular representations of out-valency two for finite simple groups
Gabriel Verret, Binzhou Xia . . . . . . . . . . . . . . . . . . . . . . . . . 115
New strong divisibility sequences
Zhibin Du, Darko Dimitrov, Carlos M. da Fonseca . . . . . . . . . . . . . . 121
Maximal order group actions on Riemann surfaces
Jay Zimmerman, Coy L. May . . . . . . . . . . . . . . . . . . . . . . . . . 135
Signed graphs with two eigenvalues and vertex degree five
Zoran Stanić . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Volume 22, Number 1, Spring/Summer 2022, Pages 1–161
iii

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P1.01 / 1–22
https://doi.org/10.26493/1855-3974.2467.497
(Also available at http://amc-journal.eu)
Modifications of the Floyd-Warshall algorithm
with nearly quadratic expected-time*
Andrej Brodnik †
University of Primorska, UP FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia
University of Primorska, UP IAM, Muzejski trg 2, 6000 Koper, Slovenia
University of Ljubljana, UL FRI, Večna pot 113, 1000 Ljubljana, Slovenia
Marko Grgurovič
University of Primorska, UP FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia
University of Primorska, UP IAM, Muzejski trg 2, 6000 Koper, Slovenia
Rok Požar ‡
University of Primorska, UP FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia
University of Primorska, UP IAM, Muzejski trg 2, 6000 Koper, Slovenia
IMFM, Jadranska 19, 1000 Ljubljana, Slovenia
Received 25 October 2020, accepted 10 April 2021, published online 25 November 2021
Abstract
The paper describes two relatively simple modifications of the well-known Floyd-
Warshall algorithm for computing all-pairs shortest paths. A fundamental difference of
both modifications in comparison to the Floyd-Warshall algorithm is that the relaxation is
done in a smart way. We show that the expected-case time complexity of both algorithms
is O(n2 log2 n) for the class of complete directed graphs on n vertices with arc weights
selected independently at random from the uniform distribution on [0, 1]. Theoretically
best known algorithms for this class of graphs are all based on Dijkstra’s algorithm and
obtain a better expected-case bound. However, by conducting an empirical evaluation we
prove that our algorithms are at least competitive in practice with best know algorithms
and, moreover, outperform most of them. The reason for the practical efficiency of the
presented algorithms is the absence of use of priority queue.
*A preliminary version of this work has been published in Shortest Path Solvers: From Software to Wetware,
volume 32 of Emergence, Complexity and Computation (2018). The authors would like to thank the reviewer for
excellent comments that substantially improved the quality of the paper.
†This work is sponsored in part by the Slovenian Research Agency (research program P2-0359 and research
projects J1-2481, J2-2504, and N2-0171).
‡Corresponding author. This work is supported in part by the Slovenian Research Agency (research program
P1-0285 and research projects N1-0062, J1-9110, J1-9187, J1-1694, N1-0159, J1-2451).
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 22 (2022) #P1.01 / 1–22
Keywords: All-pairs shortest paths, probabilistic analysis.
Math. Subj. Class. (2020): 05C85, 68W40
1 Introduction
Finding shortest paths in graphs is a classic problem in algorithmic graph theory. Given a
(directed) graph in which arcs are assigned weights, a shortest path between pair of vertices
is such a path that infimizes the sum of the weights of its constituent arcs. The problem
pops up very frequently also in practice in areas like bioinformatics, logistics, VLSI design
(for a more comprehensive list of applications see e.g. [2]). Two of the most common
problem’s variants are the single-source shortest path problem and the all-pairs shortest
path problem (APSP). In the first variant of the problem, we are searching for paths from
a fixed vertex to every other vertex, while the APSP asks for a shortest path between every
pair of vertices. In this paper we focus exclusively on the APSP variant of the problem.
In general, the APSP can be solved by using the technique of relaxation. The relaxation
consists of testing whether we can improve the weight of the shortest path from u to v
found so far by going via w, and updating it if necessary. In fact, the number of attempts
to perform relaxation corresponds to the time complexity under the RAM model. A trivial
text-book relaxation-based solution to the APSP is a dynamic programming Floyd-Warshall
algorithm [11] running in O(n3) time on graphs with n vertices.
Moreover, also Dijkstra’s algorithm [10] solving single-source shortest path problem
is relaxation-based. However, since the order in which the relaxations are performed is
greedy, it uses an additional priority queue data structure. Obviously we can solve the
APSP running Dijkstra’s algorithm from each vertex of the graph obtaining O(mn log n)
solution where m is the number of arcs in the graph, provided we use the binary heap imple-
mentation of the priority queue. This is an improvement over the Floyd-Warshal solution
for sparse graphs. Asymptotically we get an even better solution by using Fibonacci heaps
over binary heaps yielding O(n2 log n+mn) time complexity. We refer to such approaches
as a Dijkstra-like, which inherently use some kind of a priority queue implementation.
However, the described solutions to the APSP using the Dijkstra’s algorithm have at
least two limitations. The first one is that all arc weights have to be non-negative. This
can be alleviated by using Johnson’s approach [15], which reweighs all arc weights making
them non-negative. On such a graph we can now run Dijkstra’s algorithm. The second lim-
itation is related to the efficiency of the solution implementation. Namely, due to computer
architecture efficient implementations exploit the issue of data locality; i.e. in consecutive
memory accesses they try to access memory locations that are “close together”. A simi-
lar observation is made in [5] for the solutions to the single-source shortest path problem,
where authors show that a Fibonacci heap, as asymptotically better implementation of a
priority queue, in practice underperform simple binary heap.
For dense graphs, a slightly better worst-case running time of O(n3 log log n/ log2 n)
over the O(n3)-time Floyd-Warshall algorithm can be achieved by using an efficient matrix
multiplication technique [13]. For sparse graphs on n vertices and with m non-negative
weighted arcs fastest known solution [20] runs in time O(mn+ n2 log log n).
E-mail addresses: andrej.brodnik@upr.si (Andrej Brodnik), marko.grgurovic@famnit.upr.si (Marko
Grgurovič), rok.pozar@upr.si (Rok Požar)
A. Brodnik et al.: Modifications of the Floyd-Warshall algorithm with nearly quadratic . . . 3
Considering expected-case running-time of APSP algorithms we can find in the litera-
ture a number of good solutions assuming that input instances are generated according to a
probability model on the set of complete directed graphs with arc weights. In the uniform
model, arc weights are drawn at random, independently of each other, according to a com-
mon probability distribution. A more general model is the endpoint-independent model
[3, 24], where, for each vertex v, a sequence of n−1 non-negative arc weights is generated
by a deterministic or stochastic process and then randomly permuted and assigned to the
outgoing arcs of v. In the vertex potential model [5, 6], arc weights can be both positive and
negative. This is a probability model with arbitrary real arc weights but without negative
cycles.
In the uniform model with arc weights drawn from the uniform distribution on [0, 1],
the O(n2 log n) expected time complexity algorithms for solving the APSP were pre-
sented by Karger et al. [16] and Demetrescu and Italiano [8, 9], where the latter was
inspired by the former. Another algorithm with the same expected time complexity was
presented by Brodnik and Grgurovič [4]. Peres et al. [19] improved the Demetrescu and
Italiano algorithm to theoretically optimal O(n2) by replacing the priority queue imple-
mentation with a more involved data structure yielding theoretically desired time complex-
ity. In the endpoint-independent model, Spira [24] proved an expected-case time bound of
O(n2 log2 n), which was improved by several authors. Takaoka and Moffat [25] improved
the bound to O(n2 log n log log n). Bloniarz [3] described an algorithm with expected-
case running time O(n2 log n log∗ n). Finally, Moffat and Takaoka [18] and Mehlhorn
and Priebe [17] improved the running time to O(n2 log n). In the vertex potential model,
Cooper et al. [6] gave an algorithm with an expected-case running time O(n2 log n). All
the above algorithms use Dijkstra-like approach.
In this paper, we present two modifications of the Floyd-Warshall algorithm, which we
name the Tree algorithm and the Hourglass algorithm. A fundamental difference of both
modifications in relation to the Floyd-Warshall algorithm is a smarter way to perform the
relaxations. This is done by introducing a tree structure that allows us to skip relaxations
that do not contribute to the result. The worst-case time complexity of both algorithms
remains O(n3), however, in the analysis we show that their expected running time is sub-
stantially better. To simplify the analysis, we consider the uniform model which gives us
the following main result.
Theorem 1.1. For complete directed graphs on n vertices with arc weights selected in-
dependently at random from the uniform distribution on [0, 1], the Tree algorithm and the
Hourglass algorithm both have an expected-case running time of O(n2 log2 n).
The proof of our main result relies on the following well-known properties of the com-
plete directed graph on n vertices with uniformly distributed arc weights on [0, 1]. First,
a maximum weight of a shortest path in such a graph is O(log n/n) with high probability;
second, a longest shortest path has O(log n) arcs with high probability; and third, the max-
imum outdegree of the subgraph consisting of all arcs that are shortest paths is O(log n)
with high probability. These properties, together with the observation that if the relaxation
on some vertex of the introduced tree structure fails, we can skip relaxations on the entire
subtree defined by this vertex (see Lemma 3.1), then give the desired result. Since theoreti-
cally best expected-case APSP algorithms are based on Dijkstra’s algorithm, it is interesting
that a competitive approach can also be obtained by a modification of the Floyd-Warshall
algorithm.
4 Ars Math. Contemp. 22 (2022) #P1.01 / 1–22
To prove the practical competitiveness of our algorithms, we supplement the theoret-
ical analysis with an empirical evaluation. It should be pointed out, that all algorithms
mentioned above with o(n2 log2 n) expected-case running time obtain a better theoretical
bound. Moreover, Brodnik and Grgurovič in [4] show, for the same family of graphs as
studied in this paper, practical supremacy of their algorithm over the algorithms due to
Karger et al. [16] and Demetrescu and Italiano [8, 9] and consequently over the algorithm
of Peres et al. [19], since its improvement of Demetrescu and Italiano solution does not
improve the practical efficiency of the original algorithm. Therefore in the practical evalu-
ation of Tree and Hourglass algorithms we compare them to the algorithm of Brodnik and
Grgurovič [4] only. The reason for the practical efficiency of the presented algorithms is
the absence of use of priority queue. Indeed, the Tree and Hourglass algorithms are simple
to implement and use only simple structures such as vectors and arrays, which also exhibit
a high data locality.
The structure of the paper is the following. Section 2 contains the necessary notation
and basic definitions to make the paper self-contained. In Section 3 we describe the Tree
and Hourglass algorithms. Properties of certain shortest paths in complete graphs with
independently and uniformly distributed arc weights are analyzed in Section 4. The proof
of the main result is presented in Section 5, while Section 6 contains empirical evaluation
of the algorithms. In Section 7 we give some concluding remarks and open problems.
2 Preliminaries
All logarithms are base e unless explicitly stated otherwise. The model of computation
used in algorithm design and analysis is the comparison-addition model, where the only
allowed operations on arc weights are comparisons and additions.
A digraph (or directed graph) G is a pair (V,A), where V is a non-empty finite set of
vertices and A ⊆ V × V a set of arcs. We assume V = {v1, v2, . . . , vn} for some n. The
two vertices joined by an arc are called its endvertices. For an arc (u, v) ∈ A, we say that
u is its tail. The outdegree of v ∈ V , is the number of arcs in A that have v as their tail.
The maximum outdegree in G is denoted by ∆(G).
A digraph G′ = (V ′, A′) is a subdigraph of the digraph G = (V,A) if V ′ ⊆ V and
A′ ⊆ A. The (vertex-)induced subdigraph with the vertex set S ⊆ V , denoted by G[S],
is the subgraph (S,C) of G, where C contains all arcs a ∈ A that have both endvertices
in S, that is, C = A ∩ (S × S). The (arc-)induced subdigraph with the arc set B ⊆ A,
denoted by G[B], is the subgraph (T,B) of G, where T is the set of all those vertices in V
that are endvertices of at least one arc in B.
A path P in a digraph G from vP,0 to vP,m is a finite sequence P = vP,0, vP,1, . . . , vP,m
of pairwise distinct vertices such that (vP,i, vP,i+1) is an arc of G, for i = 0, 1, . . . ,m− 1.
The length of a path P , denoted by |P |, is the number of vertices occurring on P . Any
vertex of P other than vP,0 or vP,m is an intermediate vertex. A k-path is a path in
which all intermediate vertices belong to the subset {v1, v2, . . . , vk} of vertices for some
k. Obviously, a 0-path has no intermediate vertices.
A weighted digraph is a digraph G = (V,A) with a weight function w : A → R that
assigns each arc a ∈ A a weight w(a). A weight function w can be extended to a path P
by w(P ) =
∑m−1
i=0 w(vP,i, vP,i+1). A shortest path from u to v, denoted by u ⇝ v, is
a path in G whose weight is infimum among all paths from u to v. The distance between
two vertices u and v, denoted by DG(u, v), is the weight of a shortest path u ⇝ v in G.
A. Brodnik et al.: Modifications of the Floyd-Warshall algorithm with nearly quadratic . . . 5
The diameter of G is maxu,v∈V DG(u, v), that is, the maximum distance between any two
vertices in G. Given a subset S ⊆ V , the distance between S and a vertex v in G, denoted
by DG(S, v), is DG(S, v) = minu∈S DG(u, v). A shortest k-path from u to v is denoted
by u k⇝v. Further, we denote the set of arcs that are shortest k-paths in G by A(k) and the
subdigraph G[A(k)] by G(k).
Finally, we will need some tools from combinatorics. In the balls-into-bins process M
balls are thrown uniformly and independently into N bins. The maximum number of balls
in any bin is called the maximum load. Let Li denote the load of bin i, i ∈ {1, 2, . . . , N}.
The next lemma, used in Subsection 4.3, provides an upper bound on the maximum load.
It is a simplified version of a standard result, c.f. [23], tailored to our present needs. For
completeness we provide the proof.
Lemma 2.1. If M balls are thrown into N bins where each ball is thrown into a bin chosen
uniformly at random, then P(max1≤i≤N Li ≥ e2(M/N + logN)) = O(1/N).
Proof. First, we have µ = E(Li) = M/N , i = 1, 2, . . . , N , and we can write each Li as
a sum Li = Xi1 + Xi2 + · · · + XiM , where Xij is a random variable taking value 1, if
ball j is in bin i, and 0 otherwise. Next, since Li is a sum of independent random variables
taking values in {0, 1}, we can apply, for any particular bin i and for every c > 1, the
multiplicative Chernoff bound [12], which states that
P(Li ≥ cµ) ≤
(
ec−1
c c
)µ
≤
(
e
c
)cµ
.
We consider two cases, depending on whether µ ≥ logN or not. Let µ ≥ logN . Take
c = e2. Then,
P(Li ≥ e2µ) ≤
(
1
e
)e2µ
≤
(
1
e
)e2 logN
=
1
Ne2
≤ 1
N2
.
Consider now µ < logN . Take c = e2 NM logN . Since x
−x ≤
(
1
e
)x
for all x ≥ e, we have
P
(
Li ≥ µe2
N
M
logN
)
= P(Li ≥ e2 logN) ≤
(
e
c
)cµ
=
((
c
e
)− ce)eµ
≤
((
1
e
) c
e
)eµ
=
(
1
e
)e2 logN
≤ 1
N2
.
Putting everything together, we get that
P(Li ≥ e2(µ+ logN)) ≤ P(Li ≥ e2µ | µ ≥ logN) + P(Li ≥ e2 logN | µ < logN)
≤ 1
N2
+
1
N2
=
2
N2
.
This, by the union bound, implies that
P
(
max
1≤i≤N
Li ≥ e2(µ+ logN)
)
≤
N∑
i=1
P(Li ≥ e2(µ+ logN)) ≤ N
2
N2
= O(1/N).
6 Ars Math. Contemp. 22 (2022) #P1.01 / 1–22
3 Speeding up the Floyd-Warshall algorithm
The Floyd-Warshall algorithm [11, 26] as presented in Algorithm 1 is a simple dynamic
programming approach to solve APSP on a graph G(V,A) represented by a weight matrix
W , where Wij = w(vi, vj) if (vi, vj) ∈ A and ∞ otherwise. Its running time is O(n3)
due to three nested for loops.
Algorithm 1 FLOYD-WARSHALL(W )
1 for k := 1 to n do
2 for i := 1 to n do
3 for j := 1 to n do
4 if Wik +Wkj < Wij then ▷ Relaxation
5 Wij := Wik +Wkj
3.1 The Tree algorithm
Let us consider iteration k, and let OUTk denote a shortest path tree rooted at vertex vk
in G(k−1). Intuitively, one might expect that the relaxation in lines 4-5 would not always
succeed in lowering the value of Wij which currently contains the weight w(vi
k−1
⇝ vj).
This is precisely the observation that we exploit to arrive at a more efficient algorithm:
instead of simply looping through every vertex of V in line 3, we perform the depth-first
traversal of OUTk. This permits us to skip iterations which provably cannot lower the
current value of Wij . As the following lemma shows, if w(vi
k
⇝vj) = w(vi
k−1
⇝ vj), then
w(vi
k
⇝vy) = w(vi
k−1
⇝ vy) for all vertices vy in the subtree of vj in OUTk.
Lemma 3.1. Let vj ∈ V \ {vk} be some non-leaf vertex in OUTk, vy ̸= vj an arbitrary
vertex in the subtree of vj in OUTk, and vi ∈ V \ {vk}. Consider the path vi
k−1
⇝ vk
k−1
⇝ vj .
If w(vi
k−1
⇝ vk
k−1
⇝ vj) ≥ w(vi
k−1
⇝ vj), then w(vi
k−1
⇝ vk
k−1
⇝ vy) ≥ w(vi
k−1
⇝ vy).
Proof. Since vj is neither a leaf nor the root of OUTk, we have j < k, and so vi
k−1
⇝ vj
k−1
⇝ vy
is a (k − 1)-path between vi and vy . Because vi
k−1
⇝ vy is a shortest (k − 1)-path between
vi and vy , we have
w(vi
k−1
⇝ vy) ≤ w(vi
k−1
⇝ vj
k−1
⇝ vy) = w(vi
k−1
⇝ vj) + w(vj
k−1
⇝ vy)
≤ w(vi
k−1
⇝ vk
k−1
⇝ vj) + w(vj
k−1
⇝ vy) = w(vi
k−1
⇝ vk
k−1
⇝ vj
k−1
⇝ vy),
where the last inequality follows by the assumption. Finally, since vy is in the subtree
rooted at vj , we have vi
k−1
⇝ vk
k−1
⇝ vj
k−1
⇝ vy = vi
k−1
⇝ vk
k−1
⇝ vy , and so the last term is equal
to w(vi
k−1
⇝ vk
k−1
⇝ vy). This completes the proof.
The pseudocode of the modified Floyd-Warshall algorithm augmented with the tree
OUTk, named the Tree algorithm, is given in Algorithm 2. To perform depth first search we
first construct the tree OUTk in line 3 using CONSTRUCTOUT given in Algorithm 3. For
the construction of tree OUTk an additional matrix π, where πij specifies the penultimate
vertex on a k-shortest path from vi to vj (i.e. the vertex “before” vj)1 is used. More
1C.f. π(k)ij in [7, Sec. 25.2].
A. Brodnik et al.: Modifications of the Floyd-Warshall algorithm with nearly quadratic . . . 7
Algorithm 2 TREE(W )
1 Initialize π, an n× n matrix, as πij := i.
2 for k := 1 to n do
3 OUTk := CONSTRUCTOUTk(π)
4 for i := 1 to n do
5 Stack := empty
6 Stack.push(vk)
7 while Stack ̸= empty do
8 vx := Stack.pop()
9 for all children vj of vx in OUTk do
10 if Wik +Wkj < Wij then ▷ Relaxation
11 Wij := Wik +Wkj
12 πij := πkj
13 Stack.push(vj)
Algorithm 3 CONSTRUCTOUTk(π)
1 Initialize n empty trees: T1, . . . , Tn.
2 for i := 1 to n do
3 Ti.Root := vi
4 for i := 1 to n do
5 if i ̸= k then
6 Make Ti a subtree of the root of Tπki .
return Tk
precisely, the tree OUTk is obtained from π by making vi a child of vπki for all i ̸= k.
This takes O(n) time. Finally, we replace the iterations in lines 3-5 in Algorithm 1 with
depth-first tree traversal of OUTk in lines 5-13 in Algorithm 2. Note that if, for a given i
and a child vj , the condition in line 10 evaluates to false we do not traverse the subtree of
vj in OUTk.
Corollary 3.2. The Tree algorithm correctly computes all-pairs shortest paths.
Proof. The correctness of the algorithm follows directly from Lemma 3.1.
Time complexity
Let Tk denote the running time of the algorithm TREE(W ) in lines 3-13 at iteration k. As
already said, line 3 requires O(n) time. To estimate the time complexity of lines 4-13,
we charge the vertex vx in line 8 by the number of its children. This pays for lines 9-13.
Furthermore, this means that on the one hand leaves are charged nothing, while on the
other hand nobody is charged for the root vk. To this end, let SP
(k)
k be the set of all shortest
k-paths that contain vk and end at some vertex in the set {v1, v2, . . . , vk}. Now vx in line
8 is charged at most |SP(k)k | times over all iterations of i. Since the number of children of
vx is bounded from above by ∆(OUTk), we can bound Tk from above by
Tk ≤ |SP(k)k | ·∆(OUTk) +O(n). (3.1)
8 Ars Math. Contemp. 22 (2022) #P1.01 / 1–22
Practical improvement
Observe that in Algorithm 2 vertices of OUTk are visited in a depth-first search (DFS) order,
which is facilitated by using the stack. However, this requires pushing and popping of each
vertex, as well as reading of all its children in OUTk. We can avoid this by precomputing
two read-only arrays dfs and skip to support the traversal of OUTk. The array dfs consists
of OUTk vertices as visited in the DFS order. On the other hand, the array skip is used to
skip OUTk subtree when relaxation in line 10 of Algorithm 2 does not succeed.
In detail, for a vertex vz , skipz contains the index in dfs of the first vertex after vz
in the DFS order that is not a descendant of vz in OUTk. Utilizing the arrays outlined
above, we traverse OUTk by scanning dfs in left-to-right order and using skip whenever
a relaxation is not made. Consequently, we perform only two read operations per visited
vertex. It should be pointed out that the asymptotic time remains the same, as this is solely
a technical optimization.
3.2 The Hourglass algorithm
We can further improve the Tree algorithm by using another tree. The second tree, denoted
by INk, is similar to OUTk, except that it is a shortest path “tree” for paths vi
k−1
⇝ vk for
each vi ∈ V \ {vk}. Strictly speaking, this is not a tree, but if we reverse the directions of
the arcs, it turns it into a tree with vk as the root. Traversal of INk is used as a replacement
of the for loop on variable i in line 4 of Algorithm 2 (in line 2 of Algorithm 1). As the
following lemma shows, if w(vi
k
⇝vj) = w(vi
k−1
⇝ vj), then w(vy
k
⇝vj) = w(vy
k−1
⇝ vj) for
all vertices vy in the subtree of vi in INk.
Lemma 3.3. Let vi ∈ V \ {vk} be some non-leaf vertex in INk and let vy ̸= vi be an arbi-
trary vertex in the subtree of vi in INk, and vj ∈ V \{vk}. Consider the path vi
k−1
⇝ vk
k−1
⇝ vj .
If w(vi
k−1
⇝ vk
k−1
⇝ vj) ≥ w(vi
k−1
⇝ vj), then w(vy
k−1
⇝ vk
k−1
⇝ vj) ≥ w(vy
k−1
⇝ vj).
Proof. Due to the choice of vi and vy we have: vy
k−1
⇝ vk = vy
k−1
⇝ vi
k−1
⇝ vk. We want to
show, that:
w(vy
k−1
⇝ vj) ≤ w(vy
k−1
⇝ vi) + w(vi
k−1
⇝ vk
k−1
⇝ vj).
Observe that i < k, since vi is neither a leaf nor the root of INk. Thus we have:
w(vy
k−1
⇝ vj) ≤ w(vy
k−1
⇝ vi) + w(vi
k−1
⇝ vj).
Putting these together we get the desired inequality:
w(vy
k−1
⇝ vj) ≤ w(vy
k−1
⇝ vi) + w(vi
k−1
⇝ vj) ≤ w(vy
k−1
⇝ vi) + w(vi
k−1
⇝ vk
k−1
⇝ vj).
The pseudocode of the modified Floyd-Warshall algorithm augmented with the trees
OUTk and INk, named the Hourglass algorithm2, is given in Algorithms 4 and 5. To con-
struct INk efficiently, we need to maintain an additional matrix ϕij which stores the second
2The hourglass name comes from placing INk tree atop the OUTk tree, which gives it an hourglass-like shape,
with vk being at the neck.
A. Brodnik et al.: Modifications of the Floyd-Warshall algorithm with nearly quadratic . . . 9
vertex on the path from vi to vj (cf. π and πij). Algorithm 6 constructs INk similarly to
the construction of OUTk, except that we use the matrix ϕik instead. The only extra space
requirement of the Hourglass algorithm that bears any significance is the matrix ϕ, which
does not deteriorate the space complexity of O(n2). The depth-first traversal on INk is
performed by a recursion on each child of vk in line 7 of Algorithm 4. In the recursive
step, given in Algorithm 5, we can prune OUTk as follows: if vi is the parent of vy in INk
and vi
k−1
⇝ vj ≤ vi
k−1
⇝ vk
k−1
⇝ vj , then the subtree of vj can be removed from OUTk, while
inspecting the subtree of vi in INk. Before the return from the recursion the tree OUTk is
reconstructed to the form it was passed as a parameter to the function.
Algorithm 4 HOURGLASS(W )
1 Initialize π, an n× n matrix, as πij := i.
2 Initialize ϕ, an n× n matrix, as ϕij := j.
3 for k := 1 to n do
4 OUTk := CONSTRUCTOUTk(π)
5 INk := CONSTRUCTINk(ϕ)
6 for all children vi of vk in INk do
7 RECURSEIN(W,π, ϕ, INk,OUTk, vi)
Algorithm 5 RECURSEIN(W,π, ϕ, INk,OUTk, vi)
1 Stack := empty
2 Stack.push(vk)
3 while Stack ̸= empty do
4 vx := Stack.pop()
5 for all children vj of vx in OUTk do
6 if Wik +Wkj < Wij then ▷ Relaxation
7 Wij := Wik +Wkj
8 πij := πkj
9 ϕij := ϕik
10 Stack.push(vj)
11 else
12 Remove the subtree of vj from OUTk.
13 for all children vi′ of vi in INk do
14 RECURSEIN(W,π, ϕ, INk,OUTk, vi′ )
15 Restore OUTk by reverting changes done by all iterations of line 12.
In practice, the recursion can be avoided by using an additional stack, which further
speeds up an implementation of the algorithm.
Corollary 3.4. The Hourglass algorithm correctly computes all-pairs shortest paths.
Proof. Observe, that lines 5-10 of Algorithm 5 are effectively the same as in Algorithm 2.
Line 12 of Algorithm 5 does not affect the correctness of the algorithm due to Lemma 3.3,
which states that, for any vi′ that is a child of vi in INk, these comparisons can be skipped,
as they cannot lead to shorter paths. However, Lemma 3.3 does not apply to a sibling vi∗
10 Ars Math. Contemp. 22 (2022) #P1.01 / 1–22
Algorithm 6 CONSTRUCTINk(ϕ)
1 Initialize n empty trees: T1, . . . , Tn.
2 for i := 1 to n do
3 Ti.Root := vi
4 for i := 1 to n do
5 if i ̸= k then
6 Make Ti a subtree of the root of Tπki .
return Tk
of vi, arising from line 6 of Algorithm 4. Therefore line 15 restores the tree OUTk, which
maintains the correctness of the algorithm.
Finally, note that the worst-case time complexity of the Hourglass (and Tree) algo-
rithm remains O(n3). The simplest example of this is when all shortest paths are the arcs
themselves, at which point all leaves are children of the root and the tree structure never
changes.
4 Properties of shortest k-paths in complete graphs
Let Kn denote a complete digraph on the vertex set V = {v1, v2, . . . , vn}.
4.1 Distances
We assume that arc weights of Kn are exponential random variables with mean 1 and that
all n(n − 1) random arc weights are independent. Due to the memoryless property, it
is easier to deal with exponentially distributed arc weights than directly with uniformly
distributed arc weights. The aim of this subsection is to show that the diameter of K(k)n ,
the subdigraph of Kn consisting of all (weighted) arcs that are shortest k-paths in Kn, is
O(log n/k) with very high probability. We note, however, that by the same argument as
given in the beginning of Subsection 4.3, all results derived in this subsection for expo-
nential arc weights also hold, asymptotically for [0, 1]-uniformly distributed arc weights as
soon as k ≥ log2 n.
We start by considering for a fixed u ∈ V , the maximum distance in K(k)n between u
and other vertices in V . To this end, let S = {u, v1, . . . , vk} ⊆ V , and let S = V \ S. We
clearly have
max
v∈V
D
K
(k)
n
(u, v) ≤ max
v∈S
DKn[S](u, v) + max
v∈S
DKn[S×S](S, v), (4.1)
that is, the maximum distance in K(k)n between u and other vertices in V is bounded above
by the sum of the maximum distance in Kn[S] between u and other vertices in S, and by
the maximum distance in Kn[S × S] between S and vertices in S. We note that Kn[S]
is a complete digraph on |S| vertices and Kn[S × S] is a complete bipartite digraph with
bipartition (S, S).
To provide an upper bound on maxv∈S DKn[S](u, v), we use the following result,
which follows from the equation (2.8) in the proof of Theorem 1.1 of Janson [14].
A. Brodnik et al.: Modifications of the Floyd-Warshall algorithm with nearly quadratic . . . 11
Theorem 4.1 ([14, Theorem 1.1]). Let u ∈ V be a fixed vertex of Kn. Then for every
a > 0, we have
P
(
max
v∈V
DKn(u, v) ≥
a log n
n
)
= O(ean2−a log2 n).
Lemma 4.2. Let 8 ≤ k ≤ n, and let S ⊆ V with |S| = k. Then, for a fixed u ∈ S and for
any constant c > 0, we have
P
(
max
v∈S
DKn[S](u, v) ≥
c log n
k
)
= O(n2−c/2 log2 n).
Proof. By Theorem 4.1, for any a > 0 we have
P
(
max
v∈S
DKn[S](u, v) ≥
a log k
k
)
= O(eak2−a log2 k).
Setting a = c log n/ log k we get
eak2−a log2 k = ec logn/ log kk2k−c logn/ log k log2 k ≤ (elogn)c/2k2(klogk n)−c log2 k.
In the last step we used the fact that 1/ log k ≤ 1/2 for k ≥ 8 and that log n/ log k =
logk n. Furthermore,
(elogn)c/2k2(klogk n)−c log2 k = nc/2k2n−c log2 k = O(n2−c/2 log2 n),
and the result follows.
Next, we provide an upper bound on maxv∈S DKn[S×S](S, v).
Lemma 4.3. Let 1 ≤ k ≤ n, let S ⊆ V with |S| = k, and let S = V \ S. Then for any
constant c > 0, we have
P
(
max
v∈S
DKn[S×S](S, v) ≥
c log n
k
)
= O(n1−c log n).
Proof. Let Z = maxv∈S DKn[S×S](S, v). Arguing similarly as in the proof of Theorem
1.1 of Janson [14], Z is distributed as
n−1∑
j=k
Xj ,
where Xj are independent exponentially distributed random variables with mean 1k(n−j) .
First, for any constant c > 0, the Chernoff bound [12] states that
P(Z ≥ c log n/k) ≤ e−tc lognE(ektZ).
Further, for −∞ < t ≤ 1, we have
E(ektZ) =
n−1∏
j=k
E(ektXj ) =
n−1∏
j=k
(
1− t
n− j
)−1
.
12 Ars Math. Contemp. 22 (2022) #P1.01 / 1–22
Using the inequality − log(1 − x) ≤ x + x2 for all 0 ≤ x ≤ 1/2, we can bound, for all
0 < t < 1 and k ≤ j ≤ n− 2, each term (1− t/(n− j))−1 as follows(
1− t
n− j
)−1
= exp
(
− log
(
1− t
n− j
))
≤ exp
(
t
n− j
+
(
t
n− j
)2)
.
This gives us
P(Z ≥ c log n/k) ≤ (1− t)−1 exp
(
− tc log n+
n−2∑
j=k
(
t
n− j
+
(
t
n− j
)2))
= (1− t)−1 exp(−tc log n+ t log(n− k) +O(1)).
Taking t = 1− 1/ log n, we finally get
P(Z ≥ c log n/k) ≤ (1/ log n)−1 exp(−c log n+ log n+O(1)) = O(n1−c log n).
We are now ready to show that the diameter of K(k)n is O(log n/k) with very high
probability.
Theorem 4.4. Let 8 ≤ k ≤ n. Then, for any constant c > 0, we have
P
(
max
u,v∈V
D
K
(k)
n
(u, v) ≥ c log n
k
)
= O(n3−c/4 log2 n).
Proof. Let S = {u, v1, . . . , vk} ⊆ V , let S = V \ S, and write α = c log n/k. Then, by
inequality (4.1), we have
P
(
max
v∈V
D
K
(k)
n
(u, v) ≥ α
)
≤ P
(
max
v∈S
DKn[S](u, v) + max
v∈S
DKn[S×S](S, v) ≥ α
)
≤ P
(
max
v∈S
DKn[S](u, v) ≥
α
2
)
+ P
(
max
v∈S
DKn[S×S](S, v) ≥
α
2
)
.
By Lemma 4.2, we have
P
(
max
v∈S
DKn[S](u, v) ≥
α
2
)
= O(n2−c/4 log2 n),
and, by Lemma 4.3,
P
(
max
u∈S
DKn[S×S](S, v) ≥
α
2
)
= O(n1−c/2 log n).
Putting everything together, we get
P
(
max
v∈V
D
K
(k)
n
(u, v) ≥ α
)
= O(n2−c/4 log2 n),
which, by the union bound, implies
P
(
max
u,v∈V
D
K
(k)
n
(u, v) ≥ α
)
≤ nP
(
max
u∈V
D
K
(k)
n
(v, u) ≥ α
)
= O(n3−c/4 log2 n).
A. Brodnik et al.: Modifications of the Floyd-Warshall algorithm with nearly quadratic . . . 13
4.2 Lengths
Let all arc weights of Kn be either independent [0, 1]-uniform random variables or inde-
pendent exponential random variables with mean 1. In this subsection, we bound the length
of the longest shortest k-path in Kn.
The proof of our next lemma follows directly from Theorem 1.1 of Addario-Berry et.
al [1] on the longest shortest path in Kn.
Theorem 4.5 ([1, Theorem 1.1]). The following two properties hold:
(i) For every t > 0, we have
P
(
max
u,v∈V
|u⇝v| ≥ α∗ log n+ t
)
≤ eα
∗+t/ logne−t,
where α∗ ≈ 3.5911 is the unique solution of α logα− α = 1.
(ii) E(maxu,v∈V |u⇝v|) = O(log n).
Lemma 4.6. The following two properties hold:
(i) For every c > 5 and 8 ≤ k ≤ n, we have P(maxu,v∈V |u
k
⇝v| ≥ c log n) =
O(n2−c/2).
(ii) E(maxu,v∈V |u
k
⇝v|) = O(log k).
Proof. Let S = {v1, v2, . . . , vk}, and let u → w
k
⇝z → v be a shortest k-path in Kn. Since
w
k
⇝z is a shortest path from w to z in Kn[S], we have
max
u,v∈V
|u k⇝v| ≤ max
w,z∈S
|w k⇝z|+ 2. (4.2)
By (i) of Theorem 4.5, for any t > 0,
P
(
max
w,z∈S
|w k⇝z| ≥ α∗ log k + t
)
≤ eα
∗+t/ log ke−t,
where α∗ ≈ 3.5911 is the unique solution of α logα−α = 1. Using t = (c−α∗) log n−2
gives us
P
(
max
w,z∈S
|w k⇝z|+ 2 ≥ α∗ log(k/n) + c log n
)
≤ eα
∗−2/ log k+2e(c−α
∗)( log nlog k −logn)
≤ eα
∗−2/ log k+2(elogn)1/2(α
∗−c)
= O(n2−c/2).
By inequality (4.2), we have
P
(
max
u,v∈V
|u k⇝v| ≥ c log n
)
≤ P
(
max
w,z∈S
|u k⇝v|+ 2 ≥ c log n
)
≤ P
(
max
w,z∈S
|w k⇝z|+ 2 ≥ α∗ log(k/n) + c log n
)
,
and (i) follows.
To prove (ii), we note that, by (ii) of Theorem 4.5, E(maxu,v∈S |u
k
⇝v|) = O(log k),
and, by inequality (4.2), the result follows.
14 Ars Math. Contemp. 22 (2022) #P1.01 / 1–22
4.3 Maximum outdegree
Let arc weights of Kn be independent [0, 1]-uniform random variables. Our goal in this
subsection is to show that the maximum outdegree of a shortest path tree OUTk in K
(k)
n is
O(log k + (n− k)/k) with high probability for all k ≥ log2 n.
Let now S = {v1, v2, . . . , vk} and S = V \ S. We can consider OUTk as consisting
of the subtree OUTk[S] to which each vertex from S is attached as a leaf. To see how
these vertices are attached to OUTk[S], let us assume for the moment that arc weights
are exponentially distributed with mean 1. Then, it is easy to see that a vertex v ∈ S is
attached to that one in S with which it forms a shortest arc, say av , between S and v.
Let (Kn[S × S])∗ be the subdigraph of Kn[S × S] with the set V of vertices and the set
{av | v ∈ S} of arcs. By observing that OUTk[S] is a subdigraph of the graph (Kn[S])(k)
consisting of all arcs that are shortest paths in Kn[S], we have
∆(OUTk) ≤ ∆((Kn[S])(k)) + ∆((Kn[S × S])∗). (4.3)
To extend the latter bound to uniform distribution, we use a standard coupling argument
as in [1]. Let U be a random variable uniform on [0, 1]. Then − log(1 − U) is an expo-
nential random variable with mean 1, and so we can couple the exponential arc weights
W ′(u, v) to uniform arc weights W (u, v) by setting W ′(u, v) = − log(1−W (u, v)). As
x ≤ − log(1 − x) ≤ x + 2x2 for all 0 ≤ x ≤ 1/2, we have that, for all arcs (u, v)
of Kn, |W ′(u, v) − W (u, v)| = O((W ′(u, v))2), uniformly for all W ′(u, v) ≤ 1/2. In
particular, if W ′(u, v) ≤ 12 log n/k, say, and k ≥ log2 n, then |W ′(u, v) − W (u, v)| =
O(1/ log2 n) for n large enough, and so for a path P with O(log n) vertices and with
W ′(P ) ≤ 12 log n/k, we have
|W ′(P )−W (P )| = O(1/ log n)
for n large enough. By Theorem 4.4, with very high probability a shortest (k − 1)-path
in Kn with the exponential arc weights has weight less than 12 log n/k, while by (i) of
Lemma 4.6, with very high probability it has O(log n) vertices. It then follows easily
that, for all n sufficiently large and k ≥ log2 n, the bound as in (4.3) holds for uniform
distribution, as well.
The following result on the maximum outdegree in the subgraph (Kn[S])(k) of the
complete graph Kn[S] on k vertices with [0, 1]-uniform arc weights can be found in Peres
et al. [19].
Lemma 4.7 ([19, Lemma 5.1]). Let 1 ≤ k ≤ n and let S ⊆ V with |S| = k. Then, for
every c > 6, we have P(∆((Kn[S])(k)) > c log k) = O(k1−c/6).
The maximum outdegree in (Kn[S × S])∗ is directly related to the maximum load in
the balls-into-bins process, which is used in the proof of the following lemma.
Lemma 4.8. Let 1 ≤ k ≤ n, let S ⊆ V with |S| = k, and let S = V \ S. Then,
P(∆((Kn[S × S])∗) ≥ e2((n− k)/k + log k)) = O(k−1).
Proof. Consider vertices from S as bins and vertices from S as balls. For v ∈ S, each arc
in S × v is equally likely to be the shortest, so v is thrown into a bin chosen uniformly at
random, and the result follows by Lemma 2.1 for N = k and M = n− k.
A. Brodnik et al.: Modifications of the Floyd-Warshall algorithm with nearly quadratic . . . 15
We are now ready to prove the main result of this subsection.
Theorem 4.9. For every k ≥ log2 n, we have
P
(
∆(OUTk) ≥ (e2 + 12) log k + e2
n− k
k
)
= O(k−1).
Proof. Let S = {v1, v2, . . . , vk} and S = V \ S. Further, let us write α = 12 log k and
β = e2((n− k)/k log k). By the inequality (4.3), for every k ≥ log2 n, we have
P(∆(OUTk) ≥ α+ β) ≤ P(∆((Kn[S])(k)) + ∆((Kn[S × S])∗) ≥ α+ β)
≤ P(∆((Kn[S])(k)) ≥ α) + P(∆((Kn[S × S])∗) ≥ β).
By Lemma 4.7, we have P(∆((Kn[S])(k)) ≥ α) ≤ 1/k. Similarly, by Lemma 4.8, we
have P(∆((Kn[S × S])∗) ≥ β) ≤ 1/k. Hence, P(∆(OUTk) ≥ α + β) ≤ 1/k + 1/k =
O(1/k).
5 Expected-case analysis
We perform an expected-case analysis of the Tree algorithm for the complete directed
graphs on n vertices with arc weights selected independently at random from the uniform
distribution on [0, 1]. Recall that SP(k)k is the set of all shortest k-paths that contain vk and
end at some vertex in the set {v1, v2, . . . , vk}. We first show that the expected number of
paths in SP(k)k is O(n log k).
Lemma 5.1. For each k = 1, 2, . . . , n, we have E(|SP(k)k |) = O(n log k).
Proof. For vi ∈ V , let SP(k)i denote the set of all shortest k-paths that contain vi and end
at some vertex in the set {v1, v2, . . . , vk}. Note that
k∑
i=1
|SP(k)i | ≤
n∑
i=1
k∑
j=1
|vi
k
⇝vj |.
By symmetry, we have E(|SP(k)i |) = E(|SP
(k)
j |) for arbitrary i, j ∈ {1, 2, . . . , k}, and
hence
kE(|SP(k)k |) =
k∑
i=1
E(|SP(k)i |) ≤
n∑
i=1
k∑
j=1
E(|vi
k
⇝vj |) ≤ knE( max
u,v∈V
|u k⇝v|).
By (ii) of Lemma 4.6, we get that E(|SP(k)k |) = O(n log k).
We are now ready to analyse the expected time of the Tree algorithm.
Theorem 5.2. The Tree algorithm has an expected-case running time of O(n2 log2 n) for
the complete directed graphs on n vertices with arc weights selected independently at ran-
dom from the uniform distribution on [0, 1].
16 Ars Math. Contemp. 22 (2022) #P1.01 / 1–22
Proof. To estimate the number of comparisons Tk at iteration k, we consider two cases.
First, for k < log2 n we bound Tk from above by n2. Second, we estimate E(Tk) for
k ≥ log2 n. For every c > 0, we have
E(Tk) = E(Tk | ∆(OUTk) < c) · P(∆(OUTk) < c)
+ E(Tk | ∆(OUTk) ≥ c) · P(∆(OUTk) ≥ c).
Using inequality (3.1) we get
E(Tk | ∆(OUTk) < c) ≤ E(|SP(k)k | ·∆(OUTk) +O(n) | ∆(OUTk) < c)
≤ c · E(|SP(k)k |) +O(n).
As Tk is always at most n2, we have E(Tk | ∆(OUTk) ≥ c) ≤ n2. Further, taking into
account that P(∆(OUTk) < c) ≤ 1, we get
E(Tk) ≤ c · E(|SP(k)k |) +O(n) + n
2 · P(∆(OUTk) ≥ c).
Take c = (e2 + 12) log k + e2 n−kk . Then, by Lemma 4.9, we have P(∆(OUTk) ≥ c) =
O(k−1). Moreover, by Lemma 5.1, we have E(|SP(k)k |) = O(n log k), which gives us
E(Tk) = O((e2 + 12)n log2 k + e2(n− k)n log k/k) +O(n) +O(n2/k)
= O(n log2 n+ n2 log n/k).
Putting everything together, we bound the expected time of the algorithm from above as
E
( n∑
k=1
Tk
)
=
log2 n−1∑
k=1
E(Tk) +
n∑
k=log2 n
E(Tk)
≤
log2 n−1∑
k=1
n2 +
n∑
k=log2 n
O(n log2 n+ n2 log n/k) = O(n2 log2 n),
as claimed.
We conclude the section with a proof of the main theorem.
Proof of Theorem 1.1. The Hourglass algorithm does not have a worse bound than the Tree
variant, so the result follows by Theorem 5.2.
6 Empirical evaluation
All algorithms were implemented in C++ and compiled using g++ -march=native
-O3. The tests were ran on an Intel(R) Core(TM) i7-2600@3.40GHz with 8GB RAM
running Windows 7 64-bit.
To make the comparison between Floyd-Warshall and its modified versions fairer, we
improved the Floyd-Warshall algorithm with a simple modification skipping combinations
of i and k where Wik = ∞, and consequently reducing the number of relaxations of the
algorithm to RFW ≤ n3.
A. Brodnik et al.: Modifications of the Floyd-Warshall algorithm with nearly quadratic . . . 17
The experiments were conducted on the following random digraphs: (i) uniform ran-
dom digraphs with arc weights uniformly distributed on the interval [0, 1], and (ii) un-
weighted random digraphs. In both cases, the digraphs were constructed by first setting
the desired vertex count and density. Then, a random Hamiltonian cycle was constructed,
ensuring the strong connectivity of the digraph. After the cycle was constructed, the re-
maining n(n − 2) arcs were put into a list and randomly permuted, and then added into
the digraph until the desired density was reached. Finally, algorithms were executed on
the instance, and their running times were recorded. Tests were conducted ten times and
averaged, with each test running on a different randomly generated graph.
6.1 Empirical comparison of the number of relaxations
Our motivation when designing the Hourglass and Tree algorithms was to skip relaxations
that are not contributing to the result. To verify the theoretical results on the expected
number of relaxations in practice we conducted two experiments in which we counted the
number of relaxations by different algorithms. For the first experiment we generated a sub-
family of digraphs from (i), mentioned above, consisting of complete digraphs of varying
size vertex set. On contrary, for the second experiment we generated another subfamily
of digraphs from (i), now consisting of sparse digraphs with fixed vertex set and variable
arc density. The results of experiments are presented in the plots relative to the number of
relaxations performed by the Floyd-Warshall algorithm; i.e. all numbers of relaxations are
divided by RFW .
The results of the first experiment, in which RFW = n3 since digraphs are complete,
are presented in Figure 1. To relate the theoretical upper bound of O(n2 lg2 n) of the Tree
algorithm and the experimental results, we added also the plot of the function 60n
2 lg2 n
n3 .
We chose the constant 60 so that the plots of the Tree algorithm and the added function start
at the same initial point, namely at 28 vertices. The results of the second experiment for
28 29 210 211 212
0
1
2
3
4
5
7.5
10
15
Number of vertices (n)
R
el
ax
at
io
ns
/n
3
Tree
Hourglass
60n
2 lg2 n
n3
Figure 1: Complete digraphs of various sizes with the number of relaxations of algorithms
divided by n3.
18 Ars Math. Contemp. 22 (2022) #P1.01 / 1–22
n = 1024 vertices and sizes of the arc set varying between n2/10 and 8n2/10 are shown
in Figure 2.
10 20 30 40 50 60 70 80
1
2
3
4
5
6
Density (100% = n2 arcs)
R
el
ax
at
io
ns
/R
F
W
n = 1024 vertices
Tree
Hourglass
Figure 2: Digraphs with n = 1024 vertices and various arc densities with the number of
relaxations of algorithms divided by RFW .
In Figure 1 we see a significant reduction of relaxations which also implies the decrease
of running time of the Tree and Hourglass algorithms. From the plot we can also see that
the experimental results indicate that the theoretical upper bound of the Tree algorithm is
asymptotic tight. The experiments on sparse digraphs (see Figure 2) also show a reduction
in relaxations as digraphs become sparser.
6.2 Empirical comparison of running times
As discussed in the introduction, we compared the Tree3 and Hourglass algorithms with the
Floyd-Warshall [11, 26] and Dijkstra [10] algorithms, as well as the algorithm of Brodnik
and Grgurovič [4], which we refer to as Propagation. These algorithms were chosen since
they proved to perform best out of a larger group of algorithms compared in [4].
It should be pointed out, that breadth-first search is extremely efficient in solving APSP
on unweighted digraphs. However, we did not include breadth-first search in comparisons,
because we consider unweighted graph instances only as the worst-case instances of the
general shortest path problem (each arc is part of at least one shortest path in such in-
stances).
The algorithms were tested on the graph families (i) and (ii) described at the beginning
of this section, with sizes of the vertex set varying between 512 and 4096, and sizes of the
arc set varying between n1.1 and n2. As the priority queue in the Dijkstra and Propaga-
tion algorithms we used pairing heaps since they are known to perform especially well in
solving APSP in practice [22], even though the amortized complexity of their decrease key
3In the tests, we used the implementation of the algorithm with improvements from Subsection 3.1.
A. Brodnik et al.: Modifications of the Floyd-Warshall algorithm with nearly quadratic . . . 19
operation takes O(22
√
lg lgn) in comparison to O(1) of Fibonacci heaps [21]. We used the
implementation of pairing heaps from the Boost Library, version 1.55.
The results for uniform random digraphs presented in Figure 3 show that both, Prop-
agation and Tree, outperform the other algorithms on all vertex and arc densities. As the
size n of graphs increases, the running time of Hourglass approaches the running time of
Tree, but the constant factors still prove to be too large for Hourglass to prevail because of
a more clever exploration strategy. Moreover, it is interesting to see that Floyd-Warshall
based Tree and Hourglass outperform Dijkstra on sparse graphs.
n1.1 n1.325 n1.55 n1.775 n2
24
25
26
27
28
Density
Ti
m
e
(m
ill
is
ec
on
ds
)
n = 512 vertices
n1.1 n1.325 n1.55 n1.775 n2
26
27
28
29
210
211
Density
Ti
m
e
(m
ill
is
ec
on
ds
)
n = 1024 vertices
n1.1 n1.325 n1.55 n1.775 n2
28
29
210
211
212
213
214
Density
Ti
m
e
(m
ill
is
ec
on
ds
)
n = 2048 vertices
n1.1 n1.325 n1.55 n1.775 n2
211
212
213
214
215
216
217
Density
Ti
m
e
(m
ill
is
ec
on
ds
)
n = 4096 vertices
Dijkstra Propagation Floyd-Warshall
Tree Hourglass
Figure 3: Experimental results on family (i) – uniform digraphs.
The results for unweighted random digraphs are shown in Figure 4. What is interesting
is that Tree and Hourglass remain competitive with Dijkstra, and even outperforming it on
smaller graphs in some instances. In contrast, the performance of Propagation falls short
of Dijkstra because each arc is part of at least one shortest path in these graphs.
20 Ars Math. Contemp. 22 (2022) #P1.01 / 1–22
n1.1 n1.325 n1.55 n1.775 n2
24
25
26
27
28
29
Density
Ti
m
e
(m
ill
is
ec
on
ds
)
n = 512 vertices
n1.1 n1.325 n1.55 n1.775 n2
25
26
27
28
29
210
211
212
213
Density
Ti
m
e
(m
ill
is
ec
on
ds
)
n = 1024 vertices
n1.1 n1.325 n1.55 n1.775 n2
28
29
210
211
212
213
214
215
216
Density
Ti
m
e
(m
ill
is
ec
on
ds
)
n = 2048 vertices
n1.1 n1.325 n1.55 n1.775 n2
210
211
212
213
214
215
216
217
218
219
220
Density
Ti
m
e
(m
ill
is
ec
on
ds
)
n = 4096 vertices
Dijkstra Propagation Floyd-Warshall
Tree Hourglass
Figure 4: Experimental results on family (ii) – unweighted digraphs.
7 Conclusion
We theoretically analyzed the Tree algorithm which is a relatively simple modification
of the Floyd-Warshall algorithm. The analysis gives its expected-case time complexity
in the uniform model of O(n2 log2 n), which also explains the algorithm’s good practical
performance presented in Section 6. We also presented the Hourglass algorithm as a further
improvement of the Tree algorithm, but it remains an open question whether its expected-
case time complexity in the uniform model is o(n2 log2 n).
Next, since both the Tree and Hourglass algorithms allow negative arc weights, it would
be interesting to analyze their expected-case running time complexity for a model that
permits negative arcs such as the vertex potential model [5, 6].
Overall, the Tree algorithm is simple to implement and offers very good performance.
The Hourglass algorithm has the potential to be even better but probably requires a more
A. Brodnik et al.: Modifications of the Floyd-Warshall algorithm with nearly quadratic . . . 21
complex implementation. It is also worthwhile to note that the space requirement of the
Tree algorithm is not worse than the space requirement of any algorithm that reports all
shortest paths. The Hourglass algorithm requires an additional matrix of size n2.
ORCID iDs
Andrej Brodnik https://orcid.org/ 0000-0001-9773-0664
Rok Požar https://orcid.org/ 0000-0002-2037-9535
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P1.02 / 23–39
https://doi.org/10.26493/1855-3974.2242.842
(Also available at http://amc-journal.eu)
Z-oriented triangulations of surfaces
Adam Tyc *
Institute of Mathematics, Polish Academy of Sciences,
Śniadeckich 8, 00-656 Warszawa, Poland
Received 5 February 2020, accepted 12 April 2021, published online 4 April 2022
Abstract
The main objects of the paper are z-oriented triangulations of connected closed 2-
dimensional surfaces. A z-orientation of a map is a minimal collection of zigzags which
double covers the set of edges. We have two possibilities for an edge – zigzags from the
z-orientation pass through this edge in different directions (type I) or in the same direction
(type II). Then there are two types of faces in a triangulation: the first type is when two
edges of the face are of type I and one edge is of type II and the second type is when all
edges of the face are of type II. We investigate z-oriented triangulations with all faces of the
first type (in the general case, any z-oriented triangulation can be shredded to a z-oriented
triangulation of such type). A zigzag is homogeneous if it contains precisely two edges of
type I after any edge of type II. We give a topological characterization of the homogeneity
of zigzags; in particular, we describe a one-to-one correspondence between z-oriented tri-
angulations with homogeneous zigzags and closed 2-cell embeddings of directed Eulerian
graphs in surfaces. At the end, we give an application to one type of the z-monodromy.
Keywords: Directed Eulerian embedding, triangulation of a surface, zigzag, z-monodromy, z-orien-
tation.
Math. Subj. Class. (2020): 05E45, 52B05
1 Introduction
Petrie polygons are well-known objects described by Coxeter [5] (see also [13]). These are
skew polygons in regular polyhedra such that any two consecutive edges, but not three, are
on the same face. Analogs of Petrie polygons for graphs embedded in surfaces are called
zigzags [7, 10] or closed left-right paths [9, 18]. These are sequences of oriented edges
defined by the rule described above. Zigzags have many applications, for example, they
are successfully exploited to enumerate all combinatorial possibilities for fullerenes [3].
*The author is grateful to Mark Pankov for useful discussions.
E-mail address: atyc@impan.pl (Adam Tyc)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
24 Ars Math. Contemp. 22 (2022) #P1.02 / 23–39
The case when a map, i.e. an embedding of a graph in a surface, has a unique zigzag is very
important [7, 9]. Following [7] we call such maps z-knotted. They have nice homological
properties and are closely connected to the Gauss code problem [6, 9, 11].
The study of zigzags in 3-regular plane graphs, in particular fullerenes, is one of the
main directions of [7]. A large class of z-knotted 3-regular plane graphs is obtained using
a computer. The dual objects, i.e. spherical triangulations, have the same zigzag structure.
Zigzags in triangulations of surfaces (not necessarily orientable) are investigated in [15, 16,
17]. By [17], every such triangulation admits a z-knotted shredding, i.e. it can be modified
to a z-knotted triangulation of the same surface by triangulating of some of its faces.
A z-orientation of a map is a minimal collection of zigzags which double covers the set
of edges [7]. In the z-knotted case, this collection contains only one zigzag and is unique
up to reversing. For every z-orientation we have the following two types of edges: an edge
is of type I if the distinguished zigzags pass through this edge in different directions and
an edge is of type II if they pass through the edge in the same direction. It is not difficult
to prove that for every face in a triangulation with fixed z-orientation one of the following
possibilities is realized: the face contains precisely two edges of type I and the third edge
is of type II (the first type of face) or all edges are of type II (the second type of face).
We observe that every z-oriented triangulation can be shredded to a triangulation where
all faces are of the first type (Section 2). In this paper, we restrict ourself to z-oriented
triangulations with all faces of the first type.
Let Γ be such a triangulation of a surface M . Then the number of edges of type I is
twice the number of edges of type II and we say that a zigzag is homogeneous if it contains
precisely two edges of type I after each edge of type II. Denote by ΓII the subgraph of Γ
formed by all edges of type II. Our first result (Theorem 3.3) states that the following three
conditions are equivalent:
(1) all zigzags of Γ are homogeneous,
(2) ΓII is a closed 2-cell embedding of a simple Eulerian digraph such that every face is
a directed cycle,
(3) each connected component of M \ ΓII is homeomorphic to an open 2-dimensional
disk.
Note that directed Eulerian spherical embeddings are known also as plane alternating
dimaps; they are investigated, for example, in [2, 8, 12]. Directed Eulerian embeddings
in arbitrary surfaces are considered in [1, 4].
We will use the following structural property of Γ (without assumption that the zigzags
are homogeneous): the connected components of M \ ΓII are open disks, cylinders or
Möbius strips (the third type of components can be realized only for the non-orientable
case) and all these possibilities are realized. We show that the existence of cylinders or
Möbius strips contradicts the homogeneity of zigzags.
A z-monodromy of face is a permutation which acts on the oriented edges of this face,
the z-monodromy of an edge e is the first oriented edge of the face which occurs in certain
zigzag after e. By [17], there are precisely 7 types of z-monodromies (M1) – (M7). For
each of the types (M3) – (M5) and (M7) there is a triangulation such that each face has
the z-monodromy of this type. The types (M1) and (M2) are exceptional: all faces with
z-monodromies of each of these types form a forest [16]. The case (M6) cannot be investi-
gated by the methods of [16] and the authors left it as an open problem. It is easy to see that
A. Tyc: Z-oriented triangulations of surfaces 25
each face with the z-monodromy (M6) is of the first type for every z-orientation. Using
this fact, we construct a series of toric triangulations where all faces have z-monodromies
of type (M6).
2 Zigzags and z-orientations of triangulations of surfaces
Let M be a connected closed 2-dimensional surface (not necessarily orientable). A trian-
gulation of M is a 2-cell embedding of a connected simple finite graph in M such that all
faces are triangles [14, Section 3.1]. Then the following assertions are fulfilled:
(1) every edge is contained in precisely two distinct faces,
(2) the intersection of two distinct faces is an edge or a vertex or empty.
Let Γ be a triangulation of M . A zigzag in Γ is a sequence of edges {ei}i∈N satisfying the
following conditions for every natural i:
• ei and ei+1 are distinct edges of a certain face (then they have a common vertex,
since every face is a triangle),
• the faces containing ei, ei+1 and ei+1, ei+2 are distinct and the edges ei and ei+2 are
non-intersecting.
Since Γ is finite, there is a natural number n > 0 such that ei+n = ei for every natural i.
In what follows, every zigzag will be presented as a cyclic sequence e1, . . . , en, where n is
the smallest number satisfying the above condition.
Every zigzag is completely determined by any pair of consecutive edges belonging to
this zigzag and for any distinct edges e and e′ on a face there is a unique zigzag containing
the sequence e, e′. If Z = {e1, . . . , en} is a zigzag, then the reversed sequence Z−1 =
{en, . . . , e1} also is a zigzag. A zigzag cannot contain a sequence e, e′, . . . , e′, e which
implies that Z ̸= Z−1 for any zigzag Z, i.e. a zigzag cannot be self-reversed (see, for
example, [17]). We say that Γ is z-knotted if it contains precisely two zigzags Z and Z−1,
in other words, there is a single zigzag up to reversing.
Example 2.1. Zigzags in the Platonic solids (three of them are triangulations of the sphere)
are skew polygons without self-intersections and are called Petrie polygons.
Example 2.2. Let BPn be the n-gonal bipyramid, where 1, . . . , n denote the consecutive
vertices of the base and the remaining two vertices are denoted by a, b (see Figure 1 for
n = 3).
3
2
1
a
b
.
Figure 1: 3-gonal bipyramid.
26 Ars Math. Contemp. 22 (2022) #P1.02 / 23–39
(a) In the case when n = 2k + 1, the bipyramid BPn is z-knotted. If k is odd, then the
unique (up to reversing) zigzag is
a1, 12, 2b, b3, . . . , a(n− 2), (n− 2)(n− 1), (n− 1)b, bn, n1,
1a, a2, 23, 3b, . . . , a(n− 1), (n− 1)n, nb,
b1, 12, 2a, a3, . . . , b(n− 2), (n− 2)(n− 1), (n− 1)a, an, n1,
1b, b2, 23, 3a, . . . , b(n− 1), (n− 1)n, na.
If k is even, then this zigzag is
a1, 12, 2b, b3, . . . , b(n− 2), (n− 2)(n− 1), (n− 1)a, an, n1,
1b, b2, 23, 3a, . . . , a(n− 1), (n− 1)n, nb,
b1, 12, 2a, a3, . . . , a(n− 2), (n− 2)(n− 1), (n− 1)b, bn, n1,
1a, a2, 23, 3b, . . . , b(n− 1), (n− 1)n, na.
(b) If n = 2k and k is odd, then the bipyramid contains precisely two zigzags (up to
reversing):
a1, 12, 2b, b3, 34, . . . , a(n− 1), (n− 1)n, nb,
b1, 12, 2a, a3, 34, . . . , b(n− 1), (n− 1)n, na
and
a2, 23, 3b, b4, 45, . . . , an, n1, 1b,
b2, 23, 3a, a4, 45, . . . , bn, n1, 1a.
(c) In the case when n = 2k and k is even, there are precisely four zigzags (up to
reversing):
a1, 12, 2b, . . . , b(n− 1), (n− 1)n, na;
b1, 12, 2a, . . . , a(n− 1), (n− 1)n, nb;
a2, 23, 3b, . . . , bn, n1, 1a;
b2, 23, 3a, . . . , an, n1, 1b.
See [15, 17] for more examples of z-knotted triangulations. Examples of z-knotted
fullerenes can be found in [7].
Suppose that Γ contains precisely k distinct zigzags up to reversing. A z-orientation
of Γ is a collection τ consisting of k distinct zigzags such that for each zigzag Z we have
Z ∈ τ or Z−1 ∈ τ . There are precisely 2k distinct z-orientations of Γ. For every z-
orientation τ = {Z1, . . . , Zk} the z-orientation τ−1 = {Z−11 , . . . , Z
−1
k } will be called
reversed to τ . The triangulation Γ with a z-orientation τ will be denoted by (Γ, τ) and
called a z-oriented triangulation.
Let τ be a z-orientation of Γ. For every edge e of Γ one of the following possibilities is
realized:
• there is a zigzag Z ∈ τ such that e occurs in this zigzag twice and other zigzags from
τ do not contain e,
A. Tyc: Z-oriented triangulations of surfaces 27
• there are two distinct zigzags Z,Z ′ ∈ τ such that e occurs in each of these zigzags
only once and other zigzags from τ do not contain e.
In the first case, we say that e is an edge of type I if Z passes through e twice in different
directions; otherwise, e is said to be an edge of type II. Similarly, in the second case: e is
an edge of type I if Z and Z ′ pass through e in different directions or e is an edge of type
II if Z and Z ′ pass through e in the same direction. In what follows, edges of type II will
be considered together with the direction defined by τ . A vertex of Γ is called of type I if it
belongs only to edges of type I; otherwise, we say that this is a vertex of type II.
The following statements hold for any z-orientation τ of Γ.
Lemma 2.3. For each vertex of type II the number of edges of type II which enter this
vertex is equal to the number of edges of type II which leave it.
Proof. The number of times that the zigzags from τ enter a vertex is equal to the number
of times that these zigzags leave this vertex.
Proposition 2.4. For every face of (Γ, τ) one of the following possibilities is realized:
(I) the face contains two edges of type I and the third edge is of type II, see Figure 2(a);
(II) all edges of the face are of type II and form a directed cycle, see Figure 2(b).
A face in a triangulation is said to be of type I or of type II if the corresponding possi-
bility is realized.
(a) (b)
Figure 2: Two types of faces.
Proof of Proposition 2.4. Consider a face whose edges are denoted by e1, e2, e3. Without
loss of generality we can assume that the zigzag containing the sequence e1, e2 belongs to
τ . Let Z and Z ′ be the zigzags containing the sequences e2, e3 and e3, e1, respectively.
Then Z ∈ τ or Z−1 ∈ τ and Z ′ ∈ τ or Z ′−1 ∈ τ . An easy verification shows that for each
of these four cases we obtain (I) or (II).
Example 2.5. If n is odd, then the bipyramid BPn has a unique z-orientation (up to re-
versing), see Example 2.2(a). The edges ai and bi, i ∈ {1, . . . , n} are of type I and the
edges on the base of the bipyramid are of type II. The vertices a, b are of type I and the
vertices on the base are of type II. All faces are of type I. The same happens for the case
when n = 2k and k is odd if the z-orientation is defined by the two zigzags presented in
Example 2.2(b); however, all faces are of type II if we replace one of these zigzags by the
reversed.
28 Ars Math. Contemp. 22 (2022) #P1.02 / 23–39
Example 2.6. Suppose that n = 2k and k is even. Let Z1, Z2, Z3, Z4 be the zigzags from
Example 2.2(c). For the z-orientation defined by these zigzags all faces are of type I. If the
z-orientation is defined by Z1, Z2 and Z−13 , Z
−1
4 , then all faces are of type II. In the case
when the z-orientation is defined by Z1, Z2, Z3 and Z−14 , there exist faces of both types.
Remark 2.7. If we replace a z-orientation by the reversed z-orientation, then the type of
every edge does not change (but all edges of type II reverse the directions), consequently,
the types of vertices and faces also do not change. For z-knotted triangulations there is a
unique z-orientation (up to reversing) and we can determine the types of edges, vertices
and faces without attaching to a z-orientation [15].
A triangulation Γ′ of M is a shredding of the triangulation Γ if it is obtained from Γ
by triangulating some faces of Γ such that all new vertices are contained in the interiors of
these faces.
Proposition 2.8. Any z-oriented triangulation admits a z-oriented shredding with all faces
of type I.
Proof. Let F be a face of type II in a z-oriented triangulation (Γ, τ) and let e1, e2, e3 be
edges of F . Suppose that the edges of F are oriented as in Figure 3 and denote by σ the
permutation (1, 2, 3).
e3 e2
e1
e3 e2
e1
e′3e
′
2
e′1
Figure 3: Triangulation of faces of type II.
Zigzags from τ passes through F precisely three times, so the face F separates them
into 3 segments of type
eσ−1(i), ei, Xij , ej , eσ(j),
where i, j ∈ {1, 2, 3} and the sequence Xij is a maximal part of a zigzag formed by edges
occuring between ei and ej . Let X be the set of all such sequences Xij for F and the
z-orientation τ . Note that every Xij ∈ X is completely determined by the beginning edge
ei and the final edge ej . Now, we triangulate the face F by adding a vertex in the interior
of F and three edges connecting this vertex with the vertices of F . We denote this new
triangulation by Γ′ and write e′i for the new edge if it does not has a common vertex with
ei (see Figure 3). Observe that for any i ∈ {1, 2, 3} there exists a zigzag in Γ′ containing a
subsequence of the form
ei, e
′
σ−1(i), e
′
i, eσ−1(i), Xσ−1(i)j
for certain j ∈ {1, 2, 3} and Xσ−1(i)j ∈ X . The edge ej which occurs in the zigzag directly
after this subsequence is the same as the edge after Xσ−1(i)j in (Γ, τ), since Xσ−1(i)j does
not contain edges of F . Therefore, zigzags of Γ′ related to the three faces not contained in
A. Tyc: Z-oriented triangulations of surfaces 29
Γ pass through the edges coming from Γ in the same way as zigzags from τ . This implies
the existence of a z-orientation of Γ′ such that all edges from Γ do not change their types
and the three new faces of Γ′ contained in F are of type I. Recursively, we eliminate all
faces of type II from (Γ, τ) and come to a z-oriented shredding of Γ with all faces of type
I and such that the type of any edge from (Γ, τ) is preserved.
3 Homogeneous zigzags in triangulations with faces of type I
In this section, we will always suppose that Γ is a triangulation with fixed z-orientation τ
such that all faces in Γ are of type I, i.e. each face contains precisely two edges of type I and
the third edge is of type II. If m is the number of faces, then there are precisely m edges
of type I and m/2 edges of type II. In other words, the number of edges of type I is twice
the number of edges of type II. We say that a zigzag of Γ is homogeneous if it is a cyclic
sequence {ei, e′i, e′′i }ni=1, where each ei is an edge of type II and all e′i, e′′i are edges of type
I. If a zigzag is homogeneous, then the reversed zigzag also is homogeneous. Denote by
ΓII the subgraph of Γ formed by all vertices and by all edges of type II.
Example 3.1. The zigzags of Γ = BPn are homogeneous if n is odd (the z-knotted case)
or n is even and the z-orientation is defined by the two zigzags from Example 2.2(b) or by
the four zigzags from Example 2.2(c). Only a and b are vertices of type I and ΓII is the
directed cycle formed by the edges of the base of the bipyramid. Conversely, if all zigzags
of Γ are homogeneous and there are precisely two vertices of type I, then Γ is a bipyramid
(this statement is an easy consequence of Theorem 3.3 which will be presented later).
Example 3.2. Let Γ′ be a triangulation of M with a z-orientation such that all faces are of
type II (see [16, Example 4] for a z-knotted triangulation of S2 whose faces are of type II).
As in the proof of Proposition 2.8, we consider the shredding Γ′′ of Γ′ which is obtained
by adding a vertex in the interior of each face and three edges connecting this vertex with
the vertices of the face. This triangulation Γ′′ admits a z-orientation such that all faces are
of type I. Every zigzag e1, e2, e3, . . . in Γ′ is extended to a zigzag
e1, e
′
1, e
′′
1 , e2, e
′
2, e
′′
2 , e3, . . .
in Γ′′ which passes through edges of Γ′ in the opposite directions. All ei are of type II and
all e′i and e
′′
i are of type I. So, all zigzags in Γ
′′ are homogeneous.
An Eulerian digraph is a connected digraph such that indegree equals outdegree for
every vertex.
Theorem 3.3. The following three conditions are equivalent:
(1) All zigzags of Γ are homogeneous.
(2) ΓII is a closed 2-cell embedding of a simple Eulerian digraph such that every face is
a directed cycle.
(3) Each connected component of M \ ΓII is homeomorphic to an open 2-dimensional
disk.
The implication (2) ⇒ (3) is obvious. The implications (1) ⇒ (2) and (3) ⇒ (1) will
be proved in Section 4 and Section 5, respectively.
30 Ars Math. Contemp. 22 (2022) #P1.02 / 23–39
4 Proof of the implication (1) ⇒ (2) in Theorem 3.3
Now, we generalize the construction described in Proposition 2.8 and Example 3.2. Let Γ′
be a closed 2-cell embedding of a connected finite simple graph in the surface M . Then all
faces of Γ′ are homeomorphic to a closed 2-dimensional disk. For each face F we take a
point vF belonging to the interior of F . We add all vF to the vertex set of Γ′ and connect
each vF with every vertex of F by an edge. We obtain a triangulation of M which will be
denoted by T(Γ′).
The assumption that our 2-cell embedding is closed cannot be omitted. Indeed, if a
certain face of Γ′ is not homeomorphic to a closed 2-dimensional disk, then there is a pair
of vertices connected by a double edge and T(Γ′) is not a triangulation in our sense.
Proposition 4.1. If all zigzags of Γ are homogeneous, then ΓII is a closed 2-cell embedding
of a simple Eulerian digraph such that every face is a directed cycle and Γ = T(ΓII).
Conversely, if Γ′ is a closed 2-cell embedding of a simple Eulerian digraph and every face
is a directed cycle, then the triangulation T(Γ′) admits a unique z-orientation such that all
zigzags of T(Γ′) are homogeneous and Γ′ is the subgraph of T(Γ′) formed by all vertices
and edges of type II.
Proof. (I): Let v be a vertex of Γ. Consider all faces containing v and take the edge on each
of these faces which does not contain v. All such edges form a cycle which will be denoted
by C(v).
Suppose that all zigzags of Γ are homogeneous and consider any edge e1 of type II. Let
v1 and v2 be the vertices of this edge such that e1 is directed from v1 to v2. We choose
one of the two faces containing e1 and take in this face the vertex v which does not belong
to e1. Let e′1 and e
′′
1 be the edges which contain v and occur in a certain zigzag Z ∈ τ
immediately after e1, see Figure 4. Denote by e2 the third edge of the face containing e′1
and e′′1 . This edge contains v2 and another one vertex, say v3. Since Z is homogeneous,
the edges e′1 and e
′′
1 are of type I, and consequently, e2 is of type II. The zigzag which
goes through e′1 from v to v2 belongs to τ (this follows easily from the fact that Z goes
through e′1 in the opposite direction and e
′
1 is an edge of type I). The latter guarantees that
the edge e2 is directed from v2 to v3. By our assumption, the edge e3 which occurs in Z
immediately after e′1 and e
′′
1 is of type II. This edge is directed from v3 to a certain vertex
v4. So, e1, e2, e3 are consecutive edges of the cycle C(v) and each ei is directed from vi
to vi+1. Consider the zigzag from τ which contains the sequence e2, e′′1 . The next edge
in this zigzag connects v and v4 (the zigzag goes from v to v4). Let e4 be the edge which
occurs in the zigzag after it. Then e4 is an edge of type II (by our assumption), it belongs
to C(v) and leaves v4. Recursively, we establish that C(v) is a directed cycle formed by
edges of type II and every edge containing v is of type I, i.e. v is a vertex of type I. Now,
v
v4
v3
v2
v1
e1
e2 e3
e4
e′1
e′′1
s
Figure 4: Cycle of edges of type II.
A. Tyc: Z-oriented triangulations of surfaces 31
we consider the other face containing e1 and take the vertex v′ of this face which does not
belong to e1. Using the same arguments, we establish that v′ is a vertex of type I and C(v′)
is a directed cycle formed by edges of type II.
For every vertex v of type I we can take a face containing v and the edge of this face
which does not contain v. This edge is of type II (since the remaining two edges of the face
are of type I). The above arguments show that the following assertions are fulfilled:
(1) vertices of type I exist and for every such vertex v the cycle C(v) is a directed cycle
formed by edges of type II;
(2) for every edge of type II there are precisely two vertices v and v′ of type I such that
this edge is contained in the cycles C(v) and C(v′).
Similarly, for every edge e of type I we take a face containing e; this face contains an edge
of type II which implies that e connects vertices of different types.
Consider ΓII. Any two vertices of type II in Γ can be connected by a path formed by
edges of type II which means that ΓII is connected. Indeed, if a path between two vertices
of type II goes through a vertex v of type I, then the edge going into v and the edge leaving
v are incident to vertices in the same cycle C(v) and so we can rewrite that part of the path
to use edges from C(v) instead of the edges through v. It is easy to see that ΓII is a 2-cell
embedding of a simple digraph such that every face is the directed cycle C(v) for a certain
vertex v of type I; in particular, this 2-cell embedding is closed. Lemma 2.3 implies that
ΓII is an Eulerian digraph. The equality Γ = T(ΓII) is obvious.
The following remark will be used to prove the second part of the theorem. The con-
ditions (1) and (2) guarantee that every zigzag of Γ containing an edge of type II is homo-
geneous. Recall that the number of edges of type I is twice the number of edges of type
II. This implies that there is no zigzag containing edges of type I only (since every edge
occurs twice in a unique zigzag from τ or it occurs ones in precisely two distinct zigzags
from τ ). Therefore, every zigzag of Γ is homogeneous if (1) and (2) hold.
(II): Suppose that Γ′ is a closed 2-cell embedding of a simple Eulerian digraph such
that every face is a directed cycle.
Let e1, . . . , en be the directed cycle formed by all edges of a certain face of Γ′. For
every i ∈ {1, . . . , n} we define j(i) = i + 2(modn) and denote by e′i and e′′i the edges
containing the vertex vF in T(Γ′) and intersecting ei and ej(i), respectively. Consider the
zigzag of T(Γ′) which contains the sequence ei, e′i, e
′′
i , ej(i). It passes through ei and ej(i)
according to the directions of these edges; and the same holds for every edge of Γ′ which
occurs in this zigzag. Such a zigzag exists for any pair formed by a face of Γ′ and an
edge on this face. The collection of all such zigzags is a z-orientation of T(Γ′) with the
following properties: all edges of Γ′ are of type II and every vF is a vertex of type I. This
implies that T(Γ′) satisfies the conditions (1) and (2) which gives the claim.
Note that the second part of Proposition 4.1 will be used to prove the implication
(3) ⇒ (1).
5 Structure of triangulations with faces of type I
In this section, we describe some structural properties of z-oriented triangulations with
faces of type I. As an immediate consequence we obtain the implication (3) ⇒ (1).
32 Ars Math. Contemp. 22 (2022) #P1.02 / 23–39
As above, we suppose that (Γ, τ) is a z-oriented triangulation of M , where all faces
are of type I. As above, we denote by ΓII the subgraph of Γ consisting of all vertices
and all edges of type II. From the previous section it follows that if the zigzags of (Γ, τ)
are homogeneous, then connected components of M \ ΓII are homeomorphic to open 2-
dimensional disks. Now, we describe the general case.
Theorem 5.1. The following assertions are fullfiled:
(1) Connected components of M \ΓII are homeomorphic to an open 2-dimensional disk,
an open Möbius strip or an open cylinder.
(2) A connected component of M \ ΓII contains a vertex of type I if and only if it is an
open 2-dimensional disk; such a vertex of type I is unique.
Proof. Consider two distinct edges e0 and e1 of type I contained in a certain face F1.
There is precisely one face containing e1 and distinct from F1. Denote this face by F2 and
write e2 for the other edge of type I on F2. Recursively, we construct sequences of edges
{ei}i∈N∪{0} and faces {Fi}i∈N such that ei−1 is the common edge of Fi−1, Fi for every
i ∈ N. For any pair of the faces Fi−1, Fi we distinguish the following two cases presented
in Figure 5. In the first case, the edges of type II of Fi−1 and Fi have a common vertex
(Figure 5(a)). In the second case (Figure 5(b)), the edges of type II are disjoint.
ei−2
ei−1
ei
Fi−1 Fi
(a)
Fi
ei−2
ei−1
ei
Fi−1
(b)
Figure 5: Two possibilities of adjacency for faces of type I.
Let n be the smallest natural number such that en = e0 (such a number exists by
finiteness). Therefore, the above sequences can be considered as cyclic sequences {ei}ni=1
and {Fi}ni=1. The union F =
⋃n
i=1 Fi will be called a component of (Γ, τ). The boundary
of F consists of (not necessarily all) edges of type II belonging to faces Fi.
Denote by eIIi the edge of type II belonging to Fi. We take n disjoint closed triangles
T1, T2, . . . , Tn. For any i = 1, 2, . . . , n there is a homeomorphism hi : Fi → Ti transfer-
ring any vertex and any edge of Fi to a vertex and an edge of Ti, respectively. We identify
hi(ei) and hi+1(ei) for any i in such a way that for every vertex v of ei the vertices hi(v)
and hi+1(v) are identified. We get a 2-dimensional surface T with boundary. The bound-
ary of T is the union of the images of all edges of type II, i.e. ∂T =
⋃n
i=1 hi(e
II
i ). Note
that F is not necessarily a surface (since it is possible that for distinct i, j the edges eIIi , eIIj
have a common vertex). The interior of surface T is homeomorphic to one of the con-
nected components of M \ ΓII and F can be obtained from T by gluing of some parts of
the boundary.
Suppose that hi(ei) and hi+1(ei) are identified only for i = 1, 2, . . . , n − 1 (but not
h1(e0) and hn(en) from T1 and Tn, respectively). Then we get a space homeomorphic to
A. Tyc: Z-oriented triangulations of surfaces 33
a closed 2-dimensional disk whose boundary contains h1(e0), hn(en). Now, to complete
the construction of T , we have to glue h1(e0) and hn(en). Precisely one of the following
possibilities is realized:
• A union of these sides is connected and by gluing of them we obtain that T is home-
omorphic to a closed 2-dimensional disk (Figure 6(1)).
• The sides are disjoint and by identification of them we get a surface homeomorphic
to a closed Möbius strip (Figure 6(2)) or a closed cylinder (Figure 6(3)).
h1(e0) hn(en)
...
(1)
h1(e0) . . . hn(en)
(2)
h1(e0) . . . hn(en)
(3)
Figure 6: Closed disk, closed Möbius strip and closed cylinder.
Let vi be the vertex of Ti corresponding to the vertex of Fi not belonging to the edge
eIIi . In the first case, the images of edges of type I have the common vertex which is the
image of all hi(vi); it is clear that this vertex corresponds to the vertex of type I from F ,
see Figure 6(1). In the remaining cases, any vertex hi(vi) is contained in the boundary of
T and correspond to a certain vertex of ΓII (see Figure 6(2) and 6(3)). So, we obtained the
statements (1) and (2).
If a connected component of M \ ΓII is homeomorphic to an open 2-dimensional disk,
then the corresponding component of (Γ, τ) is homeomorphic to a closed 2-dimensional
disk (if this component has some identifications at the boundary, then the vertex of type I
in this component is joined by a double edge to a certain vertex at the boundary which is
impossible, since we work with embeddings of simple graphs).
Proof of (3) ⇒ (1) in Theorem 3.3. Assume that each connected component of M \ΓII is a
disk. By the above remark, ΓII is a closed 2-cell embedding. Lemma 2.3 shows that this is
an embedding of simple Eulerian digraph. The second part of Theorem 5.1 states that each
34 Ars Math. Contemp. 22 (2022) #P1.02 / 23–39
disk contains a unique vertex of type I; as in the proof of Theorem 5.1 we establish that its
boundary is an oriented cycle. We have Γ = T(ΓII) and the second part of Proposition 4.1
gives the claim.
The following three examples show that all possibilities for connected components of
M \ ΓII are realized.
Example 5.2. Consider the following triangulation Γ of a torus T = S1×S1 (see Figure 7).
The triangulation Γ admits the z-orientation such that all faces are of type I. The subgraph
ΓII has two connected components which are 6-cycles and T\ΓII consists of two connected
components homeomorphic to open cylinders (−1, 1)× S1.
3
45
0
1 2
0
12
3
4 5
Figure 7: Toric triangulation.
In a similar way, we can construct a z-oriented toric triangulation with connected com-
ponents of T \ ΓII which are open cylinders of arbitrary length.
Example 5.3. Let n ∈ N and let Γ be the triangulation of a real projective plane obtained
by gluing of boundaries of a Möbius strip and a closed 2-dimensional disk (see Figure 8).
According to the corresponding z-orientation all faces are of type I and the graph ΓII con-
sists of all edges marked by the double arrows and their vertices. Then RP2 \ ΓII has two
connected components. One of them is homeomorphic to an open 2-dimensional disk and
the remaining to an open Möbius strip.
Example 5.4. Suppose that Γ is the triangulation of a sphere obtained by the gluing of
the two disks whose boundaries are cycles e1, e2, . . . , e6 (see Figure 9). There is a z-
orientation τ such that all faces are of type I. Then S2 \ ΓII has precisely four connected
components: three components are homeomorphic to an open 2-dimensional disk and the
remaining to an open cylinder. The components of (Γ, τ) corresponding to the first three
connected components are closed 2-dimensional disks. The fourth component of (Γ, τ) is
homeomorphic to a closed cylinder S1 × D1, where two points at one of the connected
components of the boundary are glued.
A. Tyc: Z-oriented triangulations of surfaces 35
. . .
n+1 n+2 n+3 n+4 2n
e e
1 2 3 4 n
4
3
2
1
2nn+4
n+3
n+2
n+1
n
...
...
Figure 8: Projective plane triangulation.
e4
e5
e6
e1
e2
e3
e4
e5
e6
e1
e2
e3
Figure 9: Spherical triangulation.
6 Relations to z-monodromies
Let F be a face in Γ whose vertices are a, b, c. Consider the set Ω(F ) of all oriented edges
of F
Ω(F ) = {ab, bc, ca, ac, cb, ba},
where xy is the edge from x ∈ {a, b, c} to y ∈ {a, b, c}. If e = xy then we write yx by
−e. Denote by DF the following permutation of the set Ω(F )
(ab, bc, ca)(ac, cb, ba).
36 Ars Math. Contemp. 22 (2022) #P1.02 / 23–39
The z-monodromy (see [16, 17]) of the face F is the permutation MF defined as follows.
For any e ∈ Ω(F ) we take e0 ∈ Ω(F ) such that DF (e0) = e and consider the zigzag
containing the sequence e0, e. We define MF (e) as the first element of Ω(F ) contained in
this zigzag after e.
By [17, Theorem 4.4], there are the following possibilities for the z-monodromy MF
and each of them is realized:
(M1) MF is the identity,
(M2) MF = DF ,
(M3) MF = (−e1, e2, e3)(−e3,−e2, e1),
(M4) MF = (e1,−e2)(e2,−e1), where e3 and −e3 are fixed points,
(M5) MF = (DF )−1,
(M6) MF = (−e1, e3, e2)(−e2,−e3, e1),
(M7) MF = (e1, e2)(−e1,−e2), where e3 and −e3 are fixed points,
where (e1, e2, e3) is one of the cycles in DF .
Let Gi be the subgraph of the dual Γ∗ formed by vertices corresponding to faces in Γ
whose z-monodromies are of type (Mi), two vertices of Gi are adjacent if they are adjacent
in Γ∗. By [16, Theorem 1], the subgraphs G1 and G2 are forests. For (M3), (M4), (M5)
and (M7) the above statement fails: z-monodromies of all faces of the bipyramid BPn are
of type
• (M3) for n = 2k + 1 where k is odd,
• (M4) for n = 2k + 1 where k is even,
• (M7) for n = 2k where k is odd,
• (M5) for n = 2k where k is even.
Proposition 6.1. If MF is (M6), then F is of type I for any z-orientation of Γ.
Proof. Let e1, e2, e3 be consecutive oriented edges of the face F . We suppose that the
z-monodromy of F is (M6), i.e.
MF = (−e1, e3, e2)(−e2,−e3, e1).
There are precisely two zigzags (up to reversing) which contain F
e1, e2, . . . ,−e1,−e3, . . . and e2, e3, . . . ;
since the edge corresponding to the pair {e1,−e1} is passed in two different directions by
the same zigzag, then it is of type I for any orientation of the zigzag. Therefore, F is of
type I for any z-orientation.
Lemma 6.2. Let F be a face in (Γ, τ) such that there are precisely two zigzags from τ
which contain edges from F . Then the following assertions are fulfilled:
A. Tyc: Z-oriented triangulations of surfaces 37
(1) There is a unique edge e ∈ F which occurs in one of these zigzags twice.
(2) The type of e does not depend on the choice of z-orientation.
(3) If e is of type I, then MF is (M6). If e is of type II, then MF is (M7).
Proof. (1): Any face occurs precisely thrice, as a pair of its adjacent edges, in zigzags from
the z-orientation τ . By the assumption, there are precisely two zigzags from τ which pass
through our face. This is possible only when one of these zigzags passes through it once
and the second twice.
(2): The edge e can occur in the same zigzag twice in two ways: the zigzag passes
through e the first time in one of directions and the second time in the opposite (type I) or
the zigzag passes through e twice in the same direction (type II). It is easy to see that the
type of e is the same for any z-orientation of Γ.
(3): By [17, Remark 4.9] the z-monodromy of the face F is (M6) or (M7). In the case
(M6) the statement follows from Proposition 6.1. Let e1, e2, e3 be consecutive edges of F
and MF be of type (M7), i.e.
MF = (e1, e2)(−e1,−e2).
In this case, F occurs twice in the zigzag
e2, e3, . . . , e3, e1, . . .
and e3 is of type II for any z-orientation of Γ.
Now, we can construct a class of toric triangulations, where z-monodromies are of type
(M6) for all faces. Our arguments are based on Lemma 6.2.
Example 6.3. Let n,m be odd numbers not less than 3 and let Γ0 be a n ×m grid where
the opposite sides are identified. Then Γ0 can be embedded into a torus in the natural way.
Suppose that Γ = T(Γ0) (see Figure 10 for the case n = m = 3).
e4 e5 e6
e4 e5 e6
e1
e2
e3
e1
e2
e3
Figure 10: Toric triangulation related to 3× 3 grid.
Each zigzag of Γ determines a band formed by n or m squares from the grid (see
Figure 11 for a band consisting of 5 squares) and passes through each face of this band
twice.
38 Ars Math. Contemp. 22 (2022) #P1.02 / 23–39
Figure 11: Band consisting of 5 squares.
Observe that the edges common for two consecutive squares from the grid are passed
twice (they marked on Figure 11 by the bold line) and are of type I for any z-orientation.
Remaining edges are passed by the zigzag once. Therefore, all edges of subgraph Γ0 are
of type I and all faces of Γ are of type I for any z-orientation. It is clear that any edge
incident to a vertex in the interior of a square occurs once in two different zigzags. Thus,
for any face of Γ there are precisely two zigzags which pass it. Lemma 6.2 guarantees that
z-monodromies of all faces of Γ are (M6).
ORCID iDs
Adam Tyc https://orcid.org/0000-0003-4870-5731
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P1.03 / 41–55
https://doi.org/10.26493/1855-3974.1988.220
(Also available at http://amc-journal.eu)
The cubical matching complex revisited*
Duško Jojić
University of Banja Luka, Faculty of Science,
Mladena Stojanovića 2, 78 000 Banja Luka, Bosnia and Herzegovina
Received 15 April 2019, accepted 29 April 2021, published online 4 April 2022
Abstract
Ehrenborg noted that all tilings of a bipartite planar graph are encoded by its cubical
matching complex and claimed that this complex is collapsible. We point out to an over-
sight in his proof and explain why these complexes can be the disjoint union of two or more
collapsible complexes. We also prove that all links in these complexes are suspensions up
to homotopy. Furthermore, we extend the definition of a cubical matching complex to
planar graphs that are not necessarily bipartite, and show that these complexes are either
contractible or a disjoint union of contractible complexes. For a simple connected region
that can be tiled with dominoes (2×1 and 1×2) and 2×2 squares, let fi denote the number
of tilings with exactly i squares. We prove that f0 − f1 + f2 − f3 + · · · = 1 (established
by Ehrenborg) is the only linear relation for the numbers fi.
Keywords: Domino tilings, independence complexes, matching, cubical complexes.
Math. Subj. Class. (2020): 52C20, 05C70
1 Introduction
Let G = (V,E) be a bipartite planar graph that admits a perfect matching. Assume that G
is embedded in the plane. This embedding splits the plane into the regions, the connected
components of R2 \ |G| (here |G| denotes the embedding of G into R2). An elementary
cycle of G is a cycle that encircles a single region R different from the outer region R∗.
Throughout this paper, we identify an elementary cycle with the region it encircles as well
as with its set of vertices or edges.
A tiling of G is a partition of the vertex set V into disjoint blocks of the following two
types:
(1) an edge {x, y} of G; or
*We are very grateful to the referee for numerous remarks and suggestions which considerably improved the
presentation of results in the paper.
E-mail address: dusko.jojic@pmf.unibl.org (Duško Jojić)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
42 Ars Math. Contemp. 22 (2022) #P1.03 / 41–55
(2) an elementary cycle R (the set of vertices of R).
The set of all tilings of G form a cubical complex C(G) (called the cubical matching
complex) defined by Ehrenborg in [6]. Note that C(G) depends not only on G, but also on
the choice of the embedding of that graph in the plane.
A face F of C(G) has the form F = MF ∪CF = (MF , CF ), where CF is a collection
CF = {R1, R2, . . . , Rt} of vertex-disjoint elementary cycles of G, and MF is a perfect
matching on G \
(
R1 ∪ R2 ∪ · · · ∪ Rt
)
. The dimension of F is |CF |, and the vertices of
C(G) are the perfect matchings of G.
All tilings of G covered by F = (MF , CF ) can be obtained by deleting an elementary
cycle R from CF , and adding every other edge of R into MF (there are two possibilities to
do this). Therefore, for two faces F1 = (MF1 , CF1) and F2 = (MF2 , CF2), we have that(
F1 ⊂ F2
)
⇐⇒
(
CF1 ⊂ CF2 and MF1 ⊃ MF2
)
. (1.1)
Let G◦ denote the weak dual graph of a planar graph G. The vertices of G◦ are all
bounded regions of G, and two regions that share a common edge are adjacent in G◦.
The independence complex of a graph H is a simplicial complex I(H) whose faces are
the independent subsets of vertices of H . Note that for any face F = (MF , CF ) of C(G),
the set CF contains independent vertices of G◦, i.e., CF is a face of I(G◦).
Since two elementary cycles of G sharing a common edge cannot be in a common face
of C(G), it may seem at the first glance that C(G) can be computed from the independence
complex I(G◦).
A B C
G1 G2 G3
C(G1) C(G2) C(G3)
A B C A
B
C
AC
B
A
B
C
AC
Figure 1: The three graphs with the same weak dual, but different cubical matching com-
plexes.
However, Figure 1 shows three graphs with the same weak dual but different cubical
matching complexes. The facets of the complexes on Figure 1 are labeled by corresponding
subsets of pairwise disjoint elementary regions. This example points out that the require-
ment that G \ (R1 ∪ · · · ∪Rt) admits a perfect matching is a substantial one.
D. Jojić: The cubical matching complex revisited 43
Example 1.1. Let Ln and Cn denote the independence complexes of Pn and Cn (the path
and cycle with n vertices) respectively. The homotopy types of these complexes are deter-
mined by Kozlov in [10]:
Ln ≃
{
a point , if n = 3k + 1;
S⌊
n−1
3 ⌋, otherwise.
Cn ≃
{
Sk−1, if n = 3k ± 1;
Sk−1 ∨ Sk−1, if n = 3k.
We will use these complexes later, see Corollary 2.2 and Remark 2.5. An interested reader
can find more details about combinatorial and topological properties of Ln and Cn (and
about independence complexes in general) in [7, 8] and [9].
There are some cubical complexes that cannot be realized as subcomplexes of the d-
cube Cd = [0, 1]d, see Chapter 4 of [5].
Proposition 1.2. Let G be a bipartite planar graph that has a perfect matching. If G has
d elementary regions, then its cubical matching complex C(G) can be embedded into Cd.
Proof. We use an idea from [12] to describe the coordinates of vertices of C(G) explicitly.
Let R1, R2, . . . , Rd be a fixed linear order of elementary regions of G. We choose an
arbitrary perfect matching M0 of G (a vertex of C(G)) to be the origin 0 = (0, 0, . . . , 0) in
Rd. For another vertex M of C(G) the symmetric difference M △ M0 is a disjoint union of
cycles. Now, to a given perfect matching M of G, we assign the vertex VM = (x1, . . . , xd)
of Cd, where
xi =
{
1, if Ri is contained in an odd number of cycles of M △ M0;
0, otherwise.
If M ′ and M ′′ are two perfect matchings of G such that M ′ △ M ′′ = Rj (meaning
that these two matchings differ just on an elementary region Rj), then their corresponding
vertices VM ′ and VM ′′ of Cd differ only at the j-th coordinate.
Therefore, all 1-dimensional faces of C(G) that correspond to the same region Ri are
mutually parallel edges of Cd. The face F = (MF , CF ) of C(G) is embedded in Cd as the
convex hull of its 2|C(F )| vertices.
2 The local structure of C(G)
The star of a face F in a cubical complex C is the set of all faces of C that contain F
star(F ) = {F ′ ∈ C : F ⊂ F ′}.
The link of a vertex v in a cubical complex C is the simplicial complex linkC(v) that can
be realized in C as a “small sphere” around the vertex v. More formally, the vertices of
linkC(v) are the edges of C containing v. A subset of vertices of linkC(v) is a face of
linkC(v) if and only if the corresponding edges belong to a common face of C.
The link of a face F in a cubical complex C is defined in a similar way. The set of
vertices of linkC(F ) is
{F ′ ∈ C : F ⊂ F ′ and dimF ′ = 1 + dimF},
and a subset A of the set of vertices is a face of linkC(F ) if and only if all elements of A
are contained in a same face of C.
44 Ars Math. Contemp. 22 (2022) #P1.03 / 41–55
Ehrenborg investigated the links of the cubical complexes associated to tilings of a
region by dominos or lozenges.
Here we describe the links in the cubical matching complex C(G) for any bipartite
planar graph G. For a face F = (MF , CF ) of C(G), let RF denote the set of all elementary
regions of G for which every second edge is contained in MF . Further, let GF denote the
subgraph of the weak dual graph G◦ induced with the regions from RF .
From the definition of the link in a cubical complex and (1.1), we obtain the next state-
ment.
Proposition 2.1. For any face F = (MF , CF ) of C(G) we have that
linkC(F ) ∼= I(GF ).
The above proposition explains the appearance of complexes Ln and Cn as links in
cubical the matching complexes, see Theorem 3.3 and Section 4 in [6].
Assume that all elementary regions of G are quadrilaterals. In that case, for any face
F of C(G), a region in RF has exactly two opposite edges in MF , and the degree of a
vertex in GF is at most two. Therefore, GF is a disjoint union of paths and cycles. If
the regions (quadrilaterals) R1, R2, . . . , Rt are vertices of a cycle in GF , then the edges of
these regions that are not in MF form two cycles of length t in G. As G is bipartite, the
length of any cycle in GF is even.
Corollary 2.2. If all elementary regions of G are quadrilaterals, then linkC(F ) is a join
of complexes Lp and C2q .
Theorem 2.3. Let G be a bipartite planar graph that has a perfect matching. For any face
F = (MF , CF ) of C(G) the graph GF is bipartite.
Proof. Assume that GF contains an odd cycle R1, R2, . . . , R2m+1. Recall that Ri is an
elementary region of G and the that every second edge of Ri is contained in MF . Two
neighborly regions Ri−1 and Ri have to share the odd number of edges, the first and the
last of their common edges belong to MF . Therefore, for each region Ri, there is an odd
number of common edges of Ri and Ri−1 that belong to MF . Obviously, the same holds
for Ri and Ri+1.
So, we can conclude that there is an odd number of edges of Ri that are between
Ri∩Ri−1 and Ri∩Ri+1 (the first and the last one of these edges are not in MF ). Therefore,
the union of these edges (for all regions Ri) forms an odd cycle in G. This is a contradiction
with the assumption that G is a bipartite graph.
Barmak proved in [1] (see also in [11]) that the independence complexes of bipartite
graphs are suspensions up to homotopy type. This implies the next result.
Corollary 2.4. All links in C(G) are homotopy equivalent to suspensions. Therefore, the
link of any face in C(G) has at most two connected components.
For any simplicial complex K there exists a bipartite graph G such that the indepen-
dence complex of G is homotopy equivalent to the suspension over K, see [1]. Skwarski
proved in [13] (see also [1]) that there exists a planar graph G whose independence complex
is homotopy equivalent to an iterated suspension of K.
We prove that the links of faces in cubical matching complexes are independence com-
plexes of bipartite planar graphs. What can be said about homotopy types of these com-
plexes?
D. Jojić: The cubical matching complex revisited 45
Remark 2.5. There is a natural question, posed by Ehrenborg in [6]: For what graphs G
would the cubical matching complex C(G) be pure and/or shellable? The complexes Ln
are non-pure for n > 4, and the complexes Cn are non-shellable for n > 5. Therefore, these
complexes can be used to show that the cubical matching complex of a concrete graph is
non-pure or non-shellable.
3 Collapsibility and contractibility of cubical matching complexes
The next statement that we discuss was the main result of [6]. We identify a problem with
the proof, describe counterexamples (infinitely many), recover a weaker result, and give a
generalization.
Theorem 3.1 (Theorem 1.2 in [6]). For a planar bipartite graph G that has a perfect
matching, the cubical matching complex C(G) is collapsible.
The proof of the above statement is based on the following two results:
(i) (Propp, Theorem 2 in [12]) The set of all perfect matchings of a bipartite planar
graph is a distributive lattice (under a certain ordering, the details of which may be
found in [12]).
(ii) (Kalai, see in [14], Solution to Exercise 3.47 c) The cubical complex associated (see
[14]) to a meet-distributive lattice is collapsible.
Note however that Propp in his proof of (i) assumed the following two additional con-
ditions for bipartite planar graph G:
(∗) Graph G is connected, and
(∗∗) Any edge of G is contained in some matching of G but not in others.
The next statement is the correct version of Theorem 3.1.
Theorem 3.2. For a connected planar bipartite graph G that has a perfect matching and
whose any edge is contained in some matching of G but not in others, the cubical matching
complex C(G) is collapsible.
G
C(G)
Figure 2: A bipartite planar graph G for which C(G) is not collapsible.
46 Ars Math. Contemp. 22 (2022) #P1.03 / 41–55
Example 3.3. The figure below shows a bipartite planar graph whose cubical matching
complex is not collapsible. Note that the subdividing of the edges between the inner and
outer quadrilaterals in Figure 2 gives us an infinite family of counterexamples for Theo-
rem 3.1. Also, we can use this example to obtain a graph whose cubical complex has ar-
bitrarily many connected components. Simply, we continue by inserting a new square into
the smallest quadrilateral of the already constructed graph, and connect two non-adjacent
vertices of the new square with the corresponding vertices of the old graph. This counterex-
ample is motivated by the Jockusch example (page 27 in [12]). In his example we find a
bipartite planar graph with 20 edges, but just 12 of them can be used in a perfect matching,
and its cubical matching complex is a disjoint union of four segments.
Now we prove a weaker version of Theorem 3.1.
Theorem 3.4. For a planar bipartite graph G that has a perfect matching, the cubical
matching complex C(G) is either collapsible or a disjoint union of collapsible complexes.
The proof will be established in a series of lemmas. Through these lemmas we assume
that G is a planar bipartite graph that has a perfect matching.
The edges that do not appear in any perfect matching of G (the forbidden edges) can
be deleted. Also, if the edge xy is a forced edge (xy appears in all perfect matching of G),
then we may consider the graph G− {x, y}.
e e e
Figure 3: If a new region can be included in a tiling of G− e, then e is not forbidden.
Lemma 3.5. Let e denote a forbidden edge in G and let G′ = G − e. The possible new
elementary region of G′, that appears after we delete e, can not be included in a tiling of
G′.
Proof. Assume that a new region R that contains e can be included in a tiling of G′. Then
e divides R into two regions of the old graph G, and we can find a perfect matching of G
that contains e, see Figure 3.
In a similar way we may prove that a new region appearing after deleting the vertices
of a forced edge can not be included in a tiling of new graph.
Corollary 3.6. Let G denote the graph obtained by deleting all forced and forbidden edges
from G. Then the cubical matching complexes of G and G are isomorphic.
Lemma 3.7. Assume that G is a not connected, and let G1, G2, . . . , Gk be the connected
components of G. If these components are separated (there is no component of G that is
D. Jojić: The cubical matching complex revisited 47
contained in an elementary region of another component) then C(G) ∼= C(G1)× C(G2)×
· · · × C(Gk).
Proof. For a tiling F = (MF , CF ) of G let Fi = (Mi, Ci) denote the corresponding face
of C(Gi) (i.e., Mi = MF ∩E(Gi) and Ci is the set of regions of Gi that are included into
CF ). Then we have that F ∼= F1 × F2 × · · · × Fk.
Lemma 3.8. Assume that G has two different connected components G1 and G2 such that
G1 is contained in an elementary region R of G2. Then we have that
C(G) = C(G1)× (C(G2) \ {R}) . (3.1)
If there exists a tiling of G2 that uses the region R, then C(G) is a disjoint union of col-
lapsible complexes.
Here C(G2) \ {R} denotes the cubical complex obtained from C(G2) by deleting the
faces (tilings) that contain R.
Proof. The proof of (3.1) is the same as in the previous lemma. Recall that C(G2) can
be embedded in a cube, and that the edges corresponding to R are mutually parallel, see
Proposition 1.2. Therefore, C(G2) \ {R} is a disjoint union of collapsible complexes.
Now, we consider the cubical matching complex for all planar graphs that have a perfect
matching (but that are not necessarily bipartite).
Definition 3.9. Let G be a planar graph that admits a perfect matching. A tiling of G is a
partition of the vertex set V into disjoint blocks of the following two types:
• an edge {x, y} of G; or
• the set of vertices {v1, v2, . . . , v2m} of an even elementary cycle R.
Let C(G) denote the set of all tilings of G. Note that C(G) is also a cubical complex.
Example 3.10. If G is a graph of a triangular prism (embedded in the plane so that the
outer region is a triangle), then C(G) is a union of three 1-dimensional segments that share
the same vertex, see the left side of Figure 4. Each segment of C(G) corresponds to a
rectangle of prism. The link of the common vertex of these segments is a 0-dimensional
complex with three points. This situation, where a link has 3 connected components, is not
possible in a bipartite planar graph, as shown by Corollary 2.4. Further, the planar graph on
the right-hand side on Figure 4 satisfies the conditions (∗) and (∗∗), but the corresponding
cubical complex is not collapsible, it is a union of three disjoint edges. Therefore, the
assumption that G is a bipartite graph is substantial in Theorem 3.2.
The next theorem describe the homotopy type of the cubical matching complex associ-
ated to a planar graph that admits a perfect matching.
Theorem 3.11. Let G be a planar graph that has a perfect matching. The cubical complex
C(G) is contractible or a disjoint union of contractible complexes.
While contractibility is weaker than collapsibility, we partly relax the bipartite condi-
tion and obtain a weaker version of a corrected Theorem 3.1, with a different proof.
48 Ars Math. Contemp. 22 (2022) #P1.03 / 41–55
Figure 4: Non-bipartite graphs and their cubical matching complexes.
Proof. We use induction on the number of edges of G. Let e = xy denote an edge that
belongs to the outer region R∗. Let R ̸= R∗ denote the elementary region that contains e.
If R is an odd region, then all tilings of G can be divided into two disjoint classes:
(a) The tilings of G that do not use e. These tilings are just the tilings of G \ e.
(b) The tilings of G that contain e as an edge in a partial matching correspond to the
tilings of G \ {x, y}, and the subcomplex of C(G) generated by these tilings is iso-
morphic to C(G \ {x, y}).
In that case we obtain that C(G) = C(G\{x, y})∪C(G\e) is a disjoint union of contractible
complexes by inductive assumption.
If R is an even elementary region, then some tilings of G may contain R, and these
tilings are not considered in (a) and (b). Note that there is a bijection between tilings of G
that contain R and all tilings of G \ R (the graph obtained from G by deleting all vertices
from R). The subcomplex of C(G) generated by tilings that contain R forms a prism over
C(G \ R), i.e., this subcomplex is isomorphic to Prism(C(G \ R)) = C(G \ R) × [0, 1].
Therefore, we obtain that
C(G) = C(G \ {x, y}) ∪ C(G \ e) ∪ Prism(C(G \R)). (3.2)
Let Ce denote the subcomplex of C(G\e) formed by all tilings that contain every second
edge of R (but do not contain e, obviously). Further, let Cx,y denote the subcomplex of
C(G \ {x, y}), defined by tilings that contain every second edge of R (these tilings have to
D. Jojić: The cubical matching complex revisited 49
contain e). Note that both of complexes Ce and Cx,y are isomorphic to C(G \R), and
C(G \ e) ∩ Prism(C(G \R)) = Ce and C(G \ {x, y}) ∩ Prism(C(G \R)) = Cx,y.
The complexes on the right-hand side of (3.2) are disjoint unions of contractible com-
plexes by the inductive hypothesis. Assume that
C(G \ {x, y}) = A1 ∪A2 ∪ · · · ∪As and Cx,y = B1 ∪B2 ∪ · · · ∪Bt,
where Ai and Bj denote the contractible components of corresponding complexes. Obvi-
ously, each complex Bj is contained in some Ai. Now, we need the following lemma.
Lemma 3.12. Each connected component of C(G\{x, y}) contains at most one component
of Cx,y .
Proof of Lemma 3.12. Assume that a component of C(G\{x, y}) contains two components
of Cx,y . In that case, there are two vertices of Cx,y (perfect matchings of G that contain
xy) that are in different components of Cx,y , but in the same component of C(G \ {x, y}).
Assume that M ′ and M ′′ are two such vertices, chosen so that the distance between them
in C(G \ {x, y}) is minimal. Let
M ′ = M0
R0 M1 . . . Mi
Ri Mi+1 . . . Mn
Rn Mn+1 = M
′′ (3.3)
denote the shortest path from M ′ to M ′′ in C(G \ {x, y}). The perfect matching Mi+1 is
obtained from Mi by removing the edges of Mi contained in an elementary region Ri, and
by inserting the complementary edges. In other words, we have that Mi+1 = Mi △ Ri, for
an elementary region Ri contained in RFi ∩RFi+1 .
Note that R0 must be adjacent (share a common edge) with R. Otherwise, both of
vertices M0 and M1 belong to the same component of Cx,y , and we obtain a contradiction
with the assumption that the distance between M ′ and M ′′ is minimal.
In a similar way, we obtain that for any i = 1, 2, . . . , n, the region Ri must be adjacent
with at least one of regions R,R0, R1, . . . , Ri−1. If not, we have that the perfect matching
M = M0 △ Ri belongs to Cx,y , and M and M ′ are contained in the same component of
Cx,y . In that case we obtain a contradiction, because the path
M = M0
R0 M1 . . . M i−1
Ri−1 M i+1
Ri+1 . . . Mn
Rn Mn+1 = M
′′
is shorter than (3.3). Here we let that M j+1 = M j △ Rj .
Let e′ denote a common edge of regions R0 and R that is contained in M ′. Note that
e′ is not contained in M1. However, this edge is again contained in M ′′, and we conclude
that the region R0 has to reappear again in (3.3).
Let Ri0 = R0 denote the first appearance of R0 in (3.3) after the first step. There are
the following three possible situations that enable the reappearance of R0:
(a) The regions R1, R2, . . . , Ri0−1 are disjoint with R0.
In that case, we can omit the steps in (3.3) labelled by R0 and Ri0 , and obtain a
shorter path between M ′ and M ′′.
(b) Each of regions that shares at least one edge with R0 appears an odd number of times
between R0 and Ri0 .
This is impossible, because R (that share an edge with R0) can not appear in (3.3).
50 Ars Math. Contemp. 22 (2022) #P1.03 / 41–55
(c) There is t < i0 such that the region Rt = R̄ shares an edge with R0, but the fragment
of the sequence (3.3) between R0 and Ri0 does not contain all region that shares an
edge with R0.
Then the same region R̄ has to appear again as Rs, for some s such that t < s < i0.
Again, if all regions Rj are disjoint with R̄ (for j = t + 1, . . . , s − 1), we can omit
Rt and Rs, and obtain a contradiction. If not, there exist indices t′ and s′ such that
t < t′ < s′ < s and R′t = R
′
s. We continue in the same way, and from the finiteness
of the path, obtain a shorter path than (3.3).
Proof of Theorem 3.11, continued: We built C(G) by starting with C(G \ e), that is a
disjoint union of contractible complexes by assumption. Then we glue the components of
Prism(C(G \R)) one by one.
After that, we glue all components of C(G \ {x, y}). At each step we are gluing
two contractible complexes along a contractible subcomplex, or we just add a new con-
tractible complex, disjoint with previously added components. From the Gluing Lemma
(see Lemma 10.3 in [4]) we obtain that C(G) is contractible, or a disjoint union of con-
tractible complexes.
Corollary 3.13. If G has two odd elementary regions that share the same edge e = xy,
then its cubical complex C(G) = C(G \ {x, y}) ∪ C(G \ e) has at least two connected
components. The same holds if there is an odd elementary region of G that shares an edge
with the outer region R∗.
4 The f -vector of domino tilings
The concept of tilings of a bipartite planar graph generalizes the notion of domino tilings.
Let R be a simple connected region, compound of unit squares in the plane, that can be
tiled with domino tiles 1×2 and 2×1. The set of all tilings of R by domino tiles and 2×2
squares defines a cubical complex, denoted by C(R). If we consider R as a planar graph
(all of its elementary regions are unit squares), and if G denotes the weak dual graph of R
(the unit squares of R are vertices of G), then C(R) is isomorphic to the cubical matching
complex C(G), see Section 3 in [6] for details. Note that the number of i-dimensional faces
of C(G) counts the number of tilings of R with exactly i squares 2× 2.
Ehrenborg used collapsibility of C(G) to conclude (see Corollary 3.1. in [6]) that the
entries of f -vector of f(C(G)) = (f0, f1, . . . , fd) satisfy
f0 − f1 + f2 − · · ·+ (−1)dfd = 1. (4.1)
Let G denote the weak dual graph of a region R that admits a domino tiling. Choose
a concrete perfect matching M of G, and let e = xy denote the edge in M that contains
the vertex (square) in the left corner of the top row of R. The complex C(G \ {x, y}) is
nonempty and contractible by induction. The simple connectivity of R implies that the
other two complexes that appear on the right-hand side of the relation (3.2) are either both
empty or contractible (by induction). If both of these complexes are nonempty, when we
glue them as in the proof of Theorem 3.11, we obtain that C(G) is contractible. So, we
conclude that the relation (4.1) is true in any case, disregarding possible problems with
Theorem 3.1.
In this section we will prove that (4.1) is the only linear relation for f -vectors of cubical
complexes of domino tilings. We follow the idea from [2], where Bayer and Billera deter-
mine the affine span of the flag f -vectors of polytopes by constructing polytopes whose
D. Jojić: The cubical matching complex revisited 51
flag f -vectors are affinely independent. Here we describe d + 1 simple connected regions
whose cubical complexes are d-dimensional and their f -vectors are affinely independent.
For all n ∈ N, we let Gn denote the following graph 1 2 n .
This graph (also known as the ladder graph) has 2n + 2 vertices, 3n + 1 edges and n
elementary regions (squares). For i = 1, 2, . . . , n, let Gn,i denote the graph obtained by
adding one unit square below the i-th square of Gn.
Now, we describe some recursive relations for f -vectors of C(Gn) and C(Gn,i).
Proposition 4.1. For all positive integers n the entries of f -vectors of C(Gn) and C(Gn,i)
satisfy the following recurrences:
fi(C(Gn+2)) = fi(C(Gn+1)) + fi(C(Gn)) + fi−1(C(Gn)), (4.2)
fi(C(Gn+2,i)) = fi(C(Gn+1,i)) + fi(C(Gn,i)) + fi−1(C(Gn,i)), (4.3)
fi(C(Gn+2,i)) = fi(C(Gn+1,i−1)) + fi(C(Gn,i−2)) + fi−1(C(Gn,i−2)). (4.4)
Proof. All formulas follow from relation (3.2), see the proof of Theorem 3.11. To obtain
the formula (4.2), we apply (3.2) on Gn+2. The rightmost vertical edge and the rightmost
unit square in Gn+2 act as e and R in (3.2), see the first row on the next figure.
= ∪ ∪
= ∪ ∪
= ∪ ∪
(4.2)
(4.3)
(4.4)
Figure 5: The “geometric proof” of recursive relations for f(C(Gn)) and f(C(Gn,i)).
In the same way we can prove the remaining two relations. For each relation, we choose
an adequate elementary region R, a corresponding edge e of R, and use relation (3.2), see
Figure 5.
The f -vector (f0, f1, f2, . . . , f⌈n2 ⌉) of C(Gn) can be encoded by the polynomial Fn:
Fn = FC(Gn)(x) = f0 + f1x+ f2x
2 + · · ·+ f⌈n2 ⌉x
⌈n2 ⌉.
Similarly, we define the polynomials Fn,i to encode the f -vector of C(Gn,i). Directly from
(4.2) and (4.3) we obtain that
Fn+2(x) = Fn+1(x) + (x+ 1)Fn(x), Fn+2,i(x) = Fn+1,i(x) + (x+ 1)Fn,i(x).
Now, we define new polynomials Pn and Pn,i by
Pn = Pn(x) = Fn(x− 1), Pn,i = Pn,i(x) = Fn,i(x− 1).
52 Ars Math. Contemp. 22 (2022) #P1.03 / 41–55
This is a variant of h-polynomial associated to a cubical complex.
From Proposition 4.1 it follows that the polynomials Pn and Pn,i satisfy the following
recurrences
Pn+2(x) = Pn+1(x) + xPn(x), (4.5)
Pn+2,i(x) = Pn+1,i(x) + xPn,i(x), (4.6)
Pn+2,i(x) = Pn+1,i−1(x) + xPn,i−2(x). (4.7)
Remark 4.2. We can use (4.5) to obtain the polynomials Pn explicitly
P2d−1 =
(
d
d
)
xd + · · ·+
(
d+ k
d− k
)
xk + · · ·+
(
2d− 1
1
)
x+
(
2d
0
)
, and
P2d =
(
d+ 1
d
)
xd + · · ·+
(
d+ k + 1
d− k
)
xk + · · ·+
(
2d
1
)
x+
(
2d+ 1
0
)
.
Note that the polynomials Pn are related with Fibonacci polynomials, see Section 9.4
in [3] for the definition and a combinatorial interpretation of coefficients. The coefficients
of these polynomials are positive integers and the sum of coefficients of Pn is a Fibonacci
number. Note that this is just the number of vertices in C(Gn).
Assume that we embedded C(Gn) into n-cube as in Proposition 1.2, so that the perfect
matching M0 = of Gn is the vertex in the origin. Now, the coefficient
of xk in Pn counts the number of vertices of C(Gn) for which the sum of coordinates is k,
i.e., it is the number of vertices of C(Gn) whose distance from M0 is k.
Also, following [3], we can recognize the coefficient of xk in Pn as the number of
k-element subsets of [n] that do not contain two consecutive integers. Similarly, we can
interpret the coefficient of xk in Pn,i as the number of k-element subsets of the multiset
M = {1, 2, . . . , i− 1, i, i, i+1, . . . , n} that do not contain two consecutive integers. Note
that the multiplicity of i in M is two, and all other elements have the multiplicity one.
Definition 4.3. Let Pd denote the vector space of all polynomials of degree at most d. We
define the linear map Ad : Pd → Pd+1 recursively by
Ad(x
k) = xAd−1(x
k−1) for all k > 0, (4.8)
A0(1) = 1 + 2x and Ad(1) = P2d+1 −Ad(P2d−1 − 1). (4.9)
Lemma 4.4. For any non-negative integer d, we have that
Ad(P2d−1) = P2d+1, Ad(P2d) = P2d+2 and Ad+1(P2d) = P2d+2.
Proof. From (4.9) it follows that Ad(P2d−1) = P2d+1. For the proof of the second formula
we use (4.5), (4.8) and induction
Ad(P2d) = Ad(P2d−1 + xP2d−2) = P2d+1 + xAd−1(P2d−2) = P2d+1 + xP2d = P2d+2.
The last formula in this lemma follows from (4.5) and earlier proved formulas
Ad+1(P2d) = Ad+1(P2d+1 − xP2d−1) = P2d+3 − xAd(P2d−1)
= P2d+3 − xP2d+1 = P2d+2.
D. Jojić: The cubical matching complex revisited 53
Lemma 4.5. For all integers i and d such that 1 ⩽ i ⩽ ⌊d2⌋, the following holds:
Ad(P2d−1,i) = P2d+1,i and Ad(P2d,i) = P2d+2,i.
Proof. For i = 1 and i = 2 we apply relation (3.2) in a similar way as in the proof of
Proposition 4.1. We just delete the only square in the second row of Gn,1 and Gn,2, and
obtain that
P2d−1,1 = P2d−1 + xP2d−3, P2d−1,2 = P2d−1 + xP2d−4.
By using Lemma 4.4, we obtain that
Ad(P2d−1,1) = Ad(P2d−1 + xP2d−3) = P2d+1 + xP2d−1 = P2d+1,1, and
Ad(P2d−1,2) = Ad(P2d−1 + xP2d−4) = P2d+1 + xAd−1(P2d−4)
= P2d+1 + xP2d−2 = P2d+1,2.
In a similar way, we can prove that
Ad(P2d,1) = P2d+2,1 and Ad(P2d,2) = P2d+2,2.
Assume that the statement of this lemma is true for P2d−1,j and P2d,j when j < i + 1.
Now, we use (4.7) and induction to calculate
Ad(P2d,i+1) = Ad(P2d−1,i + xP2d−2,i−1) = Ad(P2d−1,i) + xAd−1(P2d−2,i−1)
= P2d+1,i + xP2d,i−1 = P2d+2,i+1.
From (4.6) we obtain that
Ad(P2d−1,i+1) = Ad(P2d,i+1 − xP2d−2,i+1) = Ad(P2d,i+1)− xAd−1(P2d−2,i+1)
= P2d+2,i+1 − xP2d,i+1 = P2d+1,i+1.
From Definition 4.3 and Remark 4.2 we can obtain the concrete formula for the linear
map Ad.
Proposition 4.6. For all d, k ∈ N such that d ⩾ k ⩾ 1, we have that:
Ad(x
k) = xk
(
1 + 2x− x2 + 2x3 − 5x4 + 14x5 − 42x6 + · · ·+ (−1)d−kCd−kxd−k+1
)
.
Here Cm denotes the m-th Catalan number.
Proof. From (4.8) it is enough to prove that
Ad(1) = 1 + 2x− x2 + 2x3 − 5x4 + · · ·+ (−1)dCdxd+1. (4.10)
For all integers n and k such that n ⩾ k ⩾ 1 (by using the induction and the Pascal’s
Identity), we can obtain the next relation(
n
k
)
=
k∑
i=0
(−1)i
(
n+ 1 + i
k − i
)
Ci. (4.11)
54 Ars Math. Contemp. 22 (2022) #P1.03 / 41–55
Now, we assume that (4.10) is true for all positive integers less than d, and calculate Ad(1)
by definition:
Ad(1) = P2d+1 −Ad(P2d−1 − 1)
=
d+1∑
i=0
(
2d+ 2− i
i
)
xi −
d∑
i=1
(
2d− i
i
)
xiAd−i(1).
The coefficients of 1, x and x2 in Ad(1) are respectively:(
2d+ 2
0
)
= 1,
(
2d+ 1
1
)
−
(
2d− 1
1
)
= 2,
(
2d
2
)
−
(
2d− 2
2
)
− 2
(
2d− 1
1
)
= −1.
For k > 1 the coefficient of xk+1 in the polynomial Ad(1) is(
2d+ 1− k
k + 1
)
−
(
2d− k − 1
k + 1
)
− 2
(
2d− k
k
)
−
k−1∑
i=1
(−1)i
(
2d− k + i
k − i
)
Ci.
From (4.11) we obtain that the coefficient of xk+1 in Ad(1) is (−1)kCk.
Corollary 4.7. For any positive integer d the linear map Ad is injective.
Now, we consider all simple connected regions for which the degree of the associated
polynomial PR(x) = FR(x − 1) is equal to d. Let Fd denote the affine subspace of Pd
spanned by these polynomials.
Lemma 4.8. The polynomial P2d+1,d is not contained in Ad(Fd).
Proof. From (4.7) and (4.6) it follows that
P2d+1,d − P2d+1,d−1 = (P2d,d−1 + xP2d−1,d−2)− (P2d,d−1 + xP2d−1,d−1)
= −x(P2d−1,d−1 − P2d−1,d−2) = (−1)d+1(xd+1 + xd).
We know that P2d+1,d−1 = Ad(P2d−1,d−1). If there exists a polynomial p ∈ Fd such that
Ad(p) = P2d+1,d then we obtain
xd+1 + xd = ±Ad(p− P2d−1,d−1),
which is impossible from Proposition 4.6.
Theorem 4.9. The polynomials P2d−1, P2d, P2d−1,1, . . . , P2d−1,d−1 are affinely indepen-
dent in Fd.
Proof. We use induction on the degree. Assume that d polynomials P2d−3, P2d−2, P2d−3,1,
. . . , P2d−3,d−2 are affinely independent in Fd−1. From Lemmas 4.4 and 4.5 and Corol-
lary 4.7, we conclude that P2d−1, P2d, P2d−1,1, . . . , P2d−1,d−2 are affinely independent.
These polynomials span a (d − 1)-dimensional affine subspace of Fd. From Lemma 4.8
follows that P2d−1,d−1 is not contained in Ad−1(Fd−1).
Corollary 4.10. The Euler-Poincare relation (4.1) is the only linear relation for the f -
vectors of tilings.
This answer the question of Ehrenborg question about numerical relations between the
numbers of different types of tilings, see [6].
D. Jojić: The cubical matching complex revisited 55
ORCID iDs
Duško Jojić https://orcid.org/0000-0003-4572-0637
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P1.04 / 57–81
https://doi.org/10.26493/1855-3974.2318.fb9
(Also available at http://amc-journal.eu)
From Italian domination in lexicographic
product graphs to w-domination in graphs*
Abel Cabrera Martı́nez , Alejandro Estrada-Moreno ,
Juan Alberto Rodrı́guez-Velázquez
Universitat Rovira i Virgili, Departament d’Enginyeria Informàtica i Matemàtiques,
Av. Paı̈sos Catalans 26, 43007 Tarragona, Spain
Received 23 April 2020, accepted 9 May 2021, published online 18 February 2022
Abstract
In this paper, we show that the Italian domination number of every lexicographic prod-
uct graph G ◦ H can be expressed in terms of five different domination parameters of G.
These parameters can be defined under the following unified approach, which encompasses
the definition of several well-known domination parameters and introduces new ones.
Let N(v) denote the open neighbourhood of v ∈ V (G), and let w = (w0, w1, . . . , wl)
be a vector of nonnegative integers such that w0 ≥ 1. We say that a function f : V (G) →
{0, 1, . . . , l} is a w-dominating function if f(N(v)) =
∑
u∈N(v) f(u) ≥ wi for every
vertex v with f(v) = i. The weight of f is defined to be ω(f) =
∑
v∈V (G) f(v). The
w-domination number of G, denoted by γw(G), is the minimum weight among all w-
dominating functions on G.
Specifically, we show that γI(G ◦ H) = γw(G), where w ∈ {2} × {0, 1, 2}l and
l ∈ {2, 3}. The decision on whether the equality holds for specific values of w0, . . . , wl will
depend on the value of the domination number of H . This paper also provides preliminary
results on γw(G) and raises the challenge of conducting a detailed study of the topic.
Keywords: Italian domination, w-domination, k-domination, k-tuple domination, lexicographic prod-
uct graph.
Math. Subj. Class. (2020): 05C69, 05C76
*The authors would thank the anonymous reviewers for their generous time in providing detailed comments
and suggestions that helped us to improve the paper.
E-mail addresses: abel.cabrera@urv.cat (Abel Cabrera Martı́nez), alejandro.estrada@urv.cat (Alejandro
Estrada-Moreno), juanalberto.rodriguez@urv.cat (Juan Alberto Rodrı́guez-Velázquez)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
58 Ars Math. Contemp. 22 (2022) #P1.04 / 57–81
1 Introduction
Let G be a graph, l a positive integer, and f : V (G) → {0, . . . , l} a function. For every
i ∈ {0, . . . , l}, we define Vi = {v ∈ V (G) : f(v) = i}. We will identify f with the
subsets V0, . . . , Vl associated with it, and so we will use the unified notation f(V0, . . . , Vl)
for the function and these associated subsets. The weight of f is defined to be
ω(f) = f(V (G)) =
l∑
i=1
i|Vi|.
An Italian dominating function (IDF) on a graph G is a function f(V0, V1, V2) satisfy-
ing that f(N(v)) =
∑
u∈N(v) f(u) ≥ 2 for every v ∈ V0, where N(v) denotes the open
neighbourhood of v. Hence, f(V0, V1, V2) is an IDF if N(v)∩V2 ̸= ∅ or |N(v)∩V1| ≥ 2
for every v ∈ V0. The Italian domination number, denoted by γI(G), is the minimum
weight among all IDFs on G. This concept was introduced by Chellali et al. in [6] under
the name of Roman {2}-domination. The term “Italian domination” comes from a subse-
quent paper by Henning and Klostermeyer [13].
In this paper we show that the Italian domination number of every lexicographic product
graph G◦H can be expressed in terms of five different domination parameters of G. These
parameters can be defined under the following unified approach.
Let w = (w0, . . . , wl) be a vector of nonnegative integers such that w0 ≥ 1. We say
that f(V0, . . . , Vl) is a w-dominating function if f(N(v)) ≥ wi for every v ∈ Vi. The
w-domination number of G, denoted by γw(G), is the minimum weight among all w-
dominating functions on G. For simplicity, a w-dominating function f of weight ω(f) =
γw(G) will be called a γw(G)-function.
This unified approach allows us to encompass the definition of several well-known
domination parameters and introduce new ones. For instance, we would highlight the fol-
lowing particular cases of known domination parameters that we define here in terms of
w-domination.
• The domination number of G is defined to be γ(G) = γ(1,0)(G) = γ(1,0,0)(G).
Obviously, every γ(1,0,0)(G)-function f(V0, V1, V2) satisfies that V2 = ∅ and V1 is
a dominating set of cardinality |V1| = γ(G), i.e., V1 is a γ(G)-set.
• The total domination number of a graph G with no isolated vertex is defined to be
γt(G) = γ(1,1)(G) = γ(1,1,w2,...,wl)(G), for every w2, . . . , wl ∈ {0, 1}. Notice
that there exists a γ(1,1,w2,...,wl)(G)-function f(V0, V1, . . . , Vl) such that Vi = ∅ for
every i ∈ {2, . . . , l} and V1 is a total dominating set of cardinality |V1| = γt(G),
i.e., V1 is a γt(G)-set.
• Given a positive integer k, the k-domination number of a graph G is defined to be
γk(G) = γ(k,0)(G). In this case, V1 is a k-dominating set of cardinality |V1| =
γk(G), i.e., V1 is a γk(G)-set. The study of k-domination in graphs was initiated by
Fink and Jacobson [8] in 1984.
• Given a positive integer k, the k-tuple domination number of a graph G of minimum
degree δ ≥ k−1 is defined to be γ×k(G) = γ(k,k−1)(G). In this case, V1 is a k-tuple
dominating set of cardinality |V1| = γ×k(G), i.e., V1 is a γ×k(G)-set. In particular,
γ×1(G) = γ(G) and γ×2(G) is known as the double domination number of G. This
parameter was introduced by Harary and Haynes in [9].
A. Cabrera Martı́nez et al.: From Italian domination in lexicographic product graphs to . . . 59
• Given a positive integer k, the k-tuple total domination number of a graph G of
minimum degree δ ≥ k is defined to be γ×k,t(G) = γ(k,k)(G). In particular,
γ×1,t(G) = γt(G) and γ×2,t(G) is known as the double total domination number,
and V1 is a double total dominating set of cardinality |V1| = γ×2,t(G), i.e., V1 is a
γ×2,t(G)-set. The k-tuple total domination number was introduced by Henning and
Kazemi in [12].
• The Italian domination number of G is defined to be γ
I
(G) = γ(2,0,0)(G). As
mentioned earlier, this parameter was introduced by Chellali et al. in [6] under
the name of Roman {2}-domination number. The concept was studied further
in [13, 16].
• The total Italian domination number of a graph G with no isolated vertex is defined
to be γtI(G) = γ(2,1,1)(G). This parameter was introduced by Cabrera et al. in
[4], and independently by Abdollahzadeh Ahangar et al. in [1], under the name of
total Roman {2}-domination number. The total Italian domination number of lexi-
cographic product graphs was studied in [5].
• The {k}-domination number of G is defined to be γ{k}(G) = γ(k,k−1,...,1,0)(G).
This parameter was introduced by Domke et al. in [7] and studied further
in [3, 15, 17].
Notice that the concept of Y -dominating function introduced by Bange et al. [2] is quite
different from the concept of w-dominating function introduced in this paper. Given a set
Y of real numbers, a function f : V (G) → Y is a Y -dominating function if f(N [v]) =
f(v) +
∑
u∈N(v) f(u) ≥ 1 for every v ∈ V (G). The Y -domination number, denoted
by γ
Y
(G), is the minimum weight among all Y -dominating functions on G. Hence, if
Y = {0, 1, . . . , l}, then γ
Y
(G) = γ(1,0,...,0)(G) = γ(G).
For the graphs shown in Figure 1 we have the following values.
• γI(G1) = γ(2,1,0)(G1) = γ(2,2,0)(G1) = 4 < 6 = γ(2,2,1)(G1) = γ(2,2,2)(G1).
• γI(G2) = γ(2,1,0)(G2) = γ(2,2,0)(G2) = γ(2,2,1)(G2) = γ(2,2,2)(G2) = 3.
• γI(G3) = γ(2,1,0)(G3) = 6 < 8 = γ(2,2,0)(G3) = γ(2,2,1)(G3) = γ(2,2,2)(G3).
11
1 1
G1
1
1
1
G2
22
2 2
G3
Figure 1: The labels of black-coloured vertices describe a γ(2,1,0)(G1)-function, a
γ(2,2,0)(G2)-function and a γ(2,2,2)(G3)-function, respectively.
60 Ars Math. Contemp. 22 (2022) #P1.04 / 57–81
The remainder of the paper is organized as follows. In Section 2 we show that for any
graph G with no isolated vertex and any nontrivial graph H with γ(H) ̸= 3 or γ
I
(H) ̸= 3,
the Italian domination number of G◦H equals one of the following parameters: γ(2,1,0)(G),
γ(2,2,0)(G), γ(2,2,1)(G) or γ(2,2,2)(G). The specific value γI (G ◦H) takes depends on the
value of γ(H). For the cases where γ
I
(H) = γ(H) = 3, we show that γ
I
(G ◦ H) =
γ(2,2,2,0)(G). Section 3 is devoted to providing some preliminary results on w-domination.
We first describe some general properties of γw(G) and then dedicate a subsection to each
of the specific cases declared of interest in Section 2.
We assume that the reader is familiar with the basic concepts, notation and terminology
of domination in graph. If this is not the case, we suggest the textbooks [10, 11, 14]. For
the remainder of the paper, definitions will be introduced whenever a concept is needed.
2 Italian domination in lexicographic product graphs
The lexicographic product of two graphs G and H is the graph G ◦H whose vertex set is
V (G ◦ H) = V (G) × V (H) and (u, v)(x, y) ∈ E(G ◦ H) if and only if ux ∈ E(G) or
u = x and vy ∈ E(H).
Notice that for any u ∈ V (G) the subgraph of G ◦ H induced by {u} × V (H) is
isomorphic to H . For simplicity, we will denote this subgraph by Hu. Moreover, the
neighbourhood of (x, y) ∈ V (G)×V (H) will be denoted by N(x, y) instead of N((x, y)).
Analogously, for any function f on G ◦H , the image of (x, y) will be denoted by f(x, y)
instead of f((x, y)).
Lemma 2.1. For any graph G with no isolated vertex and any nontrivial graph H with
γ
I
(H) ̸= 3 or γ(H) ̸= 3, there exists a γ
I
(G◦H)-function f satisfying that f(V (Hu)) ≤ 2
for every u ∈ V (G).
Proof. Given an IDF f on G ◦H , we define the set Rf = {x ∈ V (G) : f(V (Hx)) ≥ 3}.
Let f be a γ
I
(G ◦H)-function such that |Rf | is minimum among all γI (G ◦H)-functions.
Suppose that |Rf | ≥ 1. Let u ∈ Rf such that f(V (Hu)) is maximum among all vertices
belonging to Rf . Suppose that f(V (Hu)) > γI (H). In this case we take a γI (H)-function
h and construct an IDF g defined on G ◦ H as g(u, y) = h(y) for every y ∈ V (H) and
g(x, y) = f(x, y) for every x ∈ V (G) \ {u} and y ∈ V (H). Obviously, ω(g) < ω(f),
which is a contradiction. Thus, 3 ≤ f(V (Hu)) ≤ γI (Hu) = γI (H). Now, we analyse the
following two cases.
Case 1. f(V (Hu)) ≥ 4. Let u′ ∈ N(u) and v ∈ V (H). We define a function f ′ on G ◦H
as f ′(u, v) = f ′(u′, v) = 2, f ′(u, y) = f(u′, y) = 0 for every y ∈ V (H) \ {v}, and
f ′(x, y) = f(x, y) for every x ∈ V (G) \ {u, u′} and y ∈ V (H). Notice that f ′ is an IDF
on G ◦H with ω(f ′) ≤ ω(f) and |Rf ′ | < |Rf |, which is a contradiction.
Case 2. f(V (Hu)) = 3. Suppose that γI (H) ̸= 3. Since γI (H) ≥ 4, there exist u′ ∈
N(u) and v ∈ V (H) such that f(u′, v) ≥ 1. Hence, the function f ′ defined in Case 1 is an
IDF on G ◦H with ω(f ′) ≤ ω(f) and |Rf ′ | < |Rf |, which is again a contradiction.
Thus, γ
I
(H) = 3, and so γ(H) ̸= 3, which implies that γ(H) = 2. Let {v1, v2} be
a γ(H)-set. Let u′ ∈ N(u) and v′ ∈ V (H) such that f(u′, v′) = max{f(u′, y) : y ∈
V (H)}. Consider the function f ′ defined as f ′(u, v1) = f ′(u, v2) = 1, f ′(u, y) = 0 for
every y ∈ V (H) \ {v1, v2}, f ′(u′, v′) = min{2, f(u′, v′) + 1}, f ′(u′, y) = 0 for every
y ∈ V (H) \ {v′}, and f ′(x, y) = f(x, y) for every x ∈ V (G) \ {u, u′} and y ∈ V (H).
A. Cabrera Martı́nez et al.: From Italian domination in lexicographic product graphs to . . . 61
Notice that f ′ is an IDF on G ◦ H with ω(f ′) ≤ ω(f) and |Rf ′ | < |Rf |, which is a
contradiction.
Therefore, Rf = ∅, and the result follows.
Theorem 2.2. The following statements hold for any graph G with no isolated vertex and
any nontrivial graph H with γ
I
(H) ̸= 3 or γ(H) ̸= 3.
(i) If γ(H) = 1, then γ
I
(G ◦H) = γ(2,1,0)(G).
(ii) If γ2(H) = γ(H) = 2, then γI (G ◦H) = γ(2,2,0)(G).
(iii) If γ2(H) > γ(H) = 2, then γI (G ◦H) = γ(2,2,1)(G).
(iv) If γ(H) ≥ 3, then γ
I
(G ◦H) = γ(2,2,2)(G).
Proof. Let f(V0, V1, V2) be a γI(G ◦H)-function which satisfies Lemma 2.1. Let f ′(X0,
X1, X2) be the function defined on G by X1 = {x ∈ V (G) : f(V (Hx)) = 1} and
X2 = {x ∈ V (G) : f(V (Hx)) = 2}. Notice that γI (G ◦H) = ω(f) = ω(f ′). We claim
that f ′ is a γ(w0,w1,w2)(G)-function. In order to prove this and find the values of w0, w1
and w2, we differentiate the following three cases.
Case 1. γ(H) = 1. Assume that x ∈ X0. Since f(V (Hx)) = 0, for any y ∈ V (H) we
have that f(N(x, y) \ V (Hx)) ≥ 2. Thus, f ′(N(x)) ≥ 2. Now, assume that x ∈ X1,
and let (x, y) ∈ V1 be the only vertex in V (Hx) such that f(x, y) > 0. Since γ(H) = 1,
for any z ∈ V (H) \ {y}, we have that f(N(x, z) \ V (Hx)) ≥ 1, which implies that
f ′(N(x)) ≥ 1. Therefore, f ′ is a (2, 1, 0)-dominating function on G and, as a consequence,
γ
I
(G ◦H) = ω(f) = ω(f ′) ≥ γ(2,1,0)(G).
Now, for any γ(2,1,0)(G)-function g(W0,W1,W2) and any universal vertex v of H , the
function g′(W ′0,W
′
1,W
′
2), defined by W
′
2 = W2 × {v} and W ′1 = W1 × {v}, is an IDF on
G ◦H . Therefore, γ
I
(G ◦H) ≤ ω(g′) = ω(g) = γ(2,1,0)(G).
Case 2. γ(H) = 2. As in Case 1 we conclude that f ′(N(x)) ≥ 2 for every x ∈ X0. Now,
assume that x ∈ X1, and let (x, y) ∈ V1 be the only vertex in V (Hx) such that f(x, y) >
0. Since γ(H) = 2, there exists a vertex z ∈ V (H) such that (x, z) ∈ V0 \ N(x, y).
Hence, f(N(x, z) \ V (Hx)) ≥ 2, which implies that f ′(N(x)) ≥ 2. Therefore, f ′ is a
(2, 2, 0)-dominating function on G and, as a consequence, γ
I
(G ◦H) = ω(f) = ω(f ′) ≥
γ(2,2,0)(G).
Now, if γ2(H) > γ(H) = 2, then for every x ∈ X2, there exists y ∈ V (H) such that
(x, y) ∈ V0 and f(N(x, y)∩V (Hx)) ≤ 1, which implies that f(N(x, y)\V (Hx)) ≥ 1, and
so f ′(N(x)) ≥ 1. Hence, f ′ is a (2, 2, 1)-dominating function on G and, as a consequence,
γ
I
(G ◦H) = ω(f) = ω(f ′) ≥ γ(2,2,1)(G).
On the other side, if γ2(H) = 2, then for any γ(2,2,0)(G)-function g(W0,W1,W2)
and any γ2(H)-set S = {v1, v2}, the function g′(W ′0,W ′1,W ′2), defined by W ′1 = (W1 ×
{v1}) ∪ (W2 × S) and W ′2 = ∅, is an IDF on G ◦H . Therefore, γI (G ◦H) ≤ ω(g′) =
ω(g) = γ(2,2,0)(G).
Finally, if γ2(H) > γ(H) = 2 then we take a γ(2,2,1)(G)-function h(Y0, Y1, Y2) and
a γ(H)-set S′ = {v′1, v′2}, and construct a function h′(Y ′0 , Y ′1 , Y ′2) on G ◦ H by making
Y ′1 = (Y1 × {v′1}) ∪ (Y2 × S′) and Y ′2 = ∅. Obviously, h′ is an IDF on G ◦H , and so we
can conclude that γ
I
(G ◦H) ≤ ω(h′) = ω(h) = γ(2,2,1)(G).
62 Ars Math. Contemp. 22 (2022) #P1.04 / 57–81
Case 3. γ(H) ≥ 3. In this case, for every x ∈ V (G), there exists y ∈ V (H) such
that f(N [(x, y)] ∩ V (Hx)) = 0. Hence, f(N(x, y) \ V (Hx)) ≥ 2, which implies that
f ′(N(x)) ≥ 2 for every x ∈ V (G). Therefore, f ′ is a (2, 2, 2)-dominating function on G
and, as a consequence, γ
I
(G ◦H) = ω(f) = ω(f ′) ≥ γ(2,2,2)(G).
On the other side, for any γ(2,2,2)(G)-function g(W0,W1,W2) and any v ∈ V (H), the
function g′(W ′0,W
′
1,W
′
2), defined by W
′
2 = W2 × {v} and W ′1 = W1 × {v}, is an IDF on
G ◦H . Hence, γ
I
(G ◦H) ≤ ω(g′) = ω(g) = γ(2,2,2)(G).
According to the three cases above, the result follows.
The following result considers the case γ
I
(H) = γ(H) = 3.
Theorem 2.3. If H is a graph with γ
I
(H) = γ(H) = 3, then for any graph G,
γ
I
(G ◦H) = γ(2,2,2,0)(G).
Proof. Let f(V0, V1, V2) be a γI(G ◦ H)-function, and f ′(X0, X1, X2, X3) the function
defined on G by X1 = {x ∈ V (G) : f(V (Hx)) = 1}, X2 = {x ∈ V (G) : f(V (Hx)) =
2} and X3 = {x ∈ V (G) : f(V (Hx)) ≥ 3}. We claim that f ′ is a (2, 2, 2, 0)-dominating
function on G.
Let x ∈ X0 ∪X1 ∪X2. Since f(V (Hx)) ≤ 2 and γ(H) = 3, there exists y ∈ V (H)
such that f(N [(x, y)] ∩ V (Hx)) = 0. Thus, f ′(N(x)) ≥ 2 for every x ∈ X0 ∪X1 ∪X2,
which implies that f ′ is a (2, 2, 2, 0)-dominating function on G. Therefore, γ
I
(G ◦H) =
ω(f) ≥ ω(f ′) ≥ γ(2,2,2,0)(G).
On the other side, let h(Y0, Y1, Y2, Y3) be a γ(2,2,2,0)(G)-function, h1 a γI (H)-function
and v ∈ V (H). We define a function g on G◦H by g(x, v) = h(x) for every x ∈ V (G)\Y3,
g(x, y) = 0 for every x ∈ V (G) \ Y3 and y ∈ V (H) \ {v}, and g(x, y) = h1(y) for every
(x, y) ∈ Y3 × V (H). A simple case analysis shows that g is an IDF on G ◦H . Therefore,
γ
I
(G ◦H) ≤ ω(g) = ω(h) = γ(2,2,2,0)(G).
The graph shown in Figure 2 satisfies 6 = γ(2,2,0)(G) = γ(2,2,1)(G) < 7 =
γ(2,2,2,0)(G) < γ(2,2,2)(G) = 8.
2 1 1 2 2 2 2
Figure 2: This figure shows two γ(2,2,0)(G)-functions on the same graph. The function on
the left is also a γ(2,2,1)(G)-function.
3 Preliminary results on w-domination
In this section, we fix the notation Z+ = {1, 2, 3, . . .} and N = Z+ ∪ {0} for the sets of
positive and nonnegative integers, respectively.
A. Cabrera Martı́nez et al.: From Italian domination in lexicographic product graphs to . . . 63
Throughout this section, we will repeatedly apply, without explicit mention, the follo-
wing necessary and sufficient condition for the existence of a w-dominating function.
Remark 3.1. Let G be a graph of minimum degree δ and let w = (w0, . . . , wl) ∈ Z+×Nl.
If w0 ≥ · · · ≥ wl, then there exists a w-dominating function on G if and only if wl ≤ lδ.
Proof. Let w = (w0, . . . , wl) ∈ Z+ × Nl such that w0 ≥ · · · ≥ wl. If wl ≤ lδ, then the
function f , defined by f(v) = l for every v ∈ V (G), is a w-dominating function on G, as
Vl = V (G) and for any x ∈ Vl, f(N(x)) ≥ lδ ≥ wl.
Now, suppose that wl > lδ. If g is a w-dominating function on G, then for any vertex
v of degree δ we have g(N(v)) ≤ δl < wl ≤ wl−1 ≤ · · · ≤ w0, which is a contradiction.
Therefore, the result follows.
We will show that in general the w-domination numbers satisfy a certain monotonicity.
Given two integer vectors w = (w0, . . . , wl) and w′ = (w′0, . . . , w
′
l), we say that w ≺ w′ if
wi ≤ w′i for every i ∈ {0, . . . , l}. With this notation in mind, we can state the next remark
which is direct consequence of the definition of w-domination number.
Remark 3.2. Let G be a graph of minimum degree δ and let w = (w0, . . . , wl), w′ =
(w′0, . . . , w
′
l) ∈ Z+×Nl such that wi ≥ wi+1 and w′i ≥ w′i+1 for every i ∈ {0, . . . , l−1}.
If w ≺ w′ and w′l ≤ lδ, then every w′-dominating function is a w-dominating function
and, as a consequence,
γw(G) ≤ γw′(G).
We would emphasize the following remark on the specific cases of domination param-
eters considered in Section 2. Obviously, when we write γ(2,2,2)(G) or γ(2,2,1)(G), we are
assuming that G has minimum degree δ ≥ 1.
Remark 3.3. The following statements hold.
(i) γ
I
(G) = γ(2,0,0)(G) ≤ γ(2,1,0)(G) ≤ γ(2,2,0)(G) ≤ γ(2,2,1)(G) ≤ γ(2,2,2)(G).
(ii) If w2 ∈ {1, 2}, then γ(1,0,w2)(G) = γ(1,0,0)(G) = γ(G) and γ(1,1,w2)(G) =
γ(1,1,0)(G) = γt(G).
(iii) For any integer k ≥ 3, there exists an infinite family Hk of graphs such that for every
graph G ∈ Hk, γI (G) = γ(2,0,0)(G) = γ(2,1,0)(G) = γ(2,2,0)(G) = γ(2,2,1)(G) =
γ(2,2,2)(G) = k.
(iv) There exists an infinite family of graphs such that γ
I
(G) < γ(2,1,0)(G) <
γ(2,2,0)(G) < γ(2,2,1)(G) < γ(2,2,2)(G).
In order to see that the remark above holds, we just have to construct families of graphs
satisfying (iii) and (iv), as (i) is a particular case of Remark 3.2 and (ii) is derived from
the definition of (w0, w1, w2)-domination number. In the case of (iii), we construct a fam-
ily Hk = {Gk,r : r ∈ Z+} as follows. Let k ≥ 3 be an integer, and let Nr be the
empty graph of order r. For any positive integer r we construct a graph Gk,r ∈ Hk
from a complete graph Kk and
(
k
2
)
copies of Nr, in such way that for each pair of dif-
ferent vertices {x, y} of Kk we choose one copy of Nr and connect every vertex of Nr
with x and y, making x and y vertices of degree (k − 1)(r + 1) in Gk,r. For instance,
the graph G3,1 is isomorphic to the graph G2 shown in Figure 1. It is readily seen that
64 Ars Math. Contemp. 22 (2022) #P1.04 / 57–81
γ
I
(Gk,r) = γ(2,2,2)(Gk,r) = k. On the other hand, in the case of (iv), we consider the
family of cycles of order n ≥ 10 with n ≡ 1 (mod 3). For these graphs we have that
γI(Cn) < γ(2,1,0)(Cn) < γ(2,2,0)(Cn) < γ(2,2,1)(Cn) < γ(2,2,2)(Cn). The specific values
of γ(w0,w1,w2)(Cn) will be given in Subsections 3.1 – 3.4.
Next we show a class of graphs where γ(w0,...,wl)(G) = w0γ(G) whenever l ≥ w0 ≥
· · · ≥ wl. To this end, we need to introduce some additional notation and terminology.
Given two graphs G1 and G2, the corona product graph G1 ⊙ G2 is the graph obtained
from G1 and G2, by taking one copy of G1 and |V (G1)| copies of G2 and joining by an
edge every vertex from the ith-copy of G2 with the ith-vertex of G1. For every x ∈ V (G1),
the copy of G2 in G1 ⊙G2 associated to x will be denoted by G2,x. It is well known that
γ(G1 ⊙ G2) = |V (G1)| and, if G1 does not have isolated vertices, then γt(G1 ⊙ G2) =
γ(G1 ⊙G2) = |V (G1)|.
Theorem 3.4. Let G ∼= G1 ⊙ G2 be a corona graph where G1 does not have isolated
vertices, and let w = (w0, . . . , wl) ∈ Z+ × Nl. If l ≥ w0 ≥ · · · ≥ wl and |V (G2)| ≥ w0,
then
γw(G) = w0γ(G).
Proof. Since G1 does not have isolated vertices, the upper bound γw(G) ≤ w0|V (G1)| =
w0γ(G) is straightforward, as the function f , defined by f(x) = w0 for every vertex
x ∈ V (G1) and f(x) = 0 for every x ∈ V (G) \ V (G1), is a w-dominating function on G.
On the other hand, let f be a γw(G)-function and suppose that there exists x ∈ V (G1)
such that f(V (G2,x)) + f(x) ≤ w0 − 1. In such a case, f(N [y]) ≤ w0 − 1 for every
y ∈ V (G2,x), which is a contradiction, as |V (G2)| ≥ w0. Therefore, γw(G) = ω(f) ≥
w0|V (G1)| = w0γ(G).
Proposition 3.5. Let G be a graph of order n. Let w = (w0, . . . , wl) ∈ Z+ ×Nl such that
w0 ≥ · · · ≥ wl. If G′ is a spanning subgraph of G with minimum degree δ′ ≥ wll , then
γw(G) ≤ γw(G′).
Proof. Let E− = {e1, . . . , ek} be the set of all edges of G not belonging to the edge set of
G′. Let G′0 = G and, for every i ∈ {1, . . . , k}, let Xi = {e1, . . . , ei} and G′i = G −Xi,
the edge-deletion subgraph of G induced by E(G) \Xi. Since any w-dominating function
on G′i is a w-dominating function on G
′
i−1, we can conclude that γw(G
′
i−1) ≤ γw(G′i).
Hence, γw(G) = γw(G′0) ≤ γw(G′1) ≤ · · · ≤ γw(G′k) = γw(G′).
From Proposition 3.5 we obtain the following result.
Corollary 3.6. Let G be a graph of order n and w = (w0, . . . , wl) ∈ Z+ × Nl such that
w0 ≥ · · · ≥ wl.
• If G is a Hamiltonian graph and wl ≤ 2l, then γw(G) ≤ γw(Cn).
• If G has a Hamiltonian path and wl ≤ l, then γw(G) ≤ γw(Pn).
In order to derive lower bounds on the w-domination number, we need to state the
following useful lemma.
A. Cabrera Martı́nez et al.: From Italian domination in lexicographic product graphs to . . . 65
Lemma 3.7. Let G be a graph with no isolated vertex, maximum degree ∆ and order n.
For any w-dominating function f(V0, . . . , Vl) on G such that w0 ≥ · · · ≥ wl,
∆ω(f) ≥ w0n+
l∑
i=1
(wi − w0)|Vi|.
Proof. The result follows from the simple fact that the contribution of any vertex x ∈ V (G)
to the sum
∑
x∈V (G) f(N(x)) equals deg(x)f(x), where deg(x) denotes the degree of x.
Hence,
∆ω(f) = ∆
∑
x∈V (G)
f(x)
≥
∑
x∈V (G)
deg(x)f(x)
=
∑
x∈V (G)
f(N(x))
≥ w0|V0|+
l∑
i=1
wi|Vi|
= w0n+
l∑
i=1
(wi − w0)|Vi|.
Therefore, the result follows.
Corollary 3.8. The following statements hold for k, l ∈ Z+ and a graph G with minimum
degree δ ≥ 1, maximum degree ∆ and order n.
(i) If k ≤ lδ + 1 and w = (k + l − 1, k + l − 2, . . . , k − 1︸ ︷︷ ︸
l+1
), then γw(G) ≥
⌈
(k+l−1)n
∆+1
⌉
.
(ii) If k ≤ lδ and w = (k, . . . , k︸ ︷︷ ︸
l+1
), then γw(G) ≥
⌈
kn
∆
⌉
.
(iii) If k ≤ lδ + 1 and w = (k, k − 1, . . . , k − 1︸ ︷︷ ︸
l+1
), then γw(G) ≥
⌈
kn
∆+1
⌉
.
(iv) Let w = (w0, . . . , wl) with w0 ≥ · · · ≥ wl. If lδ ≥ wl, then γw(G) ≥
⌈
w0n
∆+w0
⌉
.
In the next subsections we shall show that lower bounds above are tight. Corollary 3.8
implies the following known bounds.
γ(G) ≥
⌈
n
∆+1
⌉
, γt(G) ≥
⌈
n
∆
⌉
, γ
I
(G) ≥
⌈
2n
∆+2
⌉
, γtI(G) ≥
⌈
2n
∆+1
⌉
,
γk(G) ≥
⌈
kn
∆+k
⌉
, γ×k(G) ≥
⌈
kn
∆+1
⌉
, γ{k} ≥
⌈
kn
∆+1
⌉
, γ×k,t(G) ≥
⌈
kn
∆
⌉
.
It is readily seen that γ(w0,...,wl)(G) = 1 if and only if w0 = 1, w1 = 0 and γ(G) = 1.
Next we characterize the graphs with γ(w0,...,wl)(G) = 2.
66 Ars Math. Contemp. 22 (2022) #P1.04 / 57–81
Theorem 3.9. Let w = (w0, . . . , wl) ∈ Z+ × Nl such that w0 ≥ · · · ≥ wl. For a graph
G of order at least three, γ(w0,...,wl)(G) = 2 if and only if one of the following conditions
holds.
(i) w2 = 0, γ(G) = 1 and either w0 = 2 or w0 = w1 = 1.
(ii) w0 = 1, w1 = 0 and γ(G) = 2.
(iii) w0 = 1, w1 = 1 and γt(G) = 2.
(iv) w0 = 2, w1 = 0 and γ2(G) = 2.
(v) w0 = 2, w1 = 1 and γ×2(G) = 2.
Proof. Assume first that γ(w0,...,wl)(G) = 2 and let f(V0, . . . , Vl) be a γ(w0,...,wl)(G)-
function. Notice that w0 ∈ {1, 2} and |V2| ∈ {0, 1}. If |V2| = 1, then w2 = 0 and Vi = ∅
for every i ̸= 0, 2. Hence, γ(G) = 1 and either w0 = 2 or w0 = w1 = 1. Therefore, (i)
follows.
Now we consider the case V2 = ∅. Notice that V1 is a dominating set of cardinality
two, w1 ∈ {0, 1} and Vi = ∅ for every i ̸= 0, 1.
Assume first that w0 = 1 and w1 = 0. If γ(G) = 1, then γ(w0,...,wl)(G) = 1, which
is a contradiction. Hence, γ(G) = 2 and so (ii) follows. For w0 + w1 ≥ 2 we have the
following possibilities.
If w0 = w1 = 1, then V1 is a total dominating set of cardinality two, and so γt(G) = 2.
Therefore, (iii) follows.
If w0 = 2 and w1 = 0, then V1 is a 2-dominating set of cardinality two, which implies
that γ2(G) = 2. Therefore, (iv) follows.
If w0 = 2 and w1 = 1, then V1 is a double dominating set of cardinality two, and this
implies that γ×2(G) = 2. Therefore, (v) follows.
Conversely, if one of the five conditions holds, then it is easy to check that γ(w0,...,wl)(G)
= 2, which completes the proof.
In order to establish the following result, we need to define the following parameter.
ν(w0,...,wl)(G) = max{|V0| : f(V0, . . . , Vl) is a γ(w0,...,wl)(G)-function}.
In particular, for l = 1 and a graph G of order n, we have that ν(w0,w1)(G) = n −
γ(w0,w1)(G).
Theorem 3.10. Let G be a graph of minimum degree δ and order n. The following state-
ments hold for any (w0, . . . , wl) ∈ Z+ × Nl with w0 ≥ · · · ≥ wl.
(i) If there exists i ∈ {1, . . . , l − 1} such that iδ ≥ wi, then
γ(w0,...,wl)(G) ≤ γ(w0,...,wi)(G).
(ii) If l ≥ i+ 1 ≥ w0, then
γ(w0,...,wi,0,...,0)(G) ≤ (i+ 1)γ(G).
A. Cabrera Martı́nez et al.: From Italian domination in lexicographic product graphs to . . . 67
(iii) Let k, i ∈ Z+ such that l ≥ ki, and let (w′0, w′1, . . . , w′i) ∈ Z+ × Nl. If iδ ≥ w′i and
wkj = kw
′
j for every j ∈ {0, 1, . . . , i}, then
γ(w0,...,wl)(G) ≤ kγ(w′0,...,w′i)(G).
(iv) Let k ∈ Z+ and β1, . . . , βk ∈ Z+. If lδ ≥ k + wl > k and w0 + k ≥ β1 ≥ · · · ≥
βk ≥ w1 + k, then
γ(w0+k,β1,...,βk,w1+k,...,wl+k)(G) ≤ γ(w0,...,wl)(G) + k(n− ν(w0,...,wl)(G)).
(v) If lδ ≥ wl ≥ l ≥ 2, then
γ(w0,...,wl)(G) ≤ lγ(w0−l+1,wl−l+1)(G).
(vi) If δ ≥ 1, w0 ≤ l − 1 and wl−1 ≥ 1, then
γ(w0,...,wl−2,1)(G) ≤ γ(w0,...,wl−1,0)(G).
Proof. If there exists i ∈ {1, . . . , l − 1} such that iδ ≥ wi, then for any γ(w0,...,wi)(G)-
function f(V0, . . . , Vi) we define a (w0, . . . , wl)-dominating function g(W0, . . . ,Wl) by
Wj = Vj for every j ∈ {0, . . . , i} and Wj = ∅ for every j ∈ {i + 1, . . . , l}. Hence,
γ(w0,...,wl)(G) ≤ ω(g) = ω(f) = γ(w0,...,wi)(G). Therefore, (i) follows.
Now, assume l ≥ i + 1 ≥ w0. Let S be a γ(G)-set. Let f be the function defined
by f(v) = i + 1 for every v ∈ S and f(v) = 0 for the remaining vertices. Since f
is a (w0, . . . , wi, 0 . . . , 0)-dominating function, we conclude that γ(w0,...,wi,0...,0)(G) ≤
ω(f) = (i+ 1)|S| = (i+ 1)γ(G), which implies that (ii) follows.
In order to prove (iii), assume that l ≥ ki, iδ ≥ w′i and wkj = kw′j for every
j ∈ {0, . . . , i}. Let f ′(V ′0 , . . . , V ′i ) be a γ(w′0,...,w′i)(G)-function. We construct a func-
tion f(V0, . . . , Vl) as f(v) = kf ′(v) for every v ∈ V (G). Hence, Vkj = V ′j for every
j ∈ {0, . . . , i}, while Vj = ∅ for the remaining cases. Thus, for every v ∈ Vkj with
j ∈ {0, . . . , i} we have that f(N(v)) = kf ′(N(v)) ≥ kw′j = wkj , which implies that
f is a (w0, . . . , wl)-dominating function, and so γ(w0,...,wl)(G) ≤ ω(f) = kω(f ′) =
kγ(w′0,...,w′i)(G). Therefore, (iii) follows.
Now, assume that lδ ≥ k + wl > k and w0 + k ≥ β1 ≥ · · · ≥ βk ≥ w1 + k. Let
g(W0, . . . ,Wl) be a γ(w0,...,wl)(G)-function. We construct a function f(V0, . . . , Vl+k) as
f(v) = g(v) + k for every v ∈ V (G) \ W0 and f(v) = 0 for every v ∈ W0. Hence,
Vj+k = Wj for every j ∈ {1, . . . , l}, V0 = W0 and Vj = ∅ for the remaining cases. Thus,
if v ∈ Vj+k and j ∈ {1, . . . , l}, then f(N(v)) ≥ g(N(v)) + k ≥ wj + k, and if v ∈ V0,
then f(N(v)) ≥ g(N(v)) + k ≥ w0 + k. This implies that f is a (w0 + k, β1, . . . , βk,
w1 + k, . . . , wl + k)-dominating function, and so
γ(w0+k,β1,...,βk,w1+k,...,wl+k)(G) ≤ ω(f) = ω(g) + k
l∑
j=1
|Wj |
= γ(w0,...,wl)(G) + k(n− |W0|)
≤ γ(w0,...,wl)(G) + k(n− ν(w0,...,wl)(G)).
Therefore, (iv) follows.
68 Ars Math. Contemp. 22 (2022) #P1.04 / 57–81
Furthermore, if lδ ≥ wl ≥ l ≥ 2, then by applying (iv) for k = l − 1, we deduce that
γ(w0,...,wl)(G) ≤ γ(w0−l+1,wl−l+1)(G) + (l − 1)(n− ν(w0−l+1,wl−l+1)(G))
= lγ(w0−l+1,wl−l+1)(G).
Therefore, (v) follows.
From now on, let δ ≥ 1, w0 ≤ l − 1 and wl−1 ≥ 1. Let f(V0, . . . , Vl) be a
γ(w0,...,wl−1,0)(G)-function. Assume first Vl = ∅. Since wl−1 ≥ 1, we have that f is a
(w0, . . . , wl−2, 1)-dominating function on G, which implies that (vi) follows. Assume now
that there exists v ∈ Vl. If f(N(v)) ≥ l− 1, then the function f ′, defined by f ′(v) = l− 1
and f ′(x) = f(x) for every x ∈ V (G) \ {v}, is a (w0, . . . , wl−1, 0)-dominating function
with ω(f ′) < ω(f), which is a contradiction. Hence, f(N(v)) ≤ l − 2 for every v ∈ Vl.
Since δ ≥ 1, for each vertex x ∈ Vl, we fix one vertex x′ ∈ N(x) and we form a set S
from them such that |S| ≤ |Vl|. Let g be the function defined by g(x) = f(x) + 1 for
any x ∈ S, g(y) = l − 1 for any y ∈ Vl, and g(z) = f(z) for the remaining vertices of
G. Since g(N(x)) ≥ l − 1 ≥ wi for every x ∈ S and i ∈ {0, . . . , l − 2}, g(N(y)) ≥ 1
for every y ∈ Vl−1 ∪ Vl, and g(N(z)) ≥ wi for every z ∈ Vi \ (S ∪ Vl−1 ∪ Vl) and
i ∈ {0, . . . , l − 2}, we conclude that g is a (w0, . . . , wl−2, 1)-dominating function on G.
Therefore, γ(w0,...,wl−2,1)(G) ≤ ω(g) ≤ ω(f) = γ(w0,...,wl−1,0)(G), which completes the
proof of (vi).
In the next subsections we consider several applications of Theorem 3.10 where we
show that the bounds are tight. For instance, the following particular cases will be of
interest.
Corollary 3.11. Let G be a graph of minimum degree δ, and let k, l, w2, . . . , wl ∈ Z+ with
k ≥ w2 ≥ · · · ≥ wl.
(i) If δ ≥ k and w = (k + 1, k, w2, . . . , wl), then γw(G) ≤ γ×k(G).
(ii) If δ ≥ k and w = (k, k, w2, . . . , wl), then γw(G) ≤ γ×k,t(G).
(iii) If lδ ≥ k ≥ l ≥ 2 and w = (k + 1, k, . . . , k︸ ︷︷ ︸
l+1
), then γw(G) ≤ lγ×(k−l+2)(G).
(iv) If lδ ≥ k ≥ l ≥ 2 and w = (k, k, . . . , k︸ ︷︷ ︸
l+1
), then γw(G) ≤ lγ×(k−l+1),t(G).
(v) If l ≥ k, δ ≥ 1 and w = (k, . . . , k︸ ︷︷ ︸
l+1
), then γw(G) ≤ kγt(G).
Proof. If δ ≥ k, then by Theorem 3.10(i) we conclude that (i) and (ii) follows.
If lδ ≥ k ≥ l ≥ 2, then by Theorem 3.10(v) we deduce that
γ(k + 1, k, . . . , k︸ ︷︷ ︸
l+1
)(G) ≤ lγ(k−l+2,k−l+1)(G) = lγ×(k−l+2)(G).
Hence, (iii) follows. By analogy we derive (iv), as γ(k−l+1,k−l+1)(G) = lγ×(k−l+1),t(G).
Finally, if l ≥ k and δ ≥ 1, then by Theorem 3.10(iii) we deduce that
γ(k, . . . , k︸ ︷︷ ︸
l+1
)(G) ≤ kγ(1,1)(G) = kγt(G).
Therefore, (v) follows.
A. Cabrera Martı́nez et al.: From Italian domination in lexicographic product graphs to . . . 69
3.1 Preliminary results on (2, 2, 2)-domination
Theorem 3.12. For any graph G with no isolated vertex, order n and maximum degree ∆,⌈
2n
∆
⌉
≤ γ(2,2,2)(G) ≤ 2γt(G).
Furthermore, if G has minimum degree δ ≥ 2, then
γ(2,2,2)(G) ≤ γ×2,t(G).
Proof. From Corollary 3.8 we deduce the lower bound. The upper bound γ(2,2,2)(G) ≤
2γt(G) follows by Corollary 3.11(v), while, if δ ≥ 2, then we apply Corollary 3.11(ii) to
deduce that γ(2,2,2)(G) ≤ γ×2,t(G). Therefore, the result follows.
The bounds above are tight. For instance, for the graphs G2 and G3 shown in Figure 1
we have that
⌈
2n
∆
⌉
= γ(2,2,2)(G2) = γ×2,t(G2) = 3 and γ(2,2,2)(G3) = 2γt(G3) = 8. No-
tice that every graph Gk,r belonging to the infinite family Hk constructed after Remark 3.3
satisfies the equality γ(2,2,2)(Gk,r) = γ×2,t(Gk,r) = k. Furthermore, from Theorem 3.4
we have that for any corona graph G ∼= G1⊙G2, where G1 does not have isolated vertices,
γ(2,2,2)(G) = 2γ(G) = 2γt(G).
Notice that by Theorem 3.12 we have that γ(2,2,2)(G) ≥
⌈
2n
∆
⌉
≥ 3 for every graph G
with no isolated vertex. Next we characterize all graphs with γ(2,2,2)(G) = 3. To this end,
we need to establish the following lemma.
Lemma 3.13. For a graph G, the following statements are equivalent.
(i) γ(2,2,2)(G) = γ×2,t(G).
(ii) There exists a γ(2,2,2)(G)-function f(V0, V1, V2) such that V2 = ∅.
Proof. If γ(2,2,2)(G) = γ×2,t(G), then for any γ×2,t(G)-set D, the function g(W0,W1,
W2), defined by W1 = D and W0 = V (G) \D, is a γ(2,2,2)(G)-function. Therefore, (ii)
follows.
Conversely, if there exists a γ(2,2,2)(G)-function f(V0, V1, V2) such that V2 = ∅, then
V1 is a double total dominating set of G, and so γ×2,t(G) ≤ |V1| = ω(f) = γ(2,2,2)(G).
Therefore, Theorem 3.12 leads to γ(2,2,2)(G) = γ×2,t(G).
Theorem 3.14. For a graph G, the following statements are equivalent.
(i) γ(2,2,2)(G) = 3.
(ii) γ×2,t(G) = 3.
Proof. Assume first that γ(2,2,2)(G) = 3, and let f(V0, V1, V2) be a γ(2,2,2)(G)-function.
Suppose that there exists u ∈ V2. Since f(N(u)) ≥ 2, we deduce that γ(2,2,2)(G) ≥
4, which is a contradiction. Hence, V2 = ∅ and by Lemma 3.13 we conclude that
γ×2,t(G) = 3.
Conversely, if γ×2,t(G) = 3, then G has minimum degree δ ≥ 2 and so Theorem 3.12
leads to 3 ≤
⌈
2n
∆
⌉
≤ γ(2,2,2)(G) ≤ γ×2,t(G) = 3. Therefore, γ(2,2,2)(G) = 3.
Next we consider the case of graphs with γ(2,2,2)(G) = 4.
70 Ars Math. Contemp. 22 (2022) #P1.04 / 57–81
Theorem 3.15. For a graph G, γ(2,2,2)(G) = 4 if and only if at least one of the following
conditions holds.
(i) γ×2,t(G) = 4.
(ii) γt(G) = 2 and G has minimum degree δ = 1.
(iii) γt(G) = 2 and γ×2,t(G) ≥ 4.
Proof. Assume γ(2,2,2)(G) = 4. Notice that G does not have isolated vertices. Let
f(V0, V1, V2) be a γ(2,2,2)(G)-function. If V2 = ∅, then by Lemma 3.13 we obtain that
γ×2,t(G) = γ(2,2,2)(G) = 4, and so (i) follows.
From now on, assume that |V2| ∈ {1, 2}. If |V2| = 2, then V1 = ∅ and, as a result,
V2 is a total dominating set of G, which implies that γt(G) = 2. On the other side, if
|V2| = 1, then |V1| = 2 and both vertices belonging to V1 are adjacent to the vertex of
weight two, and every v ∈ V0 satisfies N(v) ∩ V2 ̸= ∅ or V1 ⊆ N(v). This implies that
the union of V2 with a singleton subset of V1 forms a total dominating set of G, and again
γt(G) = 2. Now, if δ ≥ 2, then Theorem 3.12 leads to 4 = γ(2,2,2)(G) ≤ γ×2,t(G).
Hence, by Theorem 3.14 we conclude that either δ = 1 or γ×2,t(G) ≥ 4. Therefore, either
(ii) or (iii) holds.
Conversely, if γ×2,t(G) = 4, then G has minimum degree δ ≥ 2 and by Theorem 3.12
we have that 3 ≤ γ(2,2,2)(G) ≤ 4. Hence, by Theorem 3.14 we deduce that γ(2,2,2)(G) =
4. Finally, if γt(G) = 2, then Theorem 3.12 leads to 3 ≤ γ(2,2,2)(G) ≤ 4. Therefore, if
δ = 1 or γ×2,t(G) ≥ 4, then Theorem 3.14 leads to γ(2,2,2)(G) = 4.
Theorem 3.12 implies the next result.
Corollary 3.16. For any integer n ≥ 3,
γ(2,2,2)(Cn) = n.
In order to give the value of γ(2,2,2)(Pn), we recall the following well-known result.
Proposition 3.17 ([14]). For any integer n ≥ 3,
γt(Pn) =

n
2 if n ≡ 0 (mod 4),
n+1
2 if n ≡ 1, 3 (mod 4),
n
2 + 1 if n ≡ 2 (mod 4).
Lemma 3.18. If Pn = u1u2 . . . un is a path of order n ≥ 6, then there exists a γ(2,2,2)(Pn)-
function f such that f(un) = f(un−3) = 0 and f(un−1) = f(un−2) = 2.
Proof. Let f(V0, V1, V2) be a γ(2,2,2)(Pn)-function such that |V2| is maximum. Since un
is a leaf, f(un−1) = 2. Notice that f(un) + f(un−2) ≥ 2. Hence, we can assume that
f(un−2) = 2 and f(un) = 0. Now, if f(un−3) > 0, then we can define a (2, 2, 2)-
dominating function f ′ by f ′(un−3) = 0, f ′(un−5) = min{2, f(un−5) + f(un−3)} and
f ′(ui) = f(ui) for the remaining cases. Since ω(f ′) ≤ ω(f) = γ(2,2,2)(Pn), either f ′
is a γ(2,2,2)(Pn)-function with f ′(un−3) = 0 or f(un−3) = 0. In both cases the result
follows.
A. Cabrera Martı́nez et al.: From Italian domination in lexicographic product graphs to . . . 71
Proposition 3.19. For any integer n ≥ 3,
γ(2,2,2)(Pn) = 2γt(Pn) =

n if n ≡ 0 (mod 4),
n+ 1 if n ≡ 1, 3 (mod 4),
n+ 2 if n ≡ 2 (mod 4).
Proof. Since Theorem 3.12 leads to γ(2,2,2)(Pn) ≤ 2γt(Pn), we only need to prove
that γ(2,2,2)(Pn) ≥ 2γt(Pn). We proceed by induction on n. It is easy to check that
γ(2,2,2)(Pn) = 2γt(Pn) for n = 3, 4, 5, 6. This establishes the base case. Now, we assume
that n ≥ 7 and γ(2,2,2)(Pk) ≥ 2γt(Pk) for k < n. Let f(V0, V1, V2) be a γ(2,2,2)(Pn)-
function which satisfies Lemma 3.18, and let f ′ be the restriction of f to V (Pn−4), where
Pn = u1u2 . . . un and Pn−4 = u1u2 . . . un−4. Hence, by applying the induction hypothe-
sis,
γ(2,2,2)(Pn) = ω(f) = ω(f
′) + 4 ≥ γ(2,2,2)(Pn−4) + 4 ≥ 2γt(Pn−4) + 4 ≥ 2γt(Pn).
To conclude the proof we apply Proposition 3.17.
3.2 Preliminary results on (2, 2, 1)-domination
Theorem 3.20. For any graph G with no isolated vertex, order n and maximum degree ∆,⌈
2n+ γt(G)
∆ + 1
⌉
≤ γ(2,2,1)(G) ≤ min{3γ(G), 2γt(G)}.
Furthermore, if G has minimum degree δ ≥ 2, then
γ(2,2,1)(G) ≤ γ×2,t(G).
Proof. In order to prove the upper bound γ(2,2,1)(G) ≤ 2γt(G), we apply Remark 3.2 and
Theorem 3.12, i.e., γ(2,2,1)(G) ≤ γ(2,2,2) ≤ 2γt(G).
Now, let S be a γ(G)-set. Since G does not have isolated vertex, for each vertex
x ∈ S such that N(x) ∩ S = ∅, we fix one vertex x′ ∈ N(x) and we form a set S′
from them. Hence, S ∪ S′ is a total dominating set and |S ∪ S′| = |S| + |S′| ≤ 2γ(G).
Notice that the function g(X0, X1, X2) defined by X2 = S and X1 = S′, is a (2, 2, 1)-
dominating function on G. Thus, γ(2,2,1)(G) ≤ ω(g) = 2|S| + |S′| ≤ 3γ(G), and so
γ(2,2,1)(G) ≤ min{2γt(G), 3γ(G)}.
On the other side, if G has minimum degree δ ≥ 2, then by Corollary 3.11(ii) we have
that γ(2,2,1)(G) ≤ γ×2,t(G).
In order to prove the lower bound, let f(V0, V1, V2) be a γ(2,2,1)(G)-function. Since
V1 ∪ V2 is a total dominating set, γt(G) ≤ |V1|+ |V2|. Furthermore, from Lemma 3.7 we
have, 2n − |V2| ≤ ∆γ(2,2,1)(G), which implies that 2n + γt(G) ≤ 2n + |V1| + |V2| ≤
∆γ(2,2,1)(G)+|V1|+2|V2| = (∆+1)γ(2,2,1)(G). Therefore, the lower bound follows.
The bounds above are tight. For instance, the graph in Figure 3 satisfies γ(2,2,1)(G) =
3γ(G) = 9. Next we show that the remaining two bounds are also achieved.
Corollary 3.21. Let G be a graph with no isolated vertex, order n and maximum degree
∆. If γt(G) < n+∆+1∆+1/2 , then
γ(2,2,1)(G) = 2γt(G) or γ(2,2,1)(G) =
⌈
2n+ γt(G)
∆ + 1
⌉
.
72 Ars Math. Contemp. 22 (2022) #P1.04 / 57–81
2 2 21 1 1
3 3 3
Figure 3: This figure shows a γ(2,2,1)(G)-function and a γ(2,2,2,0)(G)-function on the same
graph.
Proof. If γ(2,2,1)(G) ̸=
⌈
2n+γt(G)
∆+1
⌉
and γ(2,2,1)(G) ̸= 2γt(G), then by Theorem 3.20 we
deduce that
⌈
2n+γt(G)
∆+1
⌉
+ 1 ≤ γ(2,2,1)(G) ≤ 2γt(G) − 1, which implies that γt(G) ≥
n+∆+1
∆+1/2 . Therefore, the result follows.
For the graphs G2 and G3 illustrated in Figure 1 we have that γt(G2) = 2 < 229 =
n+∆+1
∆+1/2 and γt(G3) = 4 <
32
7 =
n+∆+1
∆+1/2 . Notice that, γ(2,2,1)(G2) = 3 =
⌈
2n+γt(G2)
∆+1
⌉
and γ(2,2,1)(G3) = 8 = 2γt(G3).
Below we characterize the graphs with γ(2,2,1)(G) = 3.
Theorem 3.22. For a graph G with no isolated vertex, the following statements are equiv-
alent.
(i) γ(2,2,1)(G) = 3.
(ii) γ(G) = 1 or γ×2,t(G) = 3.
Proof. Assume first that γ(2,2,1)(G) = 3, and let f(V0, V1, V2) be a γ(2,2,1)(G)-function.
If V2 ̸= ∅, then V2 is a dominating set of cardinality one. Hence, γ(G) = 1. Now, if
V2 = ∅, then V1 is a double total dominating set of cardinality three. Thus, γ×2,t(G) = 3.
On the other side, by Theorem 3.20 we have that 3 ≤
⌈
2n+γt(G)
∆+1
⌉
≤ γ(2,2,1)(G) ≤
3γ(G). Hence, if γ(G) = 1, then γ(2,2,1)(G) = 3. Now, if γ×2,t(G) = 3, then G has
minimum degree δ ≥ 2 and by Theorem 3.20 we have that γ(2,2,1)(G) ≤ γ×2,t(G) = 3.
Therefore, γ(2,2,1)(G) = 3.
Next we consider the case of graphs with γ(2,2,1)(G) = 4.
Theorem 3.23. For a graph G, the following statements are equivalent.
(i) γ(2,2,1)(G) = 4.
(ii) γt(G) = γ(G) = 2 or γ×2,t(G) = 4.
Proof. Assume γ(2,2,1)(G) = 4. Notice that G does not have isolated vertices and, by
Theorem 3.20, we have that γ(G) ≥ 2. Let f(V0, V1, V2) be a γ(2,2,1)(G)-function. If V2 =
A. Cabrera Martı́nez et al.: From Italian domination in lexicographic product graphs to . . . 73
∅, then V1 is a double total dominating set of cardinality four. Hence, 3 ≤ γ×2,t(G) ≤
|V1| = 4, and Theorem 3.22 implies that γ×2,t(G) = 4.
From now on, assume that |V2| ∈ {1, 2}. If |V2| = 2, then V1 = ∅ and, as a result, V2 is
a total dominating set of G, which implies that γt(G) = γ(G) = 2. Now, if |V2| = 1, then
|V1| = 2 and both vertices belonging to V1 are adjacent to the vertex of weight two, and
every v ∈ V0 satisfies N(v)∩V2 ̸= ∅ or V1 ⊆ N(v). This implies that the union of V2 with
a singleton subset of V1 forms a total dominating set of G, and again γt(G) = γ(G) = 2.
Conversely, if γ×2,t(G) = 4, then G has minimum degree δ ≥ 2 and by Theorem 3.20
we have that 3 ≤ γ(2,2,1)(G) ≤ γ×2,t(G) = 4. Hence, by Theorem 3.22 we deduce that
γ(2,2,1)(G) = 4. Finally, if γt(G) = 2, then Theorem 3.20 leads to 3 ≤ γ(2,2,1)(G) ≤ 4.
Therefore, if γ(G) = 2 then by Theorem 3.22 we conclude that γ(2,2,1)(G) = 4.
Lemma 3.24. For any integer n ≥ 3,
γ(2,2,1)(Pn) ≤
n−
⌊
n
7
⌋
+ 1 if n ≡ 1, 2 (mod 7),
n−
⌊
n
7
⌋
otherwise.
Proof. First we show how to construct a (2, 2, 1)-dominating function f on Pn for n ∈
{2, . . . , 8}.
• n = 2: f(u1) = 2 and f(u2) = 1.
• n = 3: f(u1) = 0, f(u2) = 2 and f(u3) = 1.
• n = 4: f(u1) = f(u4) = 0 and f(u2) = f(u3) = 2.
• n = 5: f(u1) = f(u5) = 0, f(u2) = f(u4) = 2 and f(u3) = 1.
• n = 6: f(u1) = f(u6) = 0, f(u2) = f(u5) = 2 and f(u3) = f(u4) = 1.
• n = 7: f(u1) = f(u4) = f(u7) = 0, f(u2) = f(u6) = 2 and f(u3) = f(u5) = 1.
• n = 8: f(u1) = f(u4) = f(u8) = 0, f(u2) = f(u6) = f(u7) = 2 and
f(u3) = f(u5) = 1.
We now proceed to describe the construction of f for any n = 7q + r, where q ≥ 1 and
0 ≤ r ≤ 6. We partition V (Pn) = {u1, . . . , un} into q sets of cardinality 7 and for r ≥ 1
one additional set of cardinality r, in such a way that the subgraph induced by all these sets
are paths.
For any r ̸= 1, the restriction of f to each of these q paths of length 7 corresponds to
the weights associated above with P7, while for the path of length r (if any) we take the
weights associated above with Pr. The case r = 1 and q ≥ 2 is slightly different, as for the
first q − 1 paths of length 7 we take the weights associated above with P7 and for the last
8 vertices of Pn we take the weights associated above with P8.
Notice that, for n ≡ 1, 2 (mod 7), we have that γ(2,2,1)(Pn) ≤ ω(f) = 6q + r + 1 =
n−
⌊
n
7
⌋
+1, while for n ̸≡ 1, 2 (mod 7) we have γ(2,2,1)(Pn) ≤ ω(f) = 6q+r = n−
⌊
n
7
⌋
.
Therefore, the result follows.
Lemma 3.25. Let P7 = x1 . . . x7 be a subgraph of Cn and X = {x1, . . . , x7}. If f is a
(2, 2, 1)-dominating function on Cn, then
f(X) ≥ 6.
74 Ars Math. Contemp. 22 (2022) #P1.04 / 57–81
Proof. Notice that f({x1, x2, x3}) ≥ 2 and f({x4, x5, x6, x7}) ≥ 3 as f is a (2, 2, 1)-
dominating function. If f({x1, x2, x3}) ≥ 3, then we are done. Hence, we assume that
f({x1, x2, x3}) = 2. In this case, it is not difficult to deduce that f({x4, x5, x6, x7}) ≥ 4,
which implies that f(X) ≥ 6, as desired. Therefore, the proof is complete.
Lemma 3.26. For any integer n ≥ 3,
γ(2,2,1)(Cn) ≥
{
n− ⌊n7 ⌋+ 1 if n ≡ 1, 2 (mod 7),
n− ⌊n7 ⌋ otherwise.
Proof. It is easy to check that γ(2,2,1)(Cn) = n for every n ∈ {3, 4, 5, 6}. Now, let
n = 7q + r, with 0 ≤ r ≤ 6 and q ≥ 1. Let f(V0, V1, V2) be a γ(2,2,1)(Cn)-function.
If r = 0, then by Lemma 3.25 we have that ω(f) ≥ 6q = n − ⌊n7 ⌋. From now on we
assume that r ≥ 1. By Proposition 3.5 and Lemma 3.24 we deduce that γ(2,2,1)(Cn) ≤
γ(2,2,1)(Pn) < n, which implies that V2 ̸= ∅, otherwise there exists u ∈ V (Cn) = V0∪V1
such that N(u) ∩ V0 ̸= ∅ and so |N(u) ∩ V1| ≤ 1, which is a contradiction. Let x ∈ V2
and, without loss of generality, we can label the vertices of Cn in such a way that x = u1,
and u2 ∈ V1 ∪ V2 whenever r ≥ 2. We partition V (Cn) into X = {u1, . . . , ur} and
Y = {ur+1, . . . , un}. Notice that Lemma 3.25 leads to f(Y ) ≥ 6q.
Now, if r ∈ {1, 2}, then f(X) ≥ r + 1, which implies that ω(f) ≥ r + 1 + 6q =
n− ⌊n7 ⌋+ 1. Analogously, if r = 3, then f(X) ≥ r and so ω(f) ≥ r + 6q = n− ⌊
n
7 ⌋.
Finally, if r ∈ {4, 5, 6}, then as f is a (2, 2, 1)-dominating function we deduce that
f(X) ≥ r, which implies that ω(f) ≥ r + 6q = n− ⌊n7 ⌋.
The following result is a direct consequence of Proposition 3.5 and Lemmas 3.24
and 3.26.
Proposition 3.27. For any integer n ≥ 3,
γ(2,2,1)(Cn) = γ(2,2,1)(Pn) =
{
n− ⌊n7 ⌋+ 1 if n ≡ 1, 2 (mod 7),
n− ⌊n7 ⌋ otherwise.
3.3 Preliminary results on (2, 2, 0)-domination
Theorem 3.28. For any graph G with no isolated vertex, order n and maximum degree ∆,⌈
2n
∆+ 1
⌉
≤ γ(2,2,0)(G) ≤ 2γ(G).
Furthermore, if G has minimum degree δ ≥ 2, then
γ(2,2,0)(G) ≤ γ×2,t(G).
Proof. The upper bound γ(2,2,0)(G) ≤ ω(g) = 2γ(G) is derived by we applying Theo-
rem 3.10(ii) for i = 1 and l = 2. Furthermore, if G has minimum degree δ ≥ 2, then by
Corollary 3.11(ii) we have that γ(2,2,0)(G) ≤ γ×2,t(G).
Now, let f(V0, V1, V2) be a γ(2,2,0)(G)-function. From Lemma 3.7 we deduce that
2(n − |V2|)) ≤ ∆γ(2,2,0)(G), which implies that 2n ≤ 2n + |V1| ≤ (∆ + 1)γ(2,2,0)(G).
Therefore, the result follows.
A. Cabrera Martı́nez et al.: From Italian domination in lexicographic product graphs to . . . 75
Theorem 3.28 implies that, if γ(G) = n∆+1 , then γ(2,2,0)(G) =
2n
∆+1 . It is easy to
see that a graph satisfies γ(G) = n∆+1 if and only if there exists a γ(G)-set S which is a
2-packing1 and every vertex in S has degree ∆. The upper bound γ(2,2,0)(G) ≤ 2γ(G)
is achieved for the graph G shown in Figure 2, which satisfies γ(2,2,0)(G) = 2γ(G) = 6.
Furthermore, by Theorem 3.4 we have that for any corona graph G ∼= G1 ⊙G2, where G1
does not have isolated vertices, γ(2,2,0)(G) = 2γ(G).
As shown in Theorem 3.9, for a graph G, γ(2,2,0)(G) = 2 if and only if γ(G) = 1.
Now we consider the case γ(2,2,0)(G) = 3.
Theorem 3.29. For a graph G, γ(2,2,0)(G) = 3 if and only if γ×2,t(G) = γ(G) + 1 = 3.
Proof. Assume γ(2,2,0)(G) = 3. By Theorem 3.9 we have that γ(G) ≥ 2. Let f(V0, V1, V2)
be a γ(2,2,0)(G)-function. If |V2| = 1 then |V1| = 1, and as f is a (2, 2, 0)-dominating
function we deduce that N [V2] = V (G), i.e., γ(G) = 1, which is a contradiction. Thus,
V2 = ∅ and |V1| = 3. Notice that V1 is a double total dominating set and since γ(G) ≥ 2,
it follows that 3 ≤ γ(G) + 1 ≤ γ×2,t(G) ≤ |V1| = 3. Hence, γ×2,t(G) = γ(G) + 1 = 3,
as required.
Conversely, assume γ×2,t(G) = γ(G) + 1 = 3. Since G has minimum degree at least
two, Theorem 3.28 leads to 2 ≤ γ(2,2,0)(G) ≤ γ×2,t(G) = 3, and so Theorem 3.9 implies
that γ(2,2,0)(G) = 3, which completes the proof.
Theorem 3.30. For a graph G, γ(2,2,0)(G) = 4 if and only if one of the following condi-
tions holds.
(i) G ∼= K1 ∪G1, where G1 is a graph with γ(G1) = 1.
(ii) γ×2,t(G) = 4.
(iii) γ(G) = 2 and G has minimum degree one.
(iv) γ(G) = 2 and γ×2,t(G) ≥ 4.
Proof. If K1 is a component of G, then by Theorem 3.9 we conclude that γ(2,2,0)(G) = 4
if and only if G ∼= K1 ∪G1, where G1 is a graph with γ(G1) = 1.
From now on, we consider the case where G is a graph with no isolated vertex. Assume
γ(2,2,0)(G) = 4 and let f(V0, V1, V2) be a γ(2,2,0)(G)-function. If V2 = ∅, then V1 is a
double total dominating set of G. In this case, G has minimum degree δ ≥ 2 and by
Theorem 3.28 we have that γ×2,t(G) ≤ |V1| = 4 = γ(2,2,0)(G) ≤ γ×2,t(G). Hence (ii)
follows.
Now, assume that |V2| ∈ {1, 2}. If |V2| = 2, then V1 = ∅, and so γ(G) ≤ 2. Now, if
|V2| = 1, then |V1| = 2 and both vertices belonging V1 are adjacent to the vertex of weight
two, and every v ∈ V0 satisfies N(v) ∩ V2 ̸= ∅ or V1 ⊆ N(v). This implies that the union
of V2 with a singleton subset of V1 forms a dominating set of G, and again γ(G) ≤ 2. Thus,
from Theorem 3.9 we deduce that γ(G) = 2. Furthermore, if δ ≥ 2, then by Theorem 3.28
we have that γ×2,t(G) ≥ γ(2,2,0) = 4. Therefore, either (iii) or (iv) holds.
Conversely, if γ×2,t(G) = 4, then Theorem 3.28 leads to 2 ≤ γ(2,2,0) ≤ γ×2,t(G) = 4.
Hence, by Theorems 3.9 and 3.29 we deduce that γ(2,2,0)(G) = 4. Analogously, if γ(G) =
2 and δ ≥ 1, then Theorem 3.28 leads to 2 ≤ γ(2,2,0) ≤ 2γ(G) = 4. Thus, by Theorem 3.9
we have that 3 ≤ γ(2,2,0) ≤ 4. In particular, if δ = 1 or γ×2,t(G) ≥ 4, then Theorem 3.29
leads to γ(2,2,0)(G) = 4, which completes the proof.
1A set S ⊆ V (G) is a 2-packing if N [u] ∩N [v] = ∅ for every pair of different vertices u, v ∈ S.
76 Ars Math. Contemp. 22 (2022) #P1.04 / 57–81
Lemma 3.31. For a graph G, the following statements are equivalent.
(i) γ(2,2,0)(G) = 2γ(G).
(ii) There exists a γ(2,2,0)(G)-function f(V0, V1, V2) such that V1 = ∅.
Proof. First, we assume that γ(2,2,0)(G) = 2γ(G) and let D be a γ(G)-set. Hence, the
function f(V0, V1, V2), defined by V2 = D and V0 = V (G) \D, is a γ(2,2,0)(G)-function
which satisfies (ii), as desired.
Finally, we assume that there exists a γ(2,2,0)(G)-function f(V0, V1, V2) such that V1 =
∅. This implies that V2 is a dominating set of G. Hence, γ(2,2,0)(G) ≤ 2γ(G) ≤ 2|V2| =
γ(2,2,0)(G), and the desired equality holds, which completes the proof.
The following result provides the (2, 2, 0)-domination number of paths and cycles.
Proposition 3.32. For any integer n ≥ 3,
γ(2,2,0)(Pn) = γ(2,2,0)(Cn) = 2
⌈n
3
⌉
.
Proof. We first prove that γ(2,2,0)(Cn) ≥ 2
⌈
n
3
⌉
. Let f(V0, V1, V2) be a γ(2,2,0)(Cn)-
function. If V1 = ∅, then by Lemma 3.31 it follows that γ(2,2,0)(Cn) = 2γ(Cn) = 2
⌈
n
3
⌉
.
If V1 ̸= ∅, then 1 + 2|V2| ≤ |V1|+ 2|V2| = γ(2,2,0)(Cn) ≤ 2γ(Cn) = 2
⌈
n
3
⌉
, which leads
to |V2| ≤
⌈
n
3
⌉
− 1. By Lemma 3.7 we have that γ(2,2,0)(Cn) ≥ n− |V2| ≥ n−
⌈
n
3
⌉
+1 ≥
2
⌈
n
3
⌉
, as desired.
Therefore, by the inequality above, Proposition 3.5 and Theorem 3.28 we deduce that
2⌈n3 ⌉ ≤ γ(2,2,0)(Cn) ≤ γ(2,2,0)(Pn) ≤ 2γ(Pn) = 2⌈
n
3 ⌉. Thus, we have equalities in the
inequality chain above, which implies that the result follows.
3.4 Preliminary results on (2, 1, 0)-domination
Given a graph G, we use the notation L(G) and S(G) for the sets of leaves and support
vertices, respectively.
Theorem 3.33. For any graph G with no isolated vertex, order n and maximum degree ∆,⌈
2n
∆+ 1
⌉
≤ γ(2,1,0)(G) ≤ min{γ×2(G)− |L(G)|+ |S(G)|, 2γ(G)}.
Proof. If f(V0, V1, V2) is a γ(2,1,0)(G)-function, then from Lemma 3.7 we conclude that
2n−|V1|−2|V2| ≤ ∆γ(2,1,0)(G). Hence, 2n ≤ ∆γ(2,1,0)(G)+ω(f) = (∆+1)γ(2,1,0)(G).
Therefore, the lower bound follows.
Let D be a γ×2(G)-set. Notice that S(G) ∪ L(G) ⊆ D. Since |N [v] ∩ D| ≥ 2 for
every v ∈ V (G), the function g(V0, V1, V2) defined by V1 = D \ (L(G) ∪ S(G)) and
V2 = S(G), is a (2, 1, 0)-dominating function. Hence, γ(2,1,0)(G) ≤ ω(g) = γ×2(G) −
|L(G)|+ |S(G)|.
By Remark 3.2, γ(2,1,0)(G) ≤ γ(2,2,0)(G), hence the upper bound γ(2,1,0)(G) ≤
2γ(G) is derived from Theorem 3.28. Therefore, γ(2,1,0)(G) ≤ min{γ×2(G)− |L(G)|+
|S(G)|, 2γ(G)}.
A. Cabrera Martı́nez et al.: From Italian domination in lexicographic product graphs to . . . 77
The bounds above are tight. For instance, for the graph G1 shown in Figure 1 we have
that γ(2,1,0)(G1) =
⌈
2n
∆+1
⌉
= γ×2(G1) = 2γ(G1) = 4. As an example of graph of
minimum degree one where γ(2,1,0)(G) = γ×2(G) − |L(G)| + |S(G)| we take the graph
G obtained from a star graph K1,r, r ≥ 3, by subdividing one edge just once. In such a
case, γ(2,1,0)(G) = 4 = γ×2(G)− |L(G)|+ |S(G)|. Another example is the graph shown
in Figure 2 which satisfies γ(2,1,0)(G) = γ×2(G)− |L(G)|+ |S(G)| = 6.
Notice that γ(2,1,0)(G) ≥
⌈
2n
∆+1
⌉
≥ 2. As shown in Theorem 3.9, γ(2,1,0)(G) = 2 if
and only if γ(G) = 1. Next we characterize the graph satisfying γ(2,1,0)(G) = 3.
Theorem 3.34. For a graph G, γ(2,1,0)(G) = 3 if and only if γ×2(G) = γ(G) + 1 = 3.
Proof. Assume γ(2,1,0)(G) = 3. By Theorem 3.9 we have that γ(G) ≥ 2. Let f(V0, V1, V2)
be a γ(2,1,0)(G)-function. If |V2| = 1 then N [V2] = V (G), i.e., γ(G) = 1, which is a con-
tradiction. Thus, V2 = ∅ and |V1| = 3, which implies that V1 is a double dominating set.
Hence, 3 ≤ γ(G) + 1 ≤ γ×2(G) ≤ |V1| = 3. Therefore, γ×2(G) = γ(G) + 1 = 3.
Conversely, assume γ×2(G) = γ(G) + 1 = 3. Notice that G has minimum degree
δ ≥ 1 and so by Theorems 3.9 and 3.33 we have that 3 ≤ γ(2,1,0)(G) ≤ γ×2(G) = 3,
which implies that γ(2,1,0)(G) = 3.
Next we consider the case of graphs with γ(2,1,0)(G) = 4.
Theorem 3.35. For a graph G, γ(2,1,0)(G) = 4 if and only if one of the following condi-
tions is satisfied.
(i) G ∼= K1 ∪G1, where G1 is a graph with γ(G1) = 1.
(ii) γ×2(G) = 4.
(iii) γ(G) = 2 and γ×2(G) ≥ 4.
Proof. If K1 is a component of G, then by Theorem 3.9 we conclude that γ(2,1,0)(G) = 4
if and only if G ∼= K1 ∪G1, where G1 is a graph with γ(G1) = 1.
From now on, we consider the case where G is a graph with no isolated vertex. Assume
γ(2,1,0)(G) = 4. By Theorem 3.33 we deduce that γ×2(G) ≥ 4 and γ(G) ≥ 2. Let
f(V0, V1, V2) be a γ(2,1,0)(G)-function. If V2 = ∅, then V1 is a double dominating set of
G, which implies that γ×2(G) ≤ |V1| = 4. Hence, (ii) follows. From now on, assume
|V2| ∈ {1, 2}. If |V2| = 2, then V1 = ∅ and so, V2 is a dominating set of G, which implies
that γ(G) = 2. If |V2| = 1, then for every v ∈ V1 we have that V2 ∪ {v} is a dominating
set of G. Hence, γ(G) = 2. Therefore, (iii) follows.
Conversely, if (ii) or (iii) holds, then by Theorems 3.33 we have that 2 ≤ γ(2,1,0)(G) ≤
4. Therefore, by Theorems 3.9 and 3.34 we deduce that γ(2,1,0)(G) = 4, which completes
the proof.
The formulas on the {k}-dominating number of cycles and paths were obtained in [17].
We present here the particular case of k = 2, as γ{2}(G) = γ(2,1,0)(G).
Proposition 3.36 ([17]). For any integer n ≥ 3,
γ{2}(Cn) =
⌈
2n
3
⌉
and γ{2}(Pn) = 2
⌈n
3
⌉
.
78 Ars Math. Contemp. 22 (2022) #P1.04 / 57–81
3.5 Preliminary results on (2, 2, 2, 0)-domination
The following result is a direct consequence of Theorem 3.10(i), (ii) and (vi).
Corollary 3.37. For any graph G with no isolated vertex,
γ(2,2,1)(G) ≤ γ(2,2,2,0)(G) ≤ min{3γ(G), γ(2,2,2)(G)}.
The bounds above are tight. For instance, every graph Gk,r belonging to the in-
finite family Hk constructed after Remark 3.3 satisfies the equalities γ(2,2,1)(Gk,r) =
γ(2,2,2)(Gk,r) = γ(2,2,2,0)(Gk,r) = k. In contrast, the graph shown in Figure 2 satis-
fies γ(2,2,1)(G) = 6 < 7 = γ(2,2,2,0)(G) < 8 = γ(2,2,2)(G). Moreover, Figure 3 illustrates
a graph G with γ(2,2,1)(G) = γ(2,2,2,0)(G) = 3γ(G) = 9.
In order to characterize the graphs with γ(2,2,2,0)(G) ∈ {3, 4}, we need to establish the
following lemma.
Lemma 3.38. For a graph G, the following statements are equivalent.
(i) γ(2,2,2,0)(G) = γ(2,2,2)(G).
(ii) There exists a γ(2,2,2,0)(G)-function f(V0, V1, V2, V3) such that V3 = ∅.
Proof. If γ(2,2,2,0)(G) = γ(2,2,2)(G), then for any γ(2,2,2)(G)-function f(V0, V1, V2), there
exists a γ(2,2,2,0)(G)-function g(W0,W1,W2,W3) defined by W0 = V0, W1 = V1, W2 =
V2 and W3 = ∅. Therefore, (i) implies (ii).
Conversely, if there exists a γ(2,2,2,0)(G)-function f(V0, V1, V2, V3) such that V3 = ∅,
then the function g(W0,W1,W2), defined by W0 = V0, W1 = V1 and W2 = V2, is a
(2, 2, 2)-dominating function on G, and so γ(2,2,2)(G) ≤ ω(g) = ω(f) = γ(2,2,2,0)(G).
Therefore, Corollary 3.37 leads to γ(2,2,2,0)(G) = γ(2,2,2)(G), which completes the proof.
Theorem 3.39. For a graph G, the following statements are equivalent.
(i) γ(2,2,2,0)(G) = 3.
(ii) γ(G) = 1 or γ×2,t(G) = 3.
Proof. Assume first that γ(2,2,2,0)(G) = 3, and let f(V0, V1, V2, V3) be a γ(2,2,2,0)(G)-
function. Notice that |V3| ∈ {0, 1}. If |V3| = 1, then V1 ∪ V2 = ∅, which implies that V3
is a dominating set of cardinality one. Hence, γ(G) = 1.
If V3 = ∅, then by Lemma 3.38 we have that γ(2,2,2)(G) = γ(2,2,2,0)(G) = 3, and by
Theorem 3.14 we deduce that γ×2,t(G) = 3.
Conversely, if γ(G) = 1, then Corollary 3.37 leads to 3 ≤ γ(2,2,2,0)(G) ≤ 3γ(G) = 3.
Moreover, if γ×2,t(G) = 3, then G has minimum degree δ ≥ 2 and so Theorem 3.10(i)
leads to 3 ≤ γ(2,2,2,0)(G) ≤ γ(2,2,2)(G) ≤ γ×2,t(G) = 3. Therefore, γ(2,2,2,0)(G) = 3.
Theorem 3.40. For a graph G, γ(2,2,2,0)(G) = 4 if and only if at least one of the following
conditions holds.
(i) γ×2,t(G) = 4.
(ii) γ(G) = γt(G) = 2 and G has minimum degree δ = 1.
A. Cabrera Martı́nez et al.: From Italian domination in lexicographic product graphs to . . . 79
(iii) γ(G) = γt(G) = 2 and γ×2,t(G) ≥ 4.
Proof. Assume γ(2,2,2,0)(G) = 4. Let f(V0, V1, V2, V3) be a γ(2,2,2,0)(G)-function. Hence,
|V3| ∈ {0, 1}. If |V3| = 1, then V3 is a dominating set of cardinality one. Hence, γ(G) = 1,
which is a contradiction with Theorem 3.39. Hence, V3 = ∅, and so, Lemma 3.38 leads to
γ(2,2,2)(G) = γ(2,2,2,0)(G) = 4. Thus, by Theorems 3.15 and 3.39 we deduce (i) – (iii).
Conversely, if conditions (i) – (iii) hold, then by Theorem 3.14 we have that
γ(2,2,2)(G) = 4. Corollary 3.37 leads to 3 ≤ γ(2,2,2,0)(G) ≤ γ(2,2,2)(G) = 4. No-
tice that if δ ≥ 2, then γ(G) ≥ 2 and γ×2,t(G) ≥ 4. Hence, Theorem 3.39 leads to
γ(2,2,2,0)(G) = 4.
Proposition 3.41. For any integer n ≥ 3,
γ(2,2,2,0)(Cn) = n.
Proof. By Corollaries 3.16 and 3.37 we have that γ(2,2,2,0)(Cn) ≤ γ(2,2,2)(Cn) = n.
We only need to prove that γ(2,2,2,0)(Cn) ≥ n. Let f(V0, V1, V2, V3) be a γ(2,2,2,0)(G)-
function such that |V3| is minimum. If V3 = ∅, then by Lemma 3.38 and Corollary 3.16
we conclude that γ(2,2,2,0)(Cn) = n. Assume V3 ̸= ∅. If v ∈ V3, then N(v) ⊆ V0 as
otherwise, by choosing one vertex u ∈ N(v) \ V0, the function f ′ defined by f ′(v) = 2,
f ′(u) = min{2, f(u) + 1} and f ′(x) = f(x) for the remaining vertices, is a (2, 2, 2, 0)-
dominating function with ω(f ′) ≤ ω(f) and |V ′3 | < |V3|, which is a contradiction. Hence,∑
x∈V3 f(N [x]) = 3|V3|. Now, we observe that
2
∑
x∈V (Cn)\N [V3]
f(x) ≥
∑
x∈V (Cn)\N [V3]
 ∑
u∈N(x)
f(u)
 ≥ 2(n− 3|V3|).
Therefore,
γ(2,2,2,0)(Cn) = ω(f) =
∑
x∈V3
f(N [x]) +
∑
x∈V (Cn)\N [V3]
f(x)
≥ 3|V3|+ (n− 3|V3|) = n,
and the result follows.
Proposition 3.42. For any integer n ≥ 3,
γ(2,2,2,0)(Pn) =
{
6 if n = 5,
n otherwise.
Proof. It is easy to check that γ(2,2,2,0)(Pn) = n for n = 3, 4, 6, 7, 8, and also
γ(2,2,2,0)(P5) = 6. From now on, assume n ≥ 9. By Propositions 3.5 and 3.41 we have that
n = γ(2,2,2,0)(Cn) ≤ γ(2,2,2,0)(Pn). Hence, we only need to prove that γ(2,2,2,0)(Pn) ≤ n.
To this end, we proceed to construct a (2, 2, 2, 0)-dominating function f(V0, V1, V2, V3) on
Pn = v1v2 . . . vn such that ω(f) = n.
• If n ≡ 0 (mod 3), then we set V3 =
⋃n/3
i=1{v3i−1} and V0 = V (G) \ V3.
• If n ≡ 1 (mod 3), then we set V3 =
⋃(n−4)/3
i=1 {v3i−1}, V2 = {vn−2, vn−1} and
V0 = V (G) \ (V2 ∪ V3).
80 Ars Math. Contemp. 22 (2022) #P1.04 / 57–81
• If n ≡ 2 (mod 3), then we set V3 =
⋃(n−8)/3
i=1 {v3i−1}, V2 = {vn−6, vn−5, vn−2,
vn−1} and V1 = ∅.
Notice that in the three cases above, f is a (2, 2, 2, 0)-dominating function of weight
ω(f) = n, as required. Therefore, the proof is complete.
ORCID iDs
Abel Cabrera Martı́nez https://orcid.org/0000-0003-2806-4842
Alejandro Estrada-Moreno https://orcid.org/0000-0001-9767-2177
Juan Alberto Rodrı́guez-Velázquez https://orcid.org/0000-0002-9082-7647
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P1.05 / 83–97
https://doi.org/10.26493/1855-3974.2083.e80
(Also available at http://amc-journal.eu)
Sum-list-colouring of θ-hypergraphs
Ewa Drgas-Burchardt * , Agata Drzystek, Elżbieta Sidorowicz
Faculty of Mathematics, Computer Science and Econometrics,
University of Zielona Góra, Prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland
Received 12 August 2019, accepted 10 May 2021, published online 21 February 2022
Abstract
Given a hypergraph H and a function f : V (H) → N, we say that H is f -choosable
if there is a proper vertex coloring ϕ of H such that ϕ(v) ∈ L(v) for all v ∈ V (H),
where L : V (H) → 2N is any assignment of f(v) colors to a vertex v. The sum choice
number χsc(H) of H is defined to be the minimum of
∑
v∈V (H) f(v) over all functions
f such that H is f -choosable. A trivial upper bound on χsc(H) is |V (H)| + |E(H)|.
The class Γsc of hypergraphs that achieve this bound is induced hereditary. We analyze
some properties of hypergraphs in Γsc as well as properties of hypergraphs in the class of
forbidden hypergraphs for Γsc. We characterize all θ-hypergraphs in Γsc, which leads to
the characterization of all θ-hypergraphs that are forbidden for Γsc.
Keywords: Hypergraphs, sum-list-colouring, θ-hypergraphs.
Math. Subj. Class. (2020): 05C15, 05C65
1 Introduction
A hypergraph is a very natural generalization of a graph. It always motivates the exten-
sion of a problem first posed in the class of graphs to the class of hypergraphs. If it is
a vertex colouring problem, then there is additional motivation. Indeed, a lot of scientists
consider different concepts of vertex colouring of graphs (for example: list-colouring, sum-
colouring, equitable-colouring), starting, in each case, from proper colouring, and next,
analyzing some improper variants, in which a graph induced by vertices of a colour class
is not necessarily edgeless. If we assume that each colour class has to induce a graph with
some property (for example: acyclic, with a bounded degree, and so on) and this property
is closed with respect to induced subgraphs, then, in each of these concepts, the problem of
improper colouring of a graph is equivalent to the problem of proper colouring of a unique
*Corresponding author.
E-mail addresses: e.drgas-burchardt@wmie.uz.zgora.pl (Ewa Drgas-Burchardt),
a.drzystek@wmie.uz.zgora.pl (Agata Drzystek), e.sidorowicz@wmie.uz.zgora.pl (Elżbieta Sidorowicz)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
84 Ars Math. Contemp. 22 (2022) #P1.05 / 83–97
hypergraph constructed for this graph. Clearly, the construction of such a hypergraph can
be difficult, but this approach gives the possibility to solve the problem. Moreover, in this
case, each of the results obtained for hypergraphs can be applied to different variants of the
same concept of graph colouring. Consequently, it can produce many special results.
The concept of sum-list-colouring of graphs is motivated by real problems and was first
introduced in [6, 8]. Erdős, Rubin and Taylor [6] considered the so called size functions
whose values for vertices of a graph represented the sizes of the lists assigned to them.
Isaak [8] was the first to analyze the minimum sum of the list sizes that guarantees the
existence of any particular proper vertex list colouring if lists are of these sizes. Such an
invariant was determined in [11], with the help of Hall’s Theorem, for complete graphs,
and then, for a few other classes of graphs [9, 12]. In [9] the upper bound on the minimum
sum of the list sizes was determined. Graphs that meet this bound, known as sc-greedy,
led themselves to a very popular line of investigation in the literature [2, 3, 7, 9, 12]. In
[5], the authors analyzed sum-list-colouring concept assuming that colour classes need
not be edgeless. This investigation shows some differences between proper and improper
cases and uses hypergraph theory tools. We continue this consideration herein, focusing
on hypergraphs, believing that the following results will be used for various variants of
the colouring concept. We extend the sc-greedy notion from graphs to hypergraphs and
characterize all θ-hypergraphs that are sc-greedy (Theorem 4.11). This yields the charac-
terization of all θ-hypergraphs that are forbidden for the class of sc-greedy hypergraphs
(Corollary 4.12).
2 Preliminaries
In general, we follow the notation and terminology of [1, 4]. A hypergraph H consists of a
non-empty finite set V (H) of vertices and a finite set E(H) of at least 2-element subsets of
V (H), called edges. A hypergraph is simple if none of its edges is a subset of another edge.
A hypergraph is linear if any two of its edges have no more than one vertex in common.
Let H be a hypergraph. A hypergraph H′ is a subhypergraph of H if V (H′) ⊆ V (H)
and E(H′) ⊆ E(H). For V ′ ⊆ V (H), a subhypergraph of a hypergraph H induced by V ′,
denoted by H[V ′], has the vertex set V ′ and the edge set {E ∈ E(H) : E ⊆ V ′}. We use
H− V ′ notation instead of H[V (H) \ V ′] and even H− v instead of H− {v}.
Let H be a hypergraph, v ∈ V (H) and E(v) = {E ∈ E(H) : v ∈ E}. By H(v) we
denote a hypergraph with the vertex set ∪E∈E(v)E and with the edge set E(v). The degree
of v in H, denoted by degH(v), is defined as the number of edges of H(v). The β-degree
of v in H, denoted by degβH(v), is the largest number of edges of a linear subhypergraph
of H(v). The H1 ∪ H2 symbol denotes the union of disjoint hypergraphs H1, H2. By the
identification of two non-adjacent vertices v1 and v2 (in a hypergraph H into a vertex w)
we mean the result of the following operations on H: the removal of vertices v1, v2, the
addition of a new vertex w, the replacement of each edge containing either v1 or v2 by an
edge in which w substitutes v1, v2, respectively, and the removal of multiple edges if the
current hypergraph has such edges. Note that v1, v2 can be vertices of different compo-
nents, say H1,H2, of H. In this case, sometimes, instead of the identification of vertices
in H1 ∪ H2 we may talk about the identification of vertices of two disjoint hypergraphs
H1,H2.
The 1-vertex hypergraph is a hypertree without edges. Next, a hypergraph that has one
edge consisting of all its vertices is a hypertree with one edge. A hypertree with m edges
E. Drgas-Burchardt et al.: Sum-list-colouring of θ-hypergraphs 85
(m ≥ 2) can be constructed from a hypertree H1 with m1 edges and a hypertree H2 with
m − m1 edges, 0 < m1 < m, by the identification of an arbitrary vertex of H1 and an
arbitrary vertex of H2. Note that each hypertree is linear.
A hypertree H is a hyperpath if there is an ordering (called canonical) of V (H) such
that each edge of H consists of some consecutive vertices (with respect to this ordering).
The length of a hyperpath is the number of its edges. By a hypercycle we mean a hypergraph
obtained from a hyperpath of length of at least three by the identification of the vertex with
the first index and the vertex with the last index in an arbitrary canonical ordering of the
vertex set of this hyperpath. The length of a hypercycle is the same as the length of a
hyperpath that was used in the construction. Moreover, if v1, . . . , vn is a canonical ordering
of the vertex set of the hyperpath, then v1, . . . , vn−1 is a canonical ordering of the vertex set
of the resulting hypercycle. Let k ∈ N. By a k-edge, k+-edge (of a hypergraph H) we mean
an edge of H consisting of k, at least k vertices, respectively. A hypergraph is k-uniform
if each of its edges is a k-edge. Thus 2-uniform hypergraphs are graphs and especially,
2-uniform hypertrees, hyperpaths, hypercycles are trees, paths, cycles, respectively.
3 Sum-choice-number of hypergraphs
Let H be a hypergraph. A proper colouring of H is a mapping ϕ : V (H) → N such
that for every edge E of H there are at least two different vertices v1, v2 in E such that
ϕ(v1) ̸= ϕ(v2). Given a mapping L : V (H) → 2N we call a mapping ϕ : V (H) → N an
L-colouring of H if for every vertex v ∈ V (H) it holds that ϕ(v) ∈ L(v). Let f be a
function from V (H) to the set of positive integers, a mapping L : V (H) → 2N such that
|L(v)| = f(v) for every vertex v in V (H) is called an f -assignment for H. The hypergraph
H is f -choosable if for each f -assignment L for H there is a proper L-coloring of H. Thus,
H is f -choosable if H is properly L-colourable for each f -assignment L for H. The sum-
choice-number χsc(H) of H is defined as the minimum of
∑
v∈V (H) f(v) taken over all f
such that H is f -choosable. Hence
χsc(H) = minf
 ∑
v∈V (H)
f(v) : H is f -choosable
 .
If H is f -choosable and χsc(H) =
∑
v∈V (H) f(v), then we say that f realizes χsc(H).
The definition of the sum-choice-number of a hypergraph implies some immediate ob-
servations.
Fact 3.1. If a hypergraph H1 is f -choosable for some function f with a domain V (H1),
then each subhypergraph H2 of H1 is f |V (H2)-choosable.
Fact 3.2. If H1,H2 are vertex disjoint hypergraphs, then
χsc(H1 ∪H2) = χsc(H1) + χsc(H2).
Fact 3.3. If H2 is a subhypergraph of a hypergraph H1, then
|V (H1)| − |V (H2)|+ χsc(H2) ≤ χsc(H1).
Applying the reasoning provided in the proof of Theorem 8 from [5] we obtain the
following theorem.
86 Ars Math. Contemp. 22 (2022) #P1.05 / 83–97
Theorem 3.4. If H is a hypergraph and v1, . . . , vn is an arbitrary ordering of V (H), then
χsc(H) ≤
n∑
i=1
degβHi(vi) + n,
where Hi = H[{v1, . . . , vi}].
Proof. Given the ordering v1, . . . , vn, let f(vi) = deg
β
Hi(vi) + 1. To finish the proof we
will show that H is f -choosable. Let L be any f -assignment for H. We colour the vertices
of H greedily, in accordance with the ordering v1, . . . , vn. Namely, in the ith step we assign
to vi the least colour from L(vi) such that for each a ∈ N the hypergraph induced by the
vertices coloured with a in the hypergraph Hi is edgeless. Note that such a colouring exists
for each of ith steps, i ∈ {1, . . . , n}, since, there are at most degβHi (vi) colours in L(vi)
for which Hi has an edge that would be monochromatic if we assigne this colour to vi.
Using the reasoning presented in the final part of the above proof we have the following
property.
Lemma 3.5. If H is a hypergraph and f is a function that realizes χsc(H), then f(v) ≤
degβH(v) + 1 for each v ∈ V (H).
Proof. Suppose, for contradiction purposes, that f satisfies the assumptions of the lemma
and there is a vertex u ∈ H such that f(u) ≥ degH(u) + 2. We will show that H is f ′-
choosable for f ′ defined by f ′(v) = f(v) for v ∈ V (H)\{u} and f ′(u) = f(u)−1, giving
a contradiction with the assumptions about f . Let H′ = H−u and let f ′-assignment L′ for
H be given. Since f ′|V (H′) = f |V (H′) we know that there is a proper L′|V (H′)-colouring
ϕ′ of H′, by Fact 3.1. Clearly ϕ′ can be extended to a proper L′-colouring of H since
f ′(u) ≥ degβH(u) + 1.
Observe that the bound given in Theorem 3.4 mostly depends on the ordering of ver-
tices. For example, consider a hypergraph H such that V (H) = {v1, . . . , v5} and E(H) =
{E1, E2, E3}, where E1 = {v1, v2, v3, v4}, E2 = {v1, v2, v3, v5} and E3 = {v1, v2, v4,
v5}. Let π1 : v1, v2, v3, v4, v5 and π2 : v3, v4, v5, v1, v2 be two different orderings of V (H).
Thus Theorem 3.4 gives the upper bound of 7 on χsc(H) when we use π1 and of 6 when
we use π2. On the other hand deg
β
H(v) ≤ degH(v) for every vertex v of a hypergraph H.
Moreover, for any ordering v1, . . . , vn of vertices of an n-vertex hypergraph H we have∑n
i=1 degHi(vi) = |E(H)|, where Hi = H[{v1, . . . , vi}]. Hence Theorem 3.4 implies the
following fact.
Fact 3.6. If H is a hypergraph, then
χsc(H) ≤ |V (H)|+ |E(H)|.
A hypergraph H is called sc-greedy if χsc(H) = |V (H)| + |E(H)|. In brief, in the
following, we denote the number |V (H)|+|E(H)| by GB(H). The notion of sc-greediness
was previously introduced for graphs in [2]. Observe that if H is an sc-greedy hypergraph,
then for every ordering v1, . . . , vn of V (H) it holds that degβHi(vi) = degHi(vi) for each
permissible i and Hi = H[{v1, . . . , vi}]. Now suppose that a hypergraph has at least two
edges E1, E2 that have at least two vertices, in common, say {v1, v2} ⊆ E1 ∩ E2. We
construct an ordering of vertices of H putting first the vertices from E1 \ {v1, v2}, next
E. Drgas-Burchardt et al.: Sum-list-colouring of θ-hypergraphs 87
the vertices from E2 \ {v1, v2}, next v1, v2, and finally the remaining vertices. Clearly,
degβHj (v2) < degHj (v2), where j = |E1 ∪ E2|. Hence we conclude the following fact.
Fact 3.7. Each sc-greedy hypergraph is linear.
The literature on sc-greediness of graphs is very rich. We try to comment this property
in the class of hypergraphs, especially those hypergraphs that are not graphs. Let Γsc
denote the family of all sc-greedy hypergraphs. First note that Γsc is not closed while
taking subhypergraphs. Indeed, the K2,3 graph is not sc-greedy, but a graph resulting from
K2,3 by the addition of an edge, which joins two vertices of degree 3, is sc-greedy [12].
On the other hand, Γsc is closed while taking induced subhypergraphs (it is a well-known
fact for sc-greedy graphs). To see it, suppose that there is an sc-greedy hypergraph H
having an induced subhypergraph H′, which is not sc-greedy. We construct a function f
such that H is f -choosable and
∑
v∈V (H) f(v) ≤ GB(H) − 1 based on the function f ′
that realizes χsc(H′). Actually, f |V (H′) = f ′ and for i ∈ {1, . . . , p} we put f(vi) =
degβHi(vi) + 1, where v1, . . . , vp is an arbitrary ordering of V (H) \ V (H
′) and Hi =
H[V (H′)∪{v1, . . . , vi}]. It implies that H is not sc-greedy, contradicting our assumption.
Thus Γsc is an induced hereditary class and there is a family C(Γsc) of hypergraphs, each
of which is not sc-greedy and whose each proper induced subhypergraph is sc-greedy. The
elements of C(Γsc) are called forbidden hypergraphs for Γsc and they uniquely determine
Γsc. Note that Γsc contains only linear hypergraphs. The class C(Γsc) does not have
this property. For example, each non-linear hypergraph H defined by V (H) = E1 ∪ E2,
E(H) = {E1, E2}, where |E1 ∩ E2| ≥ 2, E1 \ E2 ̸= ∅, E2 \ E1 ̸= ∅, is an element
of C(Γsc).
In the next part of the paper we focus our attention on linear hypergraphs in C(Γsc).
Lemma 3.8. Let H be a linear hypergraph in C(Γsc) and v ∈ V (H). If f is a function
that realizes χsc(H), then
i) f(v) ≤ degH(v), and
ii) degH(v) ≥ 2 implies f(v) ≥ 2 provided that each edge in E(H(v)) contains in H
at most two vertices of degree greater than one.
Proof. To show i) suppose that there is at least one vertex u in V (H) such that f(u) ≥
degH(u) + 1. Lemma 3.5 says that f(u) = degH(u) + 1. Now we define H′ = H − u.
Clearly H′ is a proper induced subhypergraph of H and consequently is sc-greedy, by the
definition of C(Γsc). From the construction we know that |V (H′)| = |V (H)| − 1 and
|E(H′)| = |E(H)| − degH(u). Thus χsc(H′) = GB(H′) = GB(H) − (degH(u) + 1).
As a subhypergraph of H, the hypergraph H′ is f |V (H′)-choosable, by Fact 3.1. It follows
that χsc(H′) ≤
∑
v∈V (H′) f(v) =
∑
v∈V (H) f(v) − (degH(u) + 1) ≤ GB(H) − 1 −
(degH(u)+1). Thus GB(H)−1−(degH(u)+1) ≥ χsc(H′) = GB(H)−(degH(u)+1),
i.e. a contradiction.
To show ii) suppose that H and u ∈ V (H) satisfy the assumptions and f(u) = 1. If
there is an edge E ∈ E(H(u)) that contains only one vertex of degree greater than one
(only u), then for each vertex in E the value of f is equal to one, and consequently H is not
f -choosable, a contradiction. Let {E1, . . . , Ek} = E(H(u)). Thus for each i ∈ {1, . . . , k}
there is exactly one vertex ui different from u such that ui ∈ Ei and degH(ui) ≥ 2. Let
H′ = H[(V (H)\∪ki=1Ei)∪{u1, . . . , uk}]. Note that |E(H′)| = |E(H)|−k and |V (H′)| =
88 Ars Math. Contemp. 22 (2022) #P1.05 / 83–97
|V (H)| − t for some t ∈ N. We define f ′ : V (H′) → N such that f ′(v) = f(v) if
v /∈ {u1, . . . , uk} and f ′(ui) = f(ui)−1 for i ∈ {1, . . . , k}. Note that
∑
v∈V (H′) f
′(v) ≤∑
v∈V (H) f(v) − k − t. Since H ∈ C(Γsc), it holds that
∑
v∈V (H) f(v) ≤ GB(H) − 1,
and consequently
∑
v∈V (H′) f
′(v) ≤ GB(H) − 1 − k − t = GB(H′) − 1. Thus H′
is not f ′-choosable. Let L′ be an f ′-assignment for H′ such that there is no proper L′-
colouring of H′ and let a /∈
⋃
v∈V (H′) L
′(v). We define an f -assignment L for H in the
following way: L(v) = L′(v) for v ∈ V (H′) \ {u1, . . . , uk}, next L(ui) = L′(ui) ∪ {a}
for i ∈ {1, . . . , k} and L(v) = {a} for v ∈ V (H) \ V (H′). It is very easy to see that there
is no proper L-colouring of H, which means that H is not f -choosable contradicting the
assumption. Hence f(u) ≥ 2 in this case.
Let us continue the investigation concerning sc-greedy hypergraphs. We start with the
observation that Theorem 1 in [2] (referring to graphs) can be extended to hypergraphs. In
fact, the same proof, in which the words graphs are substituted by hypergraphs, works to
obtain the following statement.
Theorem 3.9. Let H1, H2 be two disjoint hypergraphs and v1 ∈ V (H1), v2 ∈ V (H2). If
H is the hypergraph obtained by the identification of v1 and v2 in H1 ∪H2, then
χsc(H) = χsc(H1) + χsc(H2)− 1.
Note that each hypertree with one edge is sc-greedy. The recursion used in the defini-
tion of a hypertree with an arbitrary number of edges and Theorem 3.9 give us the following
consequence.
Corollary 3.10. Each hypertree is sc-greedy.
Using Lemma 3.8 concerning hypergraphs in C(Γsc), we have the next result.
Theorem 3.11. Each hypercycle is sc-greedy.
Proof. Clearly, if there is a hypercycle H that is not sc-greedy, then H ∈ C(Γsc). It follows
by Corollary 3.10 and Fact 3.2, since each component of every proper induced subhyper-
graph of H is a hypertree (actually it is a hyperpath). Suppose that f realizes χsc(H).
By Lemma 3.8 i), ii) we have f(v) ≥ degH(v) for each vertex of H, and consequently∑
v∈H f(v) ≥ GB(H), a contradiction.
Let F be the class of recursively defined hypergraphs such that: all hypercycles and all
hypertrees are in F , and, giving any two disjoint hypergraphs H1, H2 in F and vertices
v1 ∈ V (H1), v2 ∈ V (H2), a hypergraph obtained by the identification of v1 and v2 in
H1 ∪H2 is also in F .
The following result is a consequence of Corollary 3.10 and Theorems 3.11, 3.9.
Corollary 3.12. If H ∈ F , then H is sc-greedy.
4 θ-hypergraphs
Let k1, k2, k3 ∈ N. By θhk1,k2,k3 we denote the hypergraph consisting of two vertices of
degree 3 connected by three internally disjoint hyperpaths of lengths k1, k2, k3. In what
follows, we sometimes use the notion of a hyperpath of θhk1,k2,k3 of length ki, i ∈ {1, 2, 3},
E. Drgas-Burchardt et al.: Sum-list-colouring of θ-hypergraphs 89
meaning the hyperpath of length ki, used in the definition of θhk1,k2,k3 . By a θ-hypergraph
we mean an arbitrary hypergraph θhk1,k2,k3 . Observe that if at least two of the numbers
k1, k2, k3 are equal to one, then θhk1,k2,k3 is not linear. Additionally, if one of the hyperpaths
of length one is created by a 2-edge, θhk1,k2,k3 is even not simple. When θ
h
k1,k2,k3
is a graph,
we can denote it by θk1,k2,k3 since such notation is present in the literature. In [7] Heinold
found the values χsc(θk1,k2,k3) for all simple graphs θk1,k2,k3 . We recall here this result.
Theorem 4.1 ([7]). Let k1, k2, k3 ∈ N and at most one of k1, k2, k3 is equal to one. A
graph θk1,k2,k3 is not sc-greedy if and only if k1 = k2 = 2 and k3 is even. Moreover, if
θk1,k2,k3 is not sc-greedy, then
χsc(θk1,k2,k3) = GB(θk1,k2,k3)− 1.
Theorem 4.1 shows that the sum-choice-number of each simple graph θk1,k2,k3 , is al-
ways less by one or equal to the sum of the numbers of vertices and edges of this graph.
Fortunately, θ-hypergraphs have the same property.
Lemma 4.2. If k1, k2, k3 ∈ N and at most two of the numbers k1, k2, k3 are equal to one,
then χsc(θhk1,k2,k3) ≥ GB(θ
h
k1,k2,k3
)− 1.
Proof. Let H = θhk1,k2,k3 and let E be an arbitrary edge of H that is one of the edges of the
shortest hyperpath among three hyperpaths that compose H. Next let H′ be a subhyper-
graph of H such that V (H′) = V (H) and E(H′) = E(H)\{E}. Note that H′ is sc-greedy,
by Corollary 3.12 and Fact 3.2. It means χsc(H′) ≥ GB(θhk1,k2,k3)−1. Thus the statement
follows from Fact 3.3.
It is worth mentioning that χsc(θh1,1,1) = GB(θ
h
1,1,1) − 2. Indeed, let v be one of two
vertices of degree 3 in V (θh1,1,1) and let f : V (θ
h
1,1,1) → N be defined by: f(v) = 2 and
f(u) = 1 for every u ∈ V (θh1,1,1) \ {v}. Clearly, θh1,1,1 is f -choosable since, for every
f -assignment L, each colouring, in which the colours of v and the other vertex of degree
3 are different, is a proper L-colouring of θh1,1,1. Thus, χsc(θ
h
1,1,1) ≤ GB(θh1,1,1) − 2 =
|V (θh1,1,1)|+ 1. Because θh1,1,1 has edges, we have χsc(θh1,1,1) ≥ |V (θh1,1,1)|+ 1.
Let f : V (H) → N and L be an f -assignment for H. In what follows, if L(v) =
{a1, . . . , af(v)}, then we always assume that elements a1, . . . , af(v) are pairwise different.
Thus, among others, in Lemma 4.3 the integers a, b, c are pairwise different. Furthermore,
if i1, . . . , ip are consecutive integers and i1 > ip, then the set {i1, . . . , ip} is empty.
Lemma 4.3. Let k ∈ N and H be a hypercycle of length 2k. Next let v1, . . . , vn be an
arbitrary canonical ordering of V (H), where {vi1 = v1, . . . , vi2k} is the set of all vertices
of degree two in H with ij < il for j < l.
If f : V (H) → N is defined by f(v) = degH(v) and L is an f -assignment for H such
that
i) L(vij ) = {a, b} for j ∈ {1, . . . , 2k − 2}, and
ii) L(vi2k−1) = {b, c}, and
iii) L(vi2k) = {a, c}, and
iv) L(vs) = {c} for s ∈ {i2k−1 + 1, . . . , i2k − 1}, and
90 Ars Math. Contemp. 22 (2022) #P1.05 / 83–97
v) L(vs) = {a} for s ∈ {i2k + 1, . . . , n} or for s ∈ {ir + 1, . . . , ir+1 − 1} with odd r,
1 ≤ r ≤ 2k − 3, and
vi) L(vs) = {b} for s ∈ {ir + 1, . . . ir+1 − 1} with even r, 2 ≤ r ≤ 2k − 2,
then in each proper L-colouring ϕ of H it holds that ϕ(v1) = b.
Proof. Suppose that there is a proper L-colouring ϕ of H such that ϕ(v1) = a. Con-
sequently, it must be ϕ(vi2p) = b for all p ∈ {1, . . . , k − 1}, next ϕ(vi2k−1) = c and
ϕ(vi2k) = a. Hence the edge {vi2k , . . . , vn, v1} is monochromatic in ϕ, a contradic-
tion.
To avoid repetitions, we skip the simple proof of the next lemma that can be done in
the same manner as the proof of Lemma 4.3.
Lemma 4.4. Let k ∈ N and H be a hypercycle of length 2k + 1. Next let v1, . . . , vn be
an arbitrary canonical ordering of V (H), where {vi1 = v1, . . . , vi2k+1} is the set of all
vertices of degree two in H with ij < ik for j < k.
If f : V (H) → N is defined by f(v) = degH(v) and L is an f -assignment for H such
that
i) L(vij ) = {a, b} for j ∈ {1, . . . , 2k + 1}, and
ii) L(vs) = {a} for s ∈ {i2k+1 + 1, . . . , n} or for s ∈ {ir + 1, . . . , ir+1 − 1} with odd
r, 1 ≤ r ≤ 2k − 1, and
iii) L(vs) = {b} for s ∈ {ir + 1, . . . ir+1 − 1} and even r, 2 ≤ r ≤ 2k,
then in each proper L-colouring ϕ of H it holds that ϕ(v1) = b.
Lemma 4.5. Let k1, k2, k3 ∈ N. If the hyperpath of length k2 of θhk1,k2,k3 has only 2-edges,
and either
i) k1 + k2 and k2 + k3 are odd numbers and at least one of the inequalities k1 ≥ 2,
k3 ≥ 2 holds, or
ii) k1 + k2 is an odd number and k2 + k3 is an even number and k3 ≥ 3, or
iii) k1 + k2 is an even number and k2 + k3 is an even number and k1 ≥ 3 and k3 ≥ 3,
then θhk1,k2,k3 is sc-greedy.
Proof. Let H = θhk1,k2,k3 and let H satisfies the assumptions of the lemma. Observe that
at most one of integers k1, k2, k3 is equal to one since otherwise, H does not satisfy the
assumptions of the lemma, so H is linear.
Suppose, for a contradiction, that H is not sc-greedy. Since each component of each
proper induced subhypergraph of H is in F , we obtain H ∈ C(Γsc), by Corollary 3.12 and
Fact 3.2. Let f be a function that realizes χsc(H). Lemma 3.8 i) implies that f(v) = 1 if
degH(v) = 1 and Lemma 3.8 ii) implies that f(v) ≥ 2 for each vertex v of degree greater
than one. Thus f has values in {1, 2} and is fixed, since
∑
v∈V (H) f(v) = GB(H) − 1
(see Lemma 4.2).
Now we shall construct an f -assignment L for H such that H is not properly L-
colourable. Assume that P1,P2,P3 are three hyperpaths of lengths k1, k2, k3, respectively,
E. Drgas-Burchardt et al.: Sum-list-colouring of θ-hypergraphs 91
from which θhk1,k2,k3 is composed. Next let C1 be a hypercycle that is a subhypergraph of
θhk1,k2,k3 composed from vertices and edges of the hyperpaths P1,P2. Similarly, let C2 be
a hypercycle that is a subhypergraph of θhk1,k2,k3 composed from vertices and edges of the
hyperpaths P2,P3. Thus lengths of C1, C2 are k1 + k2 and k2 + k3, respectively. Now we
define canonical orderings π1 of C1 and π2 of C2, both starting with the same fixed vertex
of degree three in H, say v1, and both proceeding consecutively, first along the vertices of
P2, and next, along the vertices of either P1 or P3, respectively.
Next we construct an f -assignment L1 for C1 and π1 either in accordance with Lem-
ma 4.3 or in accordance with Lemma 4.4, depending on the parity of the length of C1 (the
parity of k1 + k2). Similarly, we construct an f -assignment L2 for C2 and π2 either in
accordance with Lemma 4.3 or Lemma 4.4, but in this case we exchange the meaning of
colours a, b. Namely, we substitute a by b and b by a in each value of L2 (given by the
corresponding lemma).
Observe that the assumptions on numbers k1, k2, k3 and the fact that P2 has only 2-
edges imply that L1 = L2 on vertices of P2. Define an f -assignment L for H such that
L(v) =

L1(v), if v ∈ V (C1) \ V (P2),
L2(v), if v ∈ V (C2) \ V (P2),
L1(v) = L2(v), if v ∈ V (P2).
Suppose, for a contradiction, that ϕ is a proper L-colouring of H. Fact 3.1 implies that
ϕ|V (C1) is a proper L1-colouring of C1 and ϕ|V (C2) is a proper L2-colouring of C2. By
Lemma 4.3 or Lemma 4.4 (depending on the parity of k1 + k2) we have ϕ(v1) = b and
by one of Lemmas 4.3, 4.4 (depending on the parity of k2 + k3) we have ϕ(v1) = a, a
contradiction.
For forthcoming Lemmas 4.6, 4.7, 4.8, 4.9, 4.10, we introduce the following notations.
Let P be a hyperpath on at least two vertices and let v1, . . . , vn be a canonical ordering of
V (P). Next let f∗ : V (P) → N be defined by f∗(v) = degP(v) for v /∈ {v1, vn} and
f∗(v1) = f
∗(vn) = 2.
Giving an f∗-assignment L for P and (α, γ) ∈ L(v1)×L(vn), we say that a pair (α, γ)
is extendable (for P) if there is a proper L-colouring ϕ of P such that ϕ(v1) = α, ϕ(vn) =
γ. The pair (α, γ) ∈ L(v1)×L(vn) that is not extendable for P is called forbidden (for P).
The next lemma is a generalization of the lemma that was proven in [3].
Lemma 4.6. Let P be a hyperpath on at least two vertices, let v1, . . . , vn be a canonical or-
dering of V (P) and let {vi1 , . . . , vik} be the set of all vertices of degree two in P with ip <
is for p < s. If L is an f∗-assignment for P such that L(v1), L(vi1), . . . , L(vik), L(vn)
are not identical, then at most one pair in L(v1)× L(vn) is forbidden for P .
Proof. We will show that at most one pair in L(v1) × L(vn) is forbidden for the path
P , where V (P ) = {v1 = vi0 , vi1 , . . . , vik , vn = vik+1} and E(P ) = {vijvij+1 : j ∈
{0, . . . , k}}. Note that P is a graph. Since every proper L|V (P )-colouring of P can be
extended to a proper L-colouring of P , the lemma will follow. We prove this statement by
the induction on the number of edges in P (the number k + 1). Observe that it is true for
a path P with one edge. Suppose that |E(P )| ≥ 2. Since the lists of v1, vi1 , . . . , vik , vn
are not identical, the lists of v1, vi1 , . . . , vik or the lists of vi1 , . . . , vik , vn are not identical
either. Say the lists of vi1 , . . . , vik , vn are not identical. Let L(v1) = {a, b}, L(vn) =
92 Ars Math. Contemp. 22 (2022) #P1.05 / 83–97
{c, d}, L(vi1) = {α, β} and assume that a ̸= β and b ̸= α. By inductive assumptions, at
least three pairs in L(vi1)×L(vn) are extendable for P−v1, say (α, c), (α, d), (β, c). Thus
pairs (b, c), (b, d), (a, c) are extendable for P , and hence, they are extendable for P .
Lemma 4.7. If P is a hyperpath with at least one 3+-edge, v1, . . . , vn is a canonical
ordering of V (P) and L is an f∗-assignment for P , then at most one pair in L(v1)×L(vn)
is forbidden for P .
Proof. We prove the assertion by the induction on the number of edges in E(P). Observe
that the lemma trivially holds when P has one edge. Suppose that P is a hyperpath with at
least one 3+-edge and |E(P)| ≥ 2. Let {vi1 , . . . , vik} be the set of all vertices of degree
two in P with ip < is for p < s. If the lists of v1, vi1 , . . . , vik , vn are not identical, then
the statement follows from Lemma 4.6. Thus we may assume that L(v1) = L(vi1) =
· · · = L(vik) = L(vn) = {a, b}. Let E1 = {v1, v2, . . . , vi1}. Renaming vertices, if it is
necessary, we may assume that P ′ = P \ {v1, . . . , vi1−1} contains a 3+-edge. Thus P ′ is
a hyperpath satisfying inductive assumptions, and so, at least three pairs in L(vi1)×L(vn)
are extendable for P ′. Note that if we colour v1 with a and vi1 with b or if we colour v1
with b and vi1 with a, then we can extend such a colouring to a proper L-colouring of P .
Hence, in L(v1)× L(vn) there are three pairs that are extendable for P .
Lemmas 4.6, 4.7 immediately imply the following fact.
Lemma 4.8. If P is a hyperpath on at least two vertices, v1, . . . , vn is a canonical ordering
of V (P) and L is an f∗-assignment for P , then at most two pairs in L(v1) × L(vn) are
forbidden for P . Moreover,
i) exactly two pairs are forbidden for P if and only if P contains only 2-edges and
L(v1) = · · · = L(vn), and
ii) if there are exactly two forbidden pairs for P and P is of even length and L(v1) =
{a, b}, then (a, a) and (b, b) are extendable for P , and
iii) if there are exactly two forbidden pairs for P and P is of odd length and L(v1) =
{a, b}, then (a, b) and (b, a) are extendable for P .
Lemma 4.9. If P is a hyperpath of length two, v1, . . . , vn is a canonical ordering of
V (P), L is an f∗-assignment for P and L(v1) = L(vn) = {a, b}, then (a, a) and (b, b)
are extendable for P .
Based on Lemmas 4.6, 4.7, 4.8, 4.9 we have the following result.
Lemma 4.10. Let k1, k2, k3 ∈ N. If θhk1,k2,k3 is sc-greedy, then one of the hyperpaths of
θhk1,k2,k3 , say the hyperpath of length k2, has only 2-edges, and, under this assumption, one
of the following conditions is satisfied:
i) k1 + k2 and k2 + k3 are odd numbers and at least one of the inequalities k1 ≥ 2,
k3 ≥ 2 holds, or
ii) k1 + k2 is an odd number and k2 + k3 is an even number and k3 ≥ 3, or
iii) k1 + k2 is an even number and k2 + k3 is an even number and k1 ≥ 3 and k3 ≥ 3.
E. Drgas-Burchardt et al.: Sum-list-colouring of θ-hypergraphs 93
Proof. Let H = θhk1,k2,k3 . Suppose that for each possible permutation ki1 , ki2 , ki3 of
numbers k1, k2, k3 either the hyperpath of H of length ki2 contains 3+-edge or θhki1 ,ki2 ,ki3
satisfies no of the conditions i), ii), iii) of the lemma. The aim is to prove that H is not
sc-greedy. We may assume that at most one of the numbers k1, k2, k3 is equal to one.
Otherwise, H is not linear and the statement follows by Fact 3.7. We define f so that
f(v) = 1 if degH(v) = 1 and f(v) = 2 if degH(v) ≥ 2. Next, we will show that for
each f -assignment L for H there is a proper L-colouring of H. Since
∑
v∈V (H) f(v) =
GB(H) − 1, the theorem will follow. Let P1,P2,P3 be the hyperpaths of H of lengths
k1, k2, k3, respectively, and let v1, vn be vertices of degree 3 in H. Let L be an arbitrary
f -assignment for H. First observe that if each hyperpath Pi, i ∈ {1, 2, 3}, has at least
one 3+-edge, then for each hyperpath Pi at most one pair in L(v1) × L(vn) is forbidden,
by Lemma 4.7. Since we have four possible pairs in L(v1) × L(vn), at least one pair can
be extended to a proper L-colouring of H. Thus at least one hyperpath contains only 2-
edges, so it is the path. Moreover, the lists of vertices of this path must be identical, by
Lemma 4.8 i). We may assume that each vertex of this path has the list {a, b}, and so,
L(v1) = L(vn) = {a, b}. Let us consider two cases.
Case 1. The numbers k1, k2, k3 are all of the same parity. Definitely, H does not fulfill
neither the property i) nor ii). Since H does not have the property iii) either, at least one of
integers k1, k2, k3 is less than or equal to two. Assume that k1 ≤ 2.
Subcase 1.1. All of the numbers k1, k2, k3 are even. Thus k1 = 2. If k2 ≥ 3 and k3 ≥ 3,
then P1 has to contain at least one 3+-edge. Otherwise, H satisfies iii) for a permutation
ki1 , ki2 , ki3 , where ki2 = k1. By our initial assumption P2 or P3 has only 2-edges and
pairs (a, a), (b, b) are extendable for this path, by Lemma 4.8 ii). Without loss of generality
assume that P2 contains only 2-edges and pairs (a, a), (b, b) are extendable for P2. Note
that pairs (a, a), (b, b) are also extendable for P1, by Lemma 4.9. From Lemma 4.8, at
most two pairs are forbidden for P3, and if exactly two pairs are forbidden for P3, then
(a, a), (b, b) are extendable for P3. Thus both pairs (a, a), (b, b) can be extended to a
proper L-colouring of H. If at most one pair is forbidden for P3, then at least one of pairs
(a, a), (b, b) can be extended to a proper L-colouring of H.
Now suppose that at least two hyperpaths have lengths two, say k1 = 2 and k2 = 2.
From Lemma 4.9, (a, a) and (b, b) are extendable for both P1 and P2. From Lemma 4.8, at
most two pairs are forbidden for P3. If exactly two pairs are forbidden for P3, then again
by Lemma 4.8 ii), both pairs (a, a), (b, b) are extendable for P3. Otherwise, at most one
pair is forbidden for P3. Thus (a, a) or (b, b) is extendable for all P1, P2, P3.
Subcase 1.2. All of the numbers k1, k2, k3 are odd. Thus k1 = 1. Since H is linear, we
have k2 ≥ 3 and k3 ≥ 3. Furthermore, P1 is a 3+-edge, otherwise, H satisfies iii). Again,
without loss of generality, assume that P2 contains only 2-edges and pairs (a, b), (b, a) are
extendable for P2, by Lemma 4.8 iii). Thus (a, b), (b, a) are extendable for both P1, P2.
If there is at most one pair forbidden for P3, then one of pairs (a, b), (b, a) is extendable
for all P1, P2, P3. Otherwise, two pairs are forbidden for P3, however, then both pairs
(a, b), (b, a) are extendable for P3, by Lemma 4.8 iii). So, both pairs (a, b), (b, a) can be
extended to a proper L-colouring of H.
Case 2. The numbers k1, k2, k3 are not of the same parity. In this case either one hyperpath
in {P1,P2,P3} is of odd length and two of them are of even length or one hyperpath in
{P1,P2,P3} is of even length and two are of odd length. So, H does not have the property
94 Ars Math. Contemp. 22 (2022) #P1.05 / 83–97
iii). The hyperpath of odd length in the first case and the hyperpath of even length in the
second case has a 3+-edge, since otherwise, H has the property i). Let us consider the
following two possible subcases.
Subcase 2.1. The number k1 is odd and P1 has a 3+-edge, k2, k3 are even. Since H
does not have the property ii), k2 = 2 or k3 = 2. Say k3 = 2. If k2 > 2, then P3 has
3+-edge, as otherwise, H has the property ii). Thus P2 must have only 2-edges and pairs
(a, a), (b, b) are extendable for P2, by Lemma 4.8 ii). From Lemma 4.9, pairs (a, a), (b, b)
are extendable for P3, so, (a, a), (b, b) are extendable for both P2, P3. By Lemma 4.7, at
most one of these pairs is forbidden for P1, so there is a proper L-colouring of H. Suppose
now that k2 = 2, so we have k2 = k3 = 2. From Lemma 4.9 both pairs (a, a), (b, b)
are extendable for both P2, P3. Since at most one of these pairs is forbidden for P1, the
statement follows.
Subcase 2.2. The number k1 is even and P1 has a 3+-edge, k2, k3 are odd. Since H does
not have the property ii), k2 = 1 or k3 = 1. Say k3 = 1. Furthermore, the previous
consideration leads to k2 ≥ 3. If P3 has exactly 2-edge, then H has the property ii). Thus
P3 must have a 3+-edge, and so, P2 must have only 2-edges. Hence, pairs (a, b), (b, a)
are extendable for P2. Thus (a, b) and (b, a) are extendable for both P2, P3. Furthermore,
at most one of pairs is forbidden for P1, and so, at least one of the pairs (a, a), (b, b) is
extendable for all P1, P2, P3, which creates a proper L-colouring of H and proves the
lemma.
Now, we are in a position to present the main result of the paper that immediately
follows from Lemmas 4.5, 4.10. Next, the consequence of this result is formulated.
Theorem 4.11. Let k1, k2, k3 ∈ N. A hypergraph θhk1,k2,k3 is sc-greedy if and only if one
of the hyperpaths of θhk1,k2,k3 , say the hyperpath of length k2, has only 2-edges and, under
this assumption, one of the following conditions holds:
i) k1 + k2 and k2 + k3 are odd numbers and at least one of the inequalities k1 ≥ 2,
k3 ≥ 2 holds, or
ii) k1 + k2 is an odd number and k2 + k3 is an even number and k3 ≥ 3, or
iii) k1 + k2 is an even number and k2 + k3 is an even number and k1 ≥ 3 and k3 ≥ 3.
Note that for arbitrary parameters k1, k2, k3 such that (k1, k2, k3) ̸= (1, 1, 1) and arbi-
trary vertex of θhk1,k2,k3 each component of θ
h
k1,k2,k3
− v is in F . Hence the following fact
is valid.
Corollary 4.12. If k1, k2, k3 ∈ N and at most two of the numbers k1, k2, k3 are equal to
one, then θhk1,k2,k3 ∈ C(Γsc) if and only if θ
h
k1,k2,k3
/∈ Γsc.
5 Concluding remarks and open problems
A connected hypergraph is 2-connected if it cannot be a result of identification of a vertex
of H1 and a vertex of H2, where H1, H2 are some disjoint hypergraphs, each on at least
two vertices.
Note that, based on Fact 3.2 and Theorem 3.9, both, the union operation and the iden-
tification operation (applied to vertices of two disjoint hypergraphs) keep sc-greediness of
E. Drgas-Burchardt et al.: Sum-list-colouring of θ-hypergraphs 95
hypergraphs. Hence, analyzing sc-greediness of hypergraphs it is enough to focus on 2-
connected ones. Additionally, each sc-greedy hypergraph must be linear, by Fact 3.7. Thus
the following question seems to be interesting.
Problem 5.1. How to characterize all hypergraphs in Γsc that are 2-connected and linear?
To support the discussion this question we start with some notes concerning graphs.
The famous theorem that characterizes all linear (equivalently, simple) 2-connected graphs
can be found in [4]. To cite it we need the following notion.
Let G be a graph on at least two vertices. By adding a G-path to G, we mean the result
of two operations of identification applied to the graph G and an arbitrary path P with a
canonical ordering v1, . . . , vn of V (P ), n ≥ 2 (G and P are disjoint). More precisely, it is
a result of identification of v1 and x and also vn and y, where x, y are two different vertices
of G.
Lemma 5.2 ([4]). A simple graph is 2-connected if and only if it can be constructed from
a cycle by successively adding G-paths to graphs G already constructed.
Observe that each cycle is 2-connected and sc-greedy. Next, adding a G-path to the
cycle G we obtain a θ-graph. Theorem 4.1 characterizes all θ-graphs that are in Γsc. The
question is, whether we should expect that an sc-greedy graph be obtained by adding a
G-path to an sc-greedy θ-graph. In [3] the authors proved that χsc(G) + 2s ≤ χsc(G1) ≤
χsc(G) + 2s + 1 if G1 is the result of adding a G-path on s + 2 vertices to an arbitrary
simple graph G, assuming that s ≥ 1. They did not consider the case when a G-path has
2 vertices, which seems to be very important. However, based on this observation, they
gave the characterization of all graphs in Γsc that are generalized θ-graphs. For r ≥ 3, a
generalized θ-graph is a simple graph, denoted by θk1,k2,...,kr , consisting of two vertices
connected by r internally vertex disjoint paths of lengths k1, k2, . . . , kr. In [3] the authors
showed that θk1,k2,...,kr is not sc-greedy if r ≥ 5. Moreover, θk1,k2,k3,k4 is not sc-greedy
if and only if it contains an induced subgraph θ2,2,ki with even ki and i ∈ {3, 4} or, if all
numbers k1, k2, k3, k4 have the same parity. It follows that starting with a θ-graph G and
adding a G-path two times, in this special case (to obtain a generalized θ-graph), we always
obtain a graph that is not sc-greedy. On the other hand, the graph presented in Figure 1 that
needs to be added a G-path 3 times is still sc-greedy (see the graph G10,12 in [10]). It leads
to formulating the following problem.
Figure 1: An sc-greedy graph that needs the application of adding a G-path 3 times.
Problem 5.3. Under which conditions can we obtain an sc-greedy graph by adding a G-
path to an sc-greedy graph G?
In a way similar for graphs we can define the operation of adding an H-path to a hy-
pergraph H and pose the problem similar to Problem 5.3 in the class of hypergraphs.
96 Ars Math. Contemp. 22 (2022) #P1.05 / 83–97
Problem 5.4. Under which conditions can we obtain an sc-greedy hypergraph by adding
an H-path to an sc-greedy hypergraph H?
Unfortunately, there are 2-connected linear hypergraphs that cannot be obtained from a
hypercycle by successively adding H-paths.
On the other hand, there is a relatively large subclass of the class of 2-connected lin-
ear hypergraphs, for which, the problem of belonging to Γsc can be solved, with help of
consideration concerning graphs.
A hypergraph G is a blown of a graph G if G is a result of a substitution of each edge
of G by a 3+-edge of G that contains vertices of substituted edge of G (see for example
Figure 2). Let F ′ be the class consisting of all hypergraphs that are all possible blowns
of 2-connected graphs, except cycles. Clearly, every hypergraph in F ′ is 2-connected, by
definition.
Figure 2: A graph and its blown.
Theorem 5.5. No hypergraph in F ′ is sc-greedy.
Proof. First observe that if H ∈ F ′, then H is a blown of some 2-connected graph H that
is not a cycle. From Lemma 5.2, H contains a subgraph that is a θ-graph. By the defini-
tion of F ′ we know that H contains an induced subhypergraph H′ that is a θ-hypergraph,
and moreover, H′ has no hyperpath having only 2-edges. Thus H′ is not sc-greedy, by
Lemma 4.10. Finally, H is not sc-greedy, since Γsc is closed when taking induced subhy-
pergraphs.
It is worth mentioning that hypergraphs in F ′ can or cannot be blowns of sc-greedy
graphs. It follows that probably, to be an sc-greedy hypergraph is a relatively rare fea-
ture in the class of hypergraphs with only 3+-edges. It motivates the following separate
subproblem of Problem 5.1.
Problem 5.6. How to characterize all hypergraphs in Γsc that are 2-connected, linear and
have only 3+-edges.
Finally, observe that each hypergraph in C(Γsc) is 2-connected, but this class includes
linear and non-linear hypergraphs. Thus, referring to the C(Γsc) class, we obtain the fol-
lowing open question and its subquestion.
Problem 5.7. How to characterize all hypergraphs in C(Γsc)?
Problem 5.8. How to characterize all non-linear hypergraphs in C(Γsc)?
E. Drgas-Burchardt et al.: Sum-list-colouring of θ-hypergraphs 97
ORCID iDs
Ewa Drgas-Burchardt https://orcid.org/0000-0001-6229-177X
Elżbieta Sidorowicz https://orcid.org/0000-0002-7774-0512
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P1.06 / 99–113
https://doi.org/10.26493/1855-3974.2332.749
(Also available at http://amc-journal.eu)
Cospectrality of multipartite graphs*
Alireza Abdollahi , Niloufar Zakeri
Department of Pure Mathematics, Faculty of Mathematics and Statistics,
University of Isfahan, Isfahan 81746-73441, Iran
Received 10 May 2020, accepted 11 June 2021, published online 14 June 2022
Abstract
Let G be a graph on n vertices and consider the adjacency spectrum of G as the or-
dered n-tuple whose entries are eigenvalues of G written decreasingly. Let G and H be
two non-isomorphic graphs on n vertices with spectra S and T , respectively. Define the
distance between the spectra of G and H as the distance of S and T to a norm N of the
n-dimensional vector space over real numbers. Define the cospectrality of G as the min-
imum of distances between the spectrum of G and spectra of all other non-isomorphic n
vertices graphs to the norm N . In this paper we investigate copsectralities of the cocktail
party graph and the complete tripartite graph with parts of the same size to the Euclidean
or Manhattan norms.
Keywords: Spectra of graphs, cospectrality of graphs, adjacency matrix of a graph, Euclidean norm,
Manhattan norm.
Math. Subj. Class. (2020): 05C50, 05C31
1 Introduction and results
All graphs considered here are simple, that is finite and undirected without loops and mul-
tiple edges. Let G be a graph with vertex set {v1, . . . , vn}. The adjacency matrix of G is
an n × n matrix A(G) = [aij ] such that aij = 1 if vi and vj are adjacent, and aij = 0
otherwise. By the eigenvalues of G, we mean those of its adjacency matrix. We denote by
Spec(G) the multiset of the eigenvalues of the graph G.
Richard Brualdi proposed in [24] the following problem:
Problem ([24, Problem AWGS.4]). Let Gn and G′n be two non-isomorphic graphs on n
vertices with spectra
λ1 ≥ λ2 ≥ · · · ≥ λn and λ′1 ≥ λ′2 ≥ · · · ≥ λ′n,
*The authors are grateful to the referee for his/her helpful comments.
E-mail addresses: a.abdollahi@math.ui.ac.ir (Alireza Abdollahi), zakeri@sci.ui.ac.ir (Niloufar Zakeri)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
100 Ars Math. Contemp. 22 (2022) #P1.06 / 99–113
respectively. Define the distance between the spectra of Gn and G′n as
λ(Gn, G
′
n) =
n∑
i=1
(λi − λ′i)2
(
or use
n∑
i=1
|λi − λ′i|
)
.
Define the cospectrality of Gn by
cs(Gn) = min{λ(Gn, G′n) : G′n not isomorphic to Gn}.
Let
csn = max{cs(Gn) : Gn a graph on n vertices}.
This function measures how far apart the spectrum of a graph with n vertices can be from
the spectrum of any other graph with n vertices.
Problem A. Investigate cs(Gn) for special classes of graphs.
Problem B. Find a good upper bound on csn.
In [15], Jovanović et al. studied the spectral distance between certain graphs to the
ℓ1-norm i.e. σ(Gn, G′n) =
∑n
i=1 |λi − λ′i|. In [1], Abdollahi et al. completely answered
Problem B to any ℓp-norm by proving that csn = 2 for all n ≥ 2, whenever 1 ≤ p < ∞
and csn = 1 to the ℓ∞-norm. In [2, 20], the authors studied Problem A to the Euclidean
norm (the ℓ2-norm) and determined the cospectralities of classes of complete graphs and
complete bipartite graphs. In [3] we compute the cospectralities to the ℓ1-norm of complete
graphs and complete bipartite graphs with parts of the same size. In [4, 10, 11, 13, 14, 16,
17, 18], Problems A or B are studied based on different matrix representations. To find
some applications of the cospectrality of graphs, we refer to [6, 25, 27].
In this paper we study Problem A and investigate the cospectralities of CPn and Kn,n,n,
(n ≥ 3), to the ℓ1- and ℓ2-norms i.e. σ(Gn, G′n) =
∑n
i=1 |λi − λ′i| and λ(Gn, G′n) =∑n
i=1(λi − λ′i)2, respectively. We find some conditions for the eigenvalues of a graph H
such that cs(G) = σ(G,H) and G is isomorphic to CPn or Kn,n,n. Also we give some
computational results and conjectures to find cs(CPn) and cs(Kn,n,n).
In the last section we consider cospectralities of null graphs, complete graphs and com-
plete bipartite graphs using the ℓp-norm for p > 2 and we see that similar known conclu-
sions using with ℓ1 and ℓ2-norms (see [2, 3, 11, 20]) hold more or less valid.
Let us first introduce some notations. For a graph G, V (G) and E(G) denote the vertex
set and edge set of G, respectively; By the order of G we mean the number of vertices;
Denote by G the complement of G. The degree of a vertex of a graph is the number of
edges that are incident with the vertex and ∆ is the maximum degree of the vertices. An
r-regular graph is a graph where all vertices have degree r.
For two graphs G and H with disjoint vertex sets, G + H denotes the graph with the
vertex set V (G) ∪ V (H) and the edge set E(G) ∪ E(H), i.e. the disjoint union of two
graphs G and H . The complete product (join) G∇H of graphs G and H is the graph
obtained from G +H by joining every vertex of G with every vertex of H . In particular,
nG denotes G+ · · ·+G︸ ︷︷ ︸
n
and ∇nG denotes G∇ · · ·∇G︸ ︷︷ ︸
n
. The coalescence G·H is obtained
by the disjoint union of two graphs G and H by identifying a vertex u of G with a vertex v
of H .
A. Abdollahi and N. Zakeri: Cospectrality of multipartite graphs 101
For positive integers n1, . . . , nℓ, Kn1,...,nℓ denotes the complete multipartite graph with
ℓ parts of sizes n1, . . . , nℓ. Let Kn denote the complete graph on n vertices, nK1 = Kn
denote the null graph on n vertices and Pn denote the path with n vertices. The cocktail
party graph CPn has 2n vertices and it is a complement of nK2. So for n = 1, CP1 = K1,1
and for n ≥ 2 we have CPn = K2, . . . , 2︸ ︷︷ ︸
n
.
Since CPn and Kn,n,n are regular graphs, by Propositions 3 and 6 of [9], CPn and
Kn,n,n are determined by their spectrum. So we can compute the values of cs(CPn) and
cs(Kn,n,n).
Our main results are as follows.
Theorem 1.1. If n ≥ 2 and cs(CPn) = σ(CPn, H) for some graph H with eigenvalues
λ1 ≥ · · · ≥ λ2n, then
(1) If H is a connected graph, then 2n− 3 ≤ λ1 < 2n− 1. Otherwise 2n− 3 ≤ λ1 <
2n− 2 and H has two connected components such that one of them is K1.
(2) 0 ≤ λ2 ≤ 1,
(3) −1 ≤ λi ≤ 12 , for any integer i, 3 ≤ i ≤ n+ 1, and if n ≥ 13, then 0 ≤ λ3 ≤
1
2 ,
(4) −3 ≤ λn+2 ≤ −1,
(5) −3 ≤ λi ≤ −32 , for any integer i, n+ 3 ≤ i ≤ 2n.
Theorem 1.2. Let n ≥ 4 and cs(Kn,n,n) = σ(Kn,n,n, H) for some graph H with eigen-
values λ1 ≥ · · · ≥ λ3n. For all ε > 0, there exists N ∈ N such that for all n ≥ N , we
have
(1) 2n−
√
3
3 −
ε
2 < λ1 < 2n+
√
3
3 +
ε
2 ,
(2)
√
2−1 < λ2 <
√
3
3 +
ε
2 or λ2 = 0 and H
∼= tK1+Kp,q,r for some positive integers
p, q and r such that at least one of them is greater than 1,
(3) 0 ≤ λ3 <
√
3
6 +
ε
4 ,
(4) −
√
3
3 −
ε
2 < λi <
√
3
6 +
ε
4 , for any integer i, 4 ≤ i ≤ 3n− 2,
(5) −n−
√
3
3 −
ε
2 < λ3n−1 < −n+
√
3
3 +
ε
2 ,
(6) −n−
√
3
3 −
ε
2 < λ3n < −n+
√
3
6 +
ε
4 .
2 Cospectrality of cocktail party graphs
In this section cs(CPn) is investigated to the ℓ1- and ℓ2-norms. We need the following
results in the sequel. The proofs of next two results are similar to those of Lemma 2.2 and
Corollary 2.3 of [18]. We give them here for the reader’s convenience.
Lemma 2.1. Let a1 ≥ a2 ≥ · · · ≥ an and b1 ≥ b2 ≥ · · · ≥ bn be two sequences with∑n
i=1 ai =
∑n
i=1 bi = 0. If there exist some 1 ≤ j ≤ n and a real positive number α such
that |aj − bj | > α, then
∑n
i=1 |ai − bi| > 2α.
102 Ars Math. Contemp. 22 (2022) #P1.06 / 99–113
Proof. Without loss of generality, we may assume that aj − bj > α. Suppose that ai1 ≥
bi1 , . . . , ais ≥ bis and ais+1 ≤ bis+1 , . . . , ain ≤ bin , then
n∑
i=1
|ai − bi| =
s∑
t=1
(ait − bit) +
n∑
t=s+1
(bit − ait)
= 2
s∑
t=1
(ait − bit)
≥ 2(aj − bj)
> 2α.
Corollary 2.2. Let a1 ≥ a2 ≥ · · · ≥ an and b1 ≥ b2 ≥ · · · ≥ bn be two sequences with∑n
i=1 ai =
∑n
i=1 bi = 0. If there exist 1 ≤ j1 ̸= j2 ≤ n and a real positive number α
such that aj1 − bj1 + aj2 − bj2 > α, then
∑n
i=1 |ai − bi| > 2α.
Proof. If either aj1 − bj1 > α or aj2 − bj2 > α, then by Lemma 2.1, the result holds. So
we may assume that 0 < aj1 − bj1 ≤ α and 0 < aj2 − bj2 ≤ α. Let a′j1 = aj1 + aj2 ,
b′j1 = bj1 + bj2 , a
′
i = ai and b
′
i = bi for i ̸= j1, j2. So
∑n
i=1,i̸=j2
a′i =
∑n
i=1,i̸=j2
b′i = 0
and a′j1 − b
′
j1
> α. Thus the result follows from Lemma 2.1.
Theorem 2.3. Let G be a graph with eigenvalues λ1 ≥ · · · ≥ λn. If cs(G) = σ(G,H) for
some graph H with eigenvalues λ′1 ≥ · · · ≥ λ′n, then for all integers i and j, 1 ≤ j < i ≤
n,
(1) |λi − λ′i| ≤ 1,
(2) λi − λ′j ≤ 12 .
Proof. By Theorem 1.1 of [1], csn = 2 for all n ≥ 2, so cs(G) ≤ 2. Now the result follows
from Lemma 2.1 and Corollary 2.2.
Theorem 2.4 ([5, Theorem 1]). Let G be a simple graph of order n without isolated ver-
tices. If λ2(G) is the second largest eigenvalue of G, then
(1) λ2(G) = −1 if and only if G is a complete graph with at least two vertices,
(2) λ2(G) = 0 if and only if G is a complete k-partite graph with 2 ≤ k ≤ n− 1,
(3) there exists no graph G such that −1 < λ2(G) < 0.
Theorem 2.5 ([21, Theorem 3.8]). Let G be a graph of order n. If λ3(G) < 0, then G has
at least n− 12 eigenvalues −1.
Theorem 2.6 ([7, Theorem 3.2.1]). Let λ1 be the greatest eigenvalue of the graph G, and
let d and ∆ be its average degree and maximum degree, respectively. Then
d ≤ λ1 ≤ ∆.
Moreover, d = λ1 if and only if G is regular. For a connected graph G, λ1 = ∆ if and only
if G is regular.
A. Abdollahi and N. Zakeri: Cospectrality of multipartite graphs 103
Proof of Theorem 1.1. Since
Spec(CPn) = {2n− 2, 0, . . . , 0︸ ︷︷ ︸
n
,−2, . . . ,−2︸ ︷︷ ︸
n−1
},
we have
σ(CPn, H) = |2n− 2− λ1|+
n+1∑
i=2
|λi|+
2n∑
i=n+2
|2 + λi|.
If cs(CPn) = σ(CPn, H), then by Theorem 1.1 of [1], cs(CPn) ≤ 2. By Theorems 2.3,
2.4, 2.5 and Corollary 2.2, we obtain (2) – (5) and 2n− 3 ≤ λ1 ≤ 2n− 1.
If H is a connected graph and λ1 = 2n − 1, then by Theorem 2.6, H ∼= K2n, a
contradiction. So 2n − 3 ≤ λ1 < 2n − 1. Now suppose that H is not connected. Let
H1, . . . ,Hk be the connected components of H . There exists an unique i, 1 ≤ i ≤ k,
such that λ1(H) = λ1(Hi). We can assume that λ1(H) = λ1(H1). Thus λ1(Hj) ≤
λ2(H) ≤ 1, for every j, 2 ≤ j ≤ k. So λ1(Hj) = 0 or λ1(Hj) = 1, 2 ≤ j ≤ k.
Since −1 ≤ λ3(H) ≤ 12 , there exists at most one connected component with λ1(Hj) = 1,
2 ≤ j ≤ k. Therefore H ∼= H1 + tK1 or H ∼= H1 +K2 + sK1, for some integers t > 0
and s ≥ 0. By Theorem 2.6, 2n − 3 ≤ λ1(H) = λ1(H1) ≤ ∆ ≤ 2n − 1, where ∆ is the
maximum degree of the vertices of H . If ∆ = 2n− 1, then, by Theorem 2.6, H1 ∼= K2n, a
contradiction. Let ∆ = 2n− 3. Therefore by Theorem 2.6, H1 ∼= K2n−2, a contradiction.
Now suppose that ∆ = 2n− 2. If λ1(H1) = 2n− 2, then by Theorem 2.6, H1 ∼= K2n−1,
a contradiction. Hence we can assume that H ∼= H1 +K1 and 2n− 3 ≤ λ1(H) < 2n− 2.
This completes the proof.
Remark 2.7. Let H be a connected graph with m edges. If cs(CPn) = σ(CPn, H), then,
by Theorem 1.1 and Theorem 1 in [26], it is not hard to see that 2n2 − 5n + 4 ≤ m <
2n2 − n.
Now we find σ(CPn, (CPn−1 ▽ K1) · K2) and λ(CPn, CPn \ e) and propose two
conjectures. We need the following results.
Theorem 2.8 ([7, Theorem 2.1.8]). If G1 is r1-regular with n1 vertices, and G2 is r2-
regular with n2 vertices, then the characteristic polynomial of the join G1 ▽ G2 is given
by
PG1▽G2(x) =
PG1(x)PG2(x)
(x− r1)(x− r2)
((x− r1)(x− r2)− n1n2).
Theorem 2.9 ([7, Theorem 2.2.3]). Let G ·H be the coalescence in which the vertex u of
G is identified with the vertex v of H . Then
PG·H(x) = PG(x)PH−v(x) + PG−u(x)PH(x)− xPG−u(x)PH−v(x).
Lemma 2.10. If (CPn−1 ▽K1) ·K2 is the coalescence of K2 with CPn−1 ▽K1 with its
vertex of maximum degree as distinguished vertex, then for n ≥ 3,
Spec((CPn−1 ▽K1) ·K2) = {x1, x2, 0, . . . , 0︸ ︷︷ ︸
n−1
, x3,−2, · · · ,−2︸ ︷︷ ︸
n−2
},
such that x1 > x2 > 0 > x3 are the roots of the polynomial x3 + (4 − 2n)x2 +
(1− 2n)x+ 2n− 4.
104 Ars Math. Contemp. 22 (2022) #P1.06 / 99–113
Proof. Since PCPn−1(x) = x
n−1(x+ 2)n−2(x− 2n+ 4) and PK1(x) = x, Theorem 2.8
implies that
PCPn−1▽K1(x) = x
n−1(x+ 2)n−2(x2 + (4− 2n)x+ 2− 2n).
Since PK2(x) = x
2 − 1, it follows from Theorem 2.9,
P(CPn−1▽K1)·K2(x) = x
n−1(x+ 2)n−2(x3 + (4− 2n)x2 + (1− 2n)x+ 2n− 4).
Thus (CPn−1 ▽ K1) · K2 has n − 1 and n − 2 eigenvalues 0 and −2, respectively. The
remaining eigenvalues are the roots of the polynomial x3+(4−2n)x2+(1−2n)x+2n−4.
If
a =
(
8n3 − 30n2 + 24n+ 8 + 3(−60n4 + 312n3 − 648n2 + 606n− 237)
1
2
) 1
3
,
b = −4
9
n2 +
10
9
n− 13
9
,
r =
(
(8n3 − 30n2 + 24n+ 8)2 + 540n4 − 2808n3 + 5832n2 − 5454n+ 2133
) 16
,
θ =
1
3
arctan
(
3(60n4 − 312n3 + 648n2 − 606n+ 237)
1
2
8n3 − 30n2 + 24n+ 8
)
.
Then
x1 =
2n
3
− 4
3
+
a
3
− 3b
a
,
x2 =
2n
3
− 4
3
+ (
3b
2r
− r
6
) cos θ −
√
3(
3b
2r
− r
6
) sin θ,
x3 =
2n
3
− 4
3
+ (
3b
2r
− r
6
) cos θ +
√
3(
3b
2r
− r
6
) sin θ.
This completes the proof.
Lemma 2.11. limn−→∞ σ
(
CPn, (CPn−1 ▽K1) ·K2
)
= 2, whenever (CPn−1 ▽K1) ·
K2 is the coalescence of K2 with CPn−1 ▽ K1 with its vertex of maximum degree as
distinguished vertex.
Proof. By Lemma 2.10 and using the symbolic computational software Maple [19] (see
https://data.amc-journal.eu/cospectrality/maplecode1.mw), the result follows.
Theorem 2.12 ([7, Theorem 2.1.5]). Let G, H be graphs with n1, n2 vertices respectively.
The characteristic polynomial of the join G▽H is given by the relation
PG▽H(x) = (−1)n2PG(x)PH(−x− 1) + (−1)
n1PH(x)PG(−x− 1)
− (−1)n1+n2PG(−x− 1)PH(−x− 1).
Lemma 2.13. For n ≥ 3 and any edge e,
Spec(CPn \ e) =
{
x1,
√
5− 1
2
, 0, . . . , 0︸ ︷︷ ︸
n−2
, x2,−
√
5 + 1
2
,−2, . . . ,−2︸ ︷︷ ︸
n−3
, x3
}
,
where x1 > 0 > x2 > x3 are the roots of the polynomial x3 − (2n− 5)x2 − (6n− 9)x−
2n+ 2.
A. Abdollahi and N. Zakeri: Cospectrality of multipartite graphs 105
Proof. For any edge e, CPn \ e = P4 ▽ CPn−2. Let G = P4 and H = CPn−2. Thus
G = G and H = (n− 2)K2. We have
PG(x) = PG(x) = x
4 − 3x2 + 1,
PH(x) = (x− 2n+ 6)xn−2(x+ 2)n−3,
PH(x) = (x
2 − 1)n−2.
Therefore
PCPn\e = PG▽H(x) = x
n−2(x+ 2)
n−3
(x2+x−1)(x3−(2n−5)x2−(6n−9)x−2n+2).
It follows CPn \ e has n− 2 and n− 3 eigenvalues 0 and −2, respectively. The remaining
eigenvalues are
√
5−1
2 , −
√
5+1
2 and the roots of x
3 − (2n− 5)x2 − (6n− 9)x− 2n+ 2. If
a =
(
64n3 − 48n2 − 312n+ 404
+ 12(−240n4 + 528n3 + 396n2 − 1740n+ 1137) 12
) 1
3 ,
b = −4
9
n2 +
2
9
(n+ 1),
r =
(
(64n3 − 48n2 − 312n+ 404)2
+ 34560n4 − 76032n3 − 57024n2 + 250560n− 163728
) 1
6 ,
θ =
1
3
arctan
(
12(240n4 − 528n3 − 396n2 + 1740n− 1137)
1
2
64n3 − 48n2 − 312n+ 404
)
.
Then
x1 =
2n
3
− 5
3
+
a
6
− 6b
a
,
x2 =
2n
3
− 5
3
+ (
3b
r
− r
12
) cos θ −
√
3(
3b
r
− r
12
) sin θ,
x3 =
2n
3
− 5
3
+ (
3b
r
− r
12
) cos θ +
√
3(
3b
r
− r
12
) sin θ,
and we are done.
Lemma 2.14. limn−→∞ λ(CPn, CPn \ e) = 10− 4
√
5.
Proof. By Lemma 2.13 and using the symbolic computational software Maple [19] (see
https://data.amc-journal.eu/cospectrality/maplecode2.mw), the result follows.
We have the following conjectures:
Conjecture 2.15. For every integer n ≥ 2, cs(CPn) = σ(CPn, H) for some graph H if
and only if H ∼= (CPn−1 ▽K1) ·K2, whenever (CPn−1 ▽K1) ·K2 is the coalescence of
K2 with CPn−1 ▽K1 with its vertex of maximum degree as distinguished vertex.
Conjecture 2.16. For every integer n ≥ 4, cs(CPn) = λ(CPn, H) for some graph H if
and only if H ∼= CPn \ e, for any edge e.
For n = 2 and n = 3, cs(CPn) = λ(CPn, H) if and only if H ∼= (CPn−1▽K1) ·K2.
Our computational results confirm Conjectures 2.15 and 2.16 for all graphs of order at most
10.
106 Ars Math. Contemp. 22 (2022) #P1.06 / 99–113
3 Cospectrality of complete tripartite graphs
In this section we investigate cs(Kn,n,n), for n ≥ 3, to the ℓ1- and ℓ2-norms. First we need
the following results.
Theorem 3.1 ([12, Theorem 9.1.1]). Let G be a graph of order n and H be an induced
subgraph of G with order m. Suppose that λ1(G) ≥ · · · ≥ λn(G) and λ1(H) ≥ · · · ≥
λm(H) are the eigenvalues of G and H , respectively. Then for every i, 1 ≤ i ≤ m,
λi(G) ≥ λi(H) ≥ λn−m+i(G).
Theorem 3.2 (See [23] and also [8, Theorem 6.7]). A graph has exactly one positive eigen-
value if and only if its non-isolated vertices form a complete multipartite graph.
Lemma 3.3 ([22, Lemma 7]). λ2((K1 +Kr,s)∇Kq) ≤
√
2− 1 (r ≤ s) if and only if one
of the conditions 1− 10 holds:
(1) r > 1, s ≥ r, q = 1;
(2) r = 1, s ≥ 1, q ≥ 2;
(3) r = 2, s ≥ 2, q = 2;
(4) r = 2, 2 ≤ s ≤ 3, q ≥ 3;
(5) r = 2, s = 4, 3 ≤ q ≤ 7;
(6) r = 2, s = 5, 3 ≤ q ≤ 4;
(7) r = 2, 6 ≤ s ≤ 8, q = 3;
(8) r = 3, s = 3, 2 ≤ q ≤ 4;
(9) r = 3, 4 ≤ s ≤ 7, q = 2;
(10) r = 4, s = 4, q = 2.
Lemma 3.4 ([22, Lemma 8]). λ2((K1 +Kr,s)∇Kp,q) ≤
√
2 − 1 (r ≤ s, p ≤ q) if and
only if one of the conditions 1− 5 holds:
(1) r = 1, s = 1, p ≥ 1, q ≥ p;
(2) r = 1, s = 2, 1 ≤ p ≤ 2, q ≤ p;
(3) r = 1, s = 2, p = 3, 3 ≤ q ≤ 7;
(4) r = 1, s = 2, p = 4, q = 4;
(5) r = 1, s = 3, p = 1, q = 1.
Theorem 3.5 ([22, Theorem]). Let G be a graph without isolated vertices and let λ2(G)
be the second largest eigenvalue of G. Then 0 < λ2(G) ≤
√
2− 1 if and only if one of the
following holds:
(1) G ∼= (∇t(K1 +K2))∇Kn1,...,nm ,
(2) G ∼= (K1 + Kr,s)∇Kq, and parameters q, r and s satisfy one of the conditions
(1) – (10) from Lemma 3.3,
(3) G ∼= (K1 +Kr,s)∇Kp,q, and parameters p, q, r and s satisfy one of the conditions
(1) – (5) from Lemma 3.4.
Lemma 3.6. Let n ≥ 3 and x1 > 0 > x2 > x3 be the roots of the polynomial x3 −
(3n2 − 1)x− 2n3 + 2n. Then
Spec(Kn−1,n,n+1) = {x1, 0, . . . , 0︸ ︷︷ ︸
3n−3
, x2, x3}.
A. Abdollahi and N. Zakeri: Cospectrality of multipartite graphs 107
Proof. Since PKn1,...,nk (x) = x
∑k
i=1 ni−k
(
1−
∑k
i=1
ni
x+ni
)∏k
i=1(x+ ni),
PKn−1,n,n+1(x) = x
3n−3(x3 − (3n2 − 1)x− 2n3 + 2n).
Thus Kn−1,n,n+1 has 3n− 3 eigenvalues 0 and 3 eigenvalues
x1 =
a2 + 9n2 − 3
3a
,
x2 =
(
−r
6
+
1− 3n2
2r
)
cos θ −
√
3
(
−r
6
+
1− 3n2
2r
)
sin θ,
x3 =
(
−r
6
+
1− 3n2
2r
)
cos θ +
√
3
(
−r
6
+
1− 3n2
2r
)
sin θ,
where
a =
(
27n3 − 27n+ 3(−81n4 + 54n2 + 3)
1
2
) 1
3
,
r =
(
(27n3 − 27n)2 + 729n4 − 486n2 − 27
) 1
6
,
θ =
1
3
arctan
(
(81n4 − 54n2 − 3)
1
2
9n3 − 9n
)
.
Lemma 3.7. limn−→∞ σ(Kn,n,n,Kn−1,n,n+1) = 2
√
3
3 .
Proof. Since Spec(Kn,n,n) = {2n, 0, . . . , 0︸ ︷︷ ︸
3n−3
,−n,−n}, by Lemma 3.6 and using the com-
putational software Maple [19] (see https://data.amc-journal.eu/cospectrality/
maplecode3.mw), the result follows.
Proof of Theorem 1.2. Note that
σ(Kn,n,n, H) = |2n− λ1|+
3n−2∑
i=2
|λi|+ |n+ λ3n−1|+ |n+ λ3n|.
By Lemma 3.7, for all ε > 0, there exists N ∈ N such that for all n ≥ N , cs(Kn,n,n) <
2
√
3
3 + ε. By Lemma 2.1, Corollary 2.2, Theorems 2.4 and 2.5, we obtain (1), (3) – (6) and
0 ≤ λ2 <
√
3
3 +
ε
2 . Suppose that 0 < λ2 ≤
√
2− 1. Hence Theorem 3.5 can be applied.
Case 1: H ∼= (∇t(K1 +K2))∇Kn1,...,nm . If t ≥ 2, then (K1 +K2)∇(K1 +K2) is an
induced subgraph of H . Since
Spec((K1 +K2)∇(K1 +K2)) = {3.73205, .41421, .26795,−1,−1,−2.41421},
by Theorem 3.1, λ3n−2 ≤ −1, a contradiction. Now, suppose that t = 1. If m =
1, then H ∼= (K1 + K2)∇K3n−3. We have PH(x) = x3n−4f(x), whenever f(x) =
108 Ars Math. Contemp. 22 (2022) #P1.06 / 99–113
x4 − (9n− 8)x2 − (6n− 6)x+3n− 3. So the non-zero eigenvalues of H are the roots of
f(x) = 0. By computing the roots, it implies that λ3n−1 = −1, a contradiction. Therefore
m ≥ 2. If n1 = · · · = nm = 1, then H ∼= (K1 +K2)∇K3n−3. So (K1 +K2)∇K2 is an
induced subgraph of H . Since
Spec((K1 +K2)∇K2) = {3.32340, .35793,−1,−1,−1.68133},
by Theorem 3.1, λ3n−2 ≤ −1, a contradiction. Now, we can assume that ni ≥ 2, for some
1 ≤ i ≤ m. Thus (K1 +K2)∇K1,2 is an induced subgraph of H . Since
Spec((K1 +K2)∇K1,2) = {4.06779, .36162, 0,−1,−1.24464,−2.18477},
by Theorem 3.1, λ3n−2 ≤ −1, a contradiction.
Case 2: H ∼= (K1 + Kr,s)∇Kq and parameters q, r and s satisfy conditions 1–10 from
Lemma 3.3. We have PH(x) = x3n−4f(x) whenever f(x) = x4− (q+ qr+ qs+ rs)x2−
2qrsx + qrs. The non-zero eigenvalues of H are determined by equation f(x) = 0. By
computing the roots, we have λ1 = −λ3n and λ2 = −λ3n−1, a contradiction.
Case 3: H ∼= (K1 +Kr,s)∇Kp,q , and parameters p, q, r and s satisfy conditions 1–5 from
Lemma 3.4. In this case, H can be isomorphic to one of these graphs: (K1+K1,2)∇K3,5,
(K1 +K1,2)∇K4,4 and (K1 +K1,1)∇Kp,q whenever q ≥ p ≥ 1 and p+ q = 3n− 3. All
of these graphs have (K1 +K1,1)∇K1,2 as an induced subgraph. Since
Spec((K1 +K1,1)∇K1,2) = {4.06779, .36162, 0,−1,−1.24464,−2.18477},
by Theorem 3.1, λ3n−2 ≤ −1, a contradiction.
So
√
2 − 1 < λ2 <
√
3
3 +
ε
2 or λ2 = 0. If λ2 = 0, then, by Theorem 3.2, there are
some positive integers k, n1, . . . , nk and an integer t ≥ 0 such that H ∼= tK1 +Kn1,...,nk .
If k = 1, then H ∼= K3n, a contradiction. If k = 2, then H ∼= tK1 +Kr,s. Since
Spec(H) = {
√
rs, 0, . . . , 0︸ ︷︷ ︸
3n−2
,−
√
rs},
λ3n−1 = 0, a contradiction. Thus k ≥ 3. Suppose that k ≥ 4. If n1 = · · · = nk = 1, then
H ∼= tK1 +K3n−t. We have
Spec(H) = {3n− t− 1, 0, . . . , 0︸ ︷︷ ︸
t
,−1, . . . ,−1︸ ︷︷ ︸
3n−t−1
}.
Hence λ3n = −1, a contradiction. If there exists an unique i, 1 ≤ i ≤ k, such that ni ≥ 2,
then K1,1,1,2 is an induced subgraph of H . Since
Spec(K1,1,1,2) = {3.64575, 0,−1,−1,−1.64575},
by Theorem 3.1, λ3n−2 ≤ −1, a contradiction. Thus there exist i and j such that ni, nj ≥
2. Hence H has K1,1,2,2 as an induced subgraph. We have
Spec(K1,1,2,2) = {4.37228, 0, 0,−1,−1.37228,−2}.
So by Theorem 3.1, λ3n−2 ≤ −1, a contradiction. Therefore we can assume that k = 3 and
H ∼= tK1+Kp,q,r, for some positive integers p, q and r. If p = q = r = 1, then, by similar
argument given in k ≥ 4, we have λ3n = −1, a contradiction. So H ∼= tK1 +Kp,q,r such
that at least one of p, q and r is greater than 1. This completes the proof.
A. Abdollahi and N. Zakeri: Cospectrality of multipartite graphs 109
Lemma 3.8. limn−→∞ λ(Kn,n,n,Kn−1,n,n+1) = 23 .
Proof. By Lemma 3.6 and using the symbolic computational software Maple [19] (see
https://data.amc-journal.eu/cospectrality/maplecode4.mw), the result follows.
The graph H in Figure 1 is the only unique graph such that σ(K3,3,3, H) and λ(K3,3,3,
H) have the minimum possible values. For n ≥ 4, we have the following conjectures:
Conjecture 3.9. For every integer n ≥ 4, cs(Kn,n,n) = σ(Kn,n,n, H) for some graph H
if and only if H ∼= Kn−1,n,n+1.
Conjecture 3.10. For every integer n ≥ 4, cs(Kn,n,n) = λ(Kn,n,n, H) for some graph
H if and only if H ∼= Kn−1,n,n+1.
Figure 1: The graph which is closest to K3,3,3 both in the ℓ1- and ℓ2-norm.
4 Cospectrality of some families of graphs using ℓp-norm for p > 2
Let p > 2 be an arbitrary positive integer. First we determine the cospectrality of the null
graphs on n vertices.
Theorem 4.1. For every integer n ≥ 2, cs(nK1) = 2. Moreover, cs(nK1) = λ(p)(nK1, H)
for some graph H if and only if H ∼= K2 + (n− 2)K1.
Proof. It is not hard to see that λ(p)(nK1,K2 + (n − 2)K1) = 2. Let H be a simple
graph of order n. Thus cs(nK1) = λ(p)(nK1, H) ≤ 2. So |λ1(H)| ≤ p
√
2, where λ1(H)
is the greatest eigenvalue of H . Since the greatest eigenvalue of a graph is always non-
negative and H ≇ nK1, we have 0 < λ1(H) ≤ p
√
2. Moreover, there is no graph whose
greatest eigenvalue lies in the intervals (0, 1) and (1,
√
2). Hence λ1(H) = 1. Thus
H ∼= K2 + (n− 2)K1.
In the following we show that the minimum value of λ(p)(Kn, H) occurs whenever
H ∼= Kn \ e, where Kn \ e is the graph obtaining from Kn by deletion one edge e. First
we need the following results.
110 Ars Math. Contemp. 22 (2022) #P1.06 / 99–113
Lemma 4.2. λ(p)(K2,K2 \ e) = 2 and for every integer n ≥ 3 and every edge e of Kn,
λ(p)(Kn,Kn \ e) < 2.
Proof. It is easy to see that λ(p)(K2,K2 \ e) = 2. By Corollary 3.4 and Lemma 3.6 in [2],
one can obtain the result.
Theorem 4.3. For every integer n ≥ 2, cs(Kn) = λ(p)(Kn, H) for some graph H if and
only if H ∼= Kn\e for any edge e, where Kn\e is the graph obtaining from Kn by deletion
one edge e.
Proof. For n = 2 and n = 3, It is easy to see that cs(Kn) = λ(p)(Kn,Kn \ e). Let n ≥ 4.
We show that if H is not isomorphic to Kn and Kn \ e, then λ(p)(Kn, H) ≥ 2.
Let λ1 ≥ · · · ≥ λn be the eigenvalues of H . Therefore
λ(p)(Kn, H) = |λ1 − n+ 1|p +
n∑
i=2
|λi + 1|p.
One can obtain this if one of the following cases holds, then λ(p)(Kn, H) ≥ 2.
Case 1: λ1 − n+ 1 ≤ − 3
√
2.
Case 2: λ2 + 1 ≥ 3
√
2.
Case 3: λ3 ≥ 0.
Now suppose that none of the above cases occurs. Thus we can assume that λ1 > n− 1−
3
√
2, λ2 <
3
√
2 − 1 and λ3 < 0. If λ2 ≤ 0, then, by Lemma 3.9 in [2], H ∼= Kn−1 +K1
and λ(p)(Kn, H) = 2.
Now suppose that λ2 > 0. Since 0 < λ2 <
3
√
2 − 1 < 13 , by Theorem 2 in [5], there
exists an integer t such that H ∼= tK1 + (K1 +K2)∇Kn−3−t where 0 ≤ t ≤ n− 4.
If n− 3− t > 1, then (K1 +K2)∇K2 is an induced subgraph of H . Since
Spec((K1 +K2)∇K2) = {2.85577, 0.32164, 0,−1,−2.17741},
by Theorem 3.1, λ3 ≥ 0, a contradiction. If n − 3 − t = 1, then H ∼= (n − 4)K1 +
(K1 +K2)∇K1. Since
Spec(H) = {2.17009, 0.31111, 0, . . . , 0︸ ︷︷ ︸
n−4
,−1,−1.48119},
λ(p)(Kn, H) > 2. Therefore by Lemma 4.2, cs(Kn) = λ(p)(Kn,Kn \ e). This completes
the proof.
In the following, we investigate the cospectrality of complete bipartite graphs. The
proofs of Lemmas 2.5 and 2.7 and Theorem 2.8 in [20] are also working for p > 2, an
arbitrary positive integer. First we need the following results, the "ℓp−version" of Lemmas
2.5 and 2.7 in [20].
Lemma 4.4. Let m and n be two positive integers and G be a graph of order m+ n. If G
has K1,1,2 or (K1 +K2)∇K1 as an induced subgraph, then λ(p)(G,Km,n) ≥ 1.
A. Abdollahi and N. Zakeri: Cospectrality of multipartite graphs 111
Lemma 4.5. Let m and n be two positive integers and G be a graph of order m + n.
Suppose that there are no positive integers r, s and a non-negative integer t such that G ∼=
Kr,s + tK1. If λ2(G) ≤
√
2− 1, then λ(p)(G,Km,n) ≥ 1.
Theorem 4.6. Let m and n be two positive integers such that (m,n) ̸= (1, 1). Then
cs(Km,n) = λ(p)(Km,n,Kr,s + tK1),
for some integers r, s ≥ 1 and t ≥ 0 such that r + s + t = m + n and r, s ̸= m,n.
Moreover, if cs(Km,n) = λ(p)(Km,n, H) for some graph H , then H ∼= Ki,j +hK1, where
i, j ≥ 1 and h ≥ 0 are some integers so that i+ j + h = m+ n.
Proof. It is easy to see that cs(K1,2) = λ(p)(K1,2,K1,1 + K1). So we can assume that
m + n ≥ 4. Let i, j ≥ 1 and h ≥ 0 be some integers such that i + j + h = m + n.
Thus λ(p)(Km,n,Ki,j + hK1) = 2|
√
mn −
√
ij|p. By Lemma 2.4 in [20], there are
some positive integers r and s such that r + s ≤ m + n and {r, s} ≠ {m,n} so that
|
√
mn−
√
rs|p < (
√
2−1√
2
)p. Let t = m+ n− r− s. Hence we obtain λ(p)(Km,n,Kr,s +
tK1) < (
√
2 − 1)p. Therefore cs(Km,n) < (
√
2 − 1)p < 1. Now suppose that H is a
graph such that cs(Km,n) = λ(p)(Km,n, H). Thus λ(p)(Km,n, H) < (
√
2 − 1)p. Let
λ2(H) be the second largest eigenvalue of H . So we have |λ2(H)| <
√
2 − 1. Since
λ(p)(Km,n, H) < 1, by Lemma 4.5, there are some integers r, s ≥ 1 and t ≥ 0 such that
H ∼= Kr,s + tK1. This completes the proof.
Theorem 4.7. Let n ≥ 1 be an integer. Then, the following hold:
(1) cs(K1,1) = λ(p)(K1,1, 2K1) = 2,
(2) cs(K1,2) = λ(p)(K1,2,K1,1 +K1) = 2|
√
2− 1|p,
(3) If n ≥ 3 is a prime number, then
cs(K1,n) = λ(p)(K1,n,K2,n+12 +
n− 3
2
K1) = 2|
√
n+ 1−
√
n|p,
(4) If n ≥ 3 is not a prime number, then
cs(K1,n) = λ(p)(K1,n,Kr,s + (n+ 1− r − s)K1) = 0,
where r and s are some positive integers such that r, s < n and n = rs.
Proof. The method is similar to that of Theorem 2.10 in [20].
By Theorem 4.6, one can easily obtain the following results.
Theorem 4.8. For every integer n ≥ 2, cs(Kn,n) = 2|n−
√
n2 − 1|p. Moreover, cs(Kn,n)
= λ(p)(Kn,n, H) for some graph H if and only if H ∼= Kn−1,n+1.
Theorem 4.9. For every integer n ≥ 2, cs(Kn,n+1) = 2|
√
n2 + n −
√
n2 + n− 2|p.
Moreover, cs(Kn,n+1) = λ(p)(Kn,n+1, H) for some graph H if and only if H ∼= Kn−1,n+2.
ORCID iDs
Alireza Abdollahi https://orcid.org/0000-0001-7277-4855
112 Ars Math. Contemp. 22 (2022) #P1.06 / 99–113
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P1.07 / 115–120
https://doi.org/10.26493/1855-3974.2558.173
(Also available at http://amc-journal.eu)
Oriented regular representations of out-valency
two for finite simple groups*
Gabriel Verret
Department of Mathematics, University of Auckland, PB 92019, Auckland, New Zealand
Binzhou Xia †
School of Mathematics and Statistics, University of Melbourne,
Parkville, VIC 3010, Australia
Received 16 February 2021, accepted 17 June 2021, published online 5 April 2022
Abstract
In this paper, we show that every finite simple group of order at least 5 admits an
oriented regular representation of out-valency 2.
Keywords: Finite simple group, DRR, ORR.
Math. Subj. Class. (2020): 05C25, 05C20, 20B25
1 Introduction
All groups and digraphs in this paper are finite. A digraph Γ consists of a set of vertices
V(Γ) and a set of arcs A(Γ), each arc being an ordered pair of vertices. A digraph is proper
if (u, v) being an arc implies that (v, u) is not an arc. The automorphisms of Γ are the
permutations of V(Γ) that preserve A(Γ). Under composition, they form the automorphism
group Aut(Γ) of Γ.
Let G be a group and S ⊆ G. The Cayley digraph Cay(G,S) on G with connection set
S is the digraph with vertex set G and (u, v) being an arc whenever vu−1 ∈ S. Note that
Cay(G,S) is a proper digraph if and only if S ∩ S−1 = ∅. Note also that every vertex u in
Cay(G,S) is contained in exactly |S| arcs of the form (u, v). We thus say that Cay(G,S)
has out-valency |S|.
It is easy to see that Aut(Cay(G,S)) contains the right regular representation of G.
If this containment is actually equality, then Cay(G,S) is called a digraphical regular
*The authors are grateful to the N.Z. Marsden Fund which helped support (via grant UOA1824) the second
author’s visit to the University of Auckland in 2019.
†Corresponding author.
E-mail addresses: g.verret@auckland.ac.nz (Gabriel Verret), binzhoux@unimelb.edu.au (Binzhou Xia)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
116 Ars Math. Contemp. 22 (2022) #P1.07 / 115–120
representation (or DRR) of G. A DRR that is a proper digraph is called an oriented regular
representation (or ORR).
Babai proved that, apart from five small groups, all groups admit a DRR [1, Theo-
rem 2.1]. He also asked which groups admit ORRs [1, Problem 2.7]. This was answered
by Morris and Spiga [9, 10, 12] who showed that apart from generalised dihedral groups
and a small list of exceptions, all groups admit ORRs.
In view of the above, a natural problem is to find “nice” DRRs and ORRs, say of “small”
out-valency. Clearly, only cyclic groups can have DRRs of out-valency 1, so out-valency 2
is the smallest interesting case. In this paper, we give the most satisfactory answer to this
question in the case of simple groups.
Theorem 1.1. Every finite simple group of order at least 5 has a ORR of out-valency 2.
A corollary of Theorem 1.1 is that every nonabelian simple group has a DRR of out-
valency 2. However, the latter conclusion is an immediate consequence of the fact that
every nonabelian simple group is generated by an involution and a non-involution (even
by an involution and an element of odd prime order, see [8, Theorem 1]. Indeed, consider
a Cayley digraph on a nonabelian simple group with connection set consisting of such a
generating pair. This digraph has out-valency 2, but one out-neighbour of every vertex is
also an in-neighbour while the other out-neighbour is not. This implies that fixing a vertex
must also fix its out-neighbours and, by connectedness, the whole digraph, and the digraph
is a DRR.
Note that Cayley digraphs of out-valency two of simple groups were previously studied
in [4]. Another interesting variant of this question would be to consider undirected graphs.
In this case, the smallest interesting valency is 3. The question of which simple groups
admit graphical regular representations of valency 3 has received some attention but is still
open [11, 13, 14, 15].
2 Preliminaries
2.1 Generation of finite simple groups
In this section we present some generation properties of finite simple groups, which will be
needed in the proof of Theorem 1.1. The following result is due to Guralnick and Kantor [7,
Corollary].
Theorem 2.1 (Guralnick-Kantor). Every nontrivial element of a finite simple group be-
longs to a pair of elements generating the group.
Note that Theorem 2.1 depends on the classification of finite simple groups.
Corollary 2.2. Let G be a finite nonabelian simple group with an element x of order 3.
Then there exists y ∈ G such that |y| ≥ 4 and G = ⟨x, y⟩.
Proof. By Theorem 2.1 there exists z ∈ G such that G = ⟨x, z⟩. Note that ⟨x, z⟩ = ⟨x, xz⟩
hence, if either z or xz has order at least 4, then the conclusion holds (by taking y = z or
y = xz). We may thus assume that z and xz both have order at most 3. This implies that
G is a quotient of the finitely presented group
⟨x, z | x3, zm, (xz)n⟩
G. Verret and B. Xia: Oriented regular representations of out-valency two for finite simple groups 117
with m,n ≤ 3. This is the “ordinary” (3,m, n) triangle group which is well known to be
solvable when m,n ≤ 3 (see for example [3]) and therefore so is G, which is a contradic-
tion.
The only nonabelian simple groups with no elements of order 3 are the Suzuki groups
(see [6, Page 8, Table I]), which we now consider. For a positive integer m and prime
number p, a prime number r is called a primitive prime divisor of pm − 1 if r divides
pm − 1 but does not divide pk − 1 for any positive integer k < m. By Zsigmondy’s
theorem [17], pm − 1 has a primitive prime divisor whenever m ≥ 3 and (p,m) ̸= (2, 6).
Proposition 2.3. Let G = Sz(q) with q = 22n+1 ≥ 8 and let r be a primitive prime divisor
of q4 − 1. Then r ≥ 5, G has an element y of order r and, for each such y, there exists
x ∈ G such that |x| = 4, |xy| ≥ 3 and G = ⟨x, y⟩.
Proof. First, recall that |G| = q2(q2 + 1)(q − 1) (see [6, Page 8, Table I]). Since r is a
primitive prime divisor of q4 − 1, it divides q4 − 1 but not q2 − 1 and thus must divide
q2+1. It follows that G has an element y of order r and that r ≥ 5. We will now prove that
there exists an element x of order 4 with the required properties, essentially by a somewhat
crude counting argument.
We denote by Eq the elementary abelian group of order q and, for an integer n ≥ 2, by
Cn the cyclic group of order n and D2n the dihedral group of order 2n.
Up to conjugation, the maximal subgroups of G are the following (see for instance [2,
Table 8.16]):
• (Eq.Eq)⋊ Cq−1,
• D2(q−1),
• Cq+√2q+1 ⋊ C4,
• Cq−√2q+1 ⋊ C4,
• Sz(q0), where q0 = q1/d > 2 for some prime divisor d of 2n+ 1.
Recall that r is odd, does not divide q − 1 nor q40 − 1 and thus does not divide its factor
(q20 + 1)(q0 − 1). This implies that r does not divide |Sz(q0)| = q20(q20 + 1)(q0 − 1). It
follows that a maximal subgroup M of G containing y must be of the form Cq±√2q+1⋊C4.
Since every subgroup of a cyclic group is characteristic, ⟨y⟩ is normal in M and thus M
is the only maximal subgroup of G containing y (for otherwise ⟨y⟩ would be normal in
another maximal subgroup N of G and thus normal in ⟨M,N⟩ = G).
Let Q be a Sylow 2-subgroup of G. Then Q = Eq.Eq and |NG(Q)| = (Eq.Eq)⋊Cq−1.
Hence the number n of Sylow 2-subgroups of G is
n =
|G|
|NG(Q)|
=
q2(q2 + 1)(q − 1)
q2(q − 1)
= q2 + 1.
Let n2 and n4 denote the numbers of elements of order 2 and 4, respectively, in G. Ac-
cording to [5, Lemma 3.2], there are q− 1 involutions and q2 − q elements of order 4 in Q,
and different conjugates of Q have trivial intersection. Then
n2 = n(q − 1) = (q2 + 1)(q − 1)
and
n4 = n(q
2 − q) = (q2 + 1)(q2 − q).
118 Ars Math. Contemp. 22 (2022) #P1.07 / 115–120
Let
I = {g ∈ G : |gy| ≤ 2}
and
J = {g ∈ G : ⟨g, y⟩ ≠ G}.
Then |I| = n2 + 1 and, since M is the unique maximal subgroup of G containing y,
|J | ≤ |M |. Since
|I|+ |J | ≤ n2 + |M |+ 1
= (q2 + 1)(q − 1) + 4(q ±
√
2q + 1) + 1
≤ (q2 + 1)(q − 1) + 4(q +
√
2q + 1) + 1
< (q2 + 1)(q2 − q) = n4,
it follows that there exists x ∈ G with |x| = 4 and x /∈ I ∪ J , as required.
2.2 Constructing ORRs of out-valency 2
Lemma 2.4. Let G = ⟨x, y⟩. If |x| = 3 and |y| ≥ 4, then Cay(G, {x, y}) is an ORR,
unless |y| = 6 and x = y4, and G ∼= C6.
Proof. Let Γ = Cay(G, {x, y}) and let A = Aut(Γ). Note that Γ is a strongly connected
proper digraph. Figure 1 shows all the directed paths of length at most 3 in Γ starting at 1.
1
x
y
x2
yx
y2
xy
x3
yx2
xyx
y2x
y3
xy2
yxy
x2y
Figure 1: All the directed paths of length at most 3 in Cay(G, {x, y}).
Since |y| ≥ 4, we have y3 ̸= 1 and y ̸= x−2. Moreover, if y2 = x−1, then |y| = 6 and
thus x = y4 and the result holds. We thus assume this is not the case. Since x3 = 1, this
implies that (1, x, x2, x3) is the only directed cycle of length 3 starting at 1. This implies
that the stabiliser A1 of the vertex 1 also fixes x. As 1 only has one out-neighbour other
than x, it must also be fixed. By vertex-transitivity, we find that fixing a vertex fixes its out
neighbours and, using connectedness, we conclude that A1 = 1 and thus Γ is an ORR.
Lemma 2.5. Let G = ⟨x, y⟩. If |x| = 4, |y| ≥ 5 and |xy| ≥ 3, then Cay(G, {x, y}) is an
ORR, unless |y| = 12 and x = y9, and G ∼= C12.
G. Verret and B. Xia: Oriented regular representations of out-valency two for finite simple groups 119
Proof. Let Γ = Cay(G, {x, y}) and let A = Aut(Γ). Note that Γ is a strongly connected
proper digraph. Figure 2 shows all the directed paths of length at most 4 in Γ starting at 1.
1
x
y
x2
yx
y2
xy
x3
yx2
xyx
y2x
y3
xy2
yxy
x2y
x4
yx3
xyx2
y2x2
x2yx
yxyx
xy2x
y3x
y4
xy3
yxy2
x2y2
y2xy
xyxy
yx2y
x3y
Figure 2: All the directed paths of length at most 4 in Cay(G, {x, y}).
Since |y| ≥ 5, we have y4 ̸= 1, y ̸= x−3 and y2 ̸= x−2. Similarly, |xy| ≥ 3 implies
that (xy)2 ̸= 1 ̸= (yx)2. Moreover, if y3 = x−1, then |y| = 12 and thus x = y9 and
the result holds. We thus assume this is not the case. Since x4 = 1, this implies that
(1, x, x2, x3, x4) is the only directed cycle of length 4 starting at 1 and, as in the previous
lemma, Γ is an ORR.
3 Proof of Theorem 1.1
Let G be a finite simple group with |G| ≥ 5. We first suppose that G = F+p for some prime
p ≥ 5. Let x, y ∈ Fp \ {0} such that x ̸= ±y and let Γ = Cay(G, {x, y}). Note that Γ is
a proper digraph of out-valency 2. By [16, Proposition 1.3 and Example 2.2], Γ is an ORR
if and only if the only solution to
{λx, λy} = {x, y} (3.1)
with λ ∈ F×p is λ = 1. Suppose otherwise, that is (3.1) holds with λ ̸= 1. This implies that
λx = y and λy = x, which yields that
λx2 = (λx)x = y(λy) = λy2,
and hence x2 = y2, contradicting x ̸= ±y. Thus we conclude that Γ is an ORR, as
required.
We may now assume that G is nonabelian. If G has an element x of order 3 then,
by Corollary 2.2 there exists y ∈ G such that |y| ≥ 4 and G = ⟨x, y⟩. By Lemma 2.4,
Cay(G, {x, y}) is an ORR. We may thus assume that G does not have an element of order
120 Ars Math. Contemp. 22 (2022) #P1.07 / 115–120
3 and thus G = Sz(q) for some q = 22n+1 ≥ 8. Let r be a primitive prime divisor of
q4 − 1. By Proposition 2.3, G contains elements x and y such that |x| = 4, |y| = r ≥ 5,
|xy| ≥ 3 and G = ⟨x, y⟩. By Lemma 2.5, Cay(G, {x, y}) is an ORR.
ORCID iDs
Gabriel Verret https://orcid.org/0000-0003-1766-4834
Binzhou Xia https://orcid.org/0000-0002-8031-2981
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P1.08 / 121–133
https://doi.org/10.26493/1855-3974.2473.f2e
(Also available at http://amc-journal.eu)
New strong divisibility sequences*
Zhibin Du
School of Software, South China Normal University, Foshan, Guangdong 528225, China
Darko Dimitrov
Faculty of Information Studies, 8000 Novo mesto, Slovenia
Carlos M. da Fonseca †
Kuwait College of Science and Technology,
Doha District, Block 4, P.O. Box 27235, Safat 13133, Kuwait, and
Chair of Computational Mathematics, University of Deusto,
48007 Bilbao, Basque Country, Spain
Received 28 October 2020, accepted 26 June 2021, published online 5 April 2022
Abstract
We provide new families of divisibility and strong divisibility sequences based on some
factorization properties of Chebyshev polynomials.
Keywords: Strong divisibility sequences, Chebyshev polynomials, tridiagonal matrices, determinant.
Math. Subj. Class. (2020): 11B37, 11B39, 11B83, 15B05, 33C45
1 Introduction
A sequence of any integer numbers {an} is said to be a divisibility sequence if
am | an, whenever m | n,
and is called a strong divisibility sequence if
gcd(am, an) = agcd(m,n).
*This work was supported by the National Natural Science Foundation of China (Grant No. 11701505).
†Corresponding author.
E-mail addresses: zhibindu@126.com (Zhibin Du), darko.dimitrov11@gmail.com (Darko Dimitrov),
c.dafonseca@kcst.edu.kw (Carlos M. da Fonseca)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
122 Ars Math. Contemp. 22 (2022) #P1.08 / 121–133
The strong divisibility sequences and its weaker version have been studied for more
than one century. Actually, the Fibonacci numbers
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, . . .
are perhaps the best known non-trivial strong divisibility sequence. For earlier questions,
open problems, and general characterizations, the reader is referred to [4, 10, 11, 12, 21,
22].
As a particular case of the general conditional recurrence sequences defined in [16],
recently it was proposed in [20] the study of the conditional recurrence sequences {fn}
satisfying the recurrence relations of integers
fn =
{
a1fn−1 + b2fn−2, if n is odd,
a2fn−1 + b1fn−2, if n is even.
for n ⩾ 2, with f0 = 1 and f1 = a1, aiming to generate new strong divisibility sequences.
Indeed, the authors were able to obtain sufficient conditions for which certain subsequences
of {fn} are strong divisible.
Theorem 1.1 ([20]). Let f̃n = f2n−1. If a1 = 1 and gcd(a1a2 + b1 + b2, b1b2) = 1, then
gcd(f̃m, f̃n) = f̃gcd(m,n).
Corollary 1.2 ([20]). Let f̃n = f2n−1. If gcd(a1a2 + b1 + b2, b1b2) = 1, then {f̃n} is a
strong divisibility sequence.
Theorem 1.3 ([20]). Let f̃n = f2n−1. Thus f̃m | f̃n, whenever m | n.
For example, setting a1 = 3, a2 = 1 = b1, and b2 = 2, we get
n 1 2 3 4 5 6 7 8 9
fn 3 4 18 22 102 124 576 700 3252
This means that the first terms of the subsequence of odd indices of {fn} are
n 1 2 3 4 5 6
f̃n 3 18 102 576 3252 18360
While {f̃n} is a divisibility sequence, it is clear that is not strong.
Another interesting result obtained in [20] is the following:
Theorem 1.4. Let f̃1 = 1 and f̃n = fn−1, for n > 1. If a1 = 1, b1 = b2, and
gcd(a2, b1) = 1, then {f̃n} is a strong divisibility sequence.
For the weaker divisibility, the following general result was obtained:
Corollary 1.5. Under the conditions of Theorem 1.4, {f̃n} is a divisibility sequence.
Z. Du et al.: New strong divisibility sequences 123
Our aim here is to extend the above results to a more general setting, namely for the
sequences of integers defined by the recurrence relations
fn =

a1fn−1 + bkfn−2, if n ≡ 1 (mod k),
a2fn−1 + b1fn−2, if n ≡ 2 (mod k),
a3fn−1 + b2fn−2, if n ≡ 3 (mod k),
· · · · · ·
ak−1fn−1 + bk−2fn−2, if n ≡ k − 1 (mod k),
akfn−1 + bk−1fn−2, if n ≡ 0 (mod k),
(1.1)
for n ⩾ 2, with f0 = 1 and f1 = a1. The previous results will be recovered by making
k = 2. Consequently, we answer to the open problem proposed in [20].
In this paper, we will relate (1.1) with the so-called periodic continuants [6, 18] (for
recent applications, the reader is referred to [1, 2, 3]). This relation is established by using
Chebyshev polynomials of the second kind. Then, from {fn} we can, under certain con-
ditions, generate new strong divisibility sequences. At the same time, we can recover the
connection between the sequences defined by recurrence relations with two terms and the
determinants of tridiagonal matrices. This is effectively in the spirit of some ideas we can
find in [15], proposed by Édouard Lucas back to 1878.
2 The determinant of a tridiagonal k-Toeplitz matrix
The matrices of the form
An =

a1 b1
c1
. . . . . .
. . . ak bk
ck a1 b1
c1
. . . . . .
. . . ak bk
ck a1 b1
c1
. . . . . .
. . .

n×n
,
i.e., tridiagonal matrices An = (aij) with entries satisfying
ai+k,j+k = aij , for i, j = 1, 2, . . . , n− k,
are known as tridiagonal k-Toeplitz. The determinant of such matrix is known as a periodic
continuant [18].
For k = 1, we get a tridiagonal Toeplitz matrix and its determinant was known in [18]
as a continuant. The characteristic polynomial of such a matrix was found by V. Lovass-
Nagy and P. Rózsa [13, 14], in 1963. Notwithstanding, the particular case when k = 2 and
the two subdiagonals are constant equal to 1, had been considered in 1947 in D. E. Ruther-
ford’s seminal paper [19], followed soon after by J. F. Elliott with his Master’s thesis [5,
124 Ars Math. Contemp. 22 (2022) #P1.08 / 121–133
Section IV.4]. In 1966, Rózsa held a seminar at the University of Hamburg on tridiagonal
k-Toeplitz matrices motivated mainly by problems of lattice dynamics, of ladder networks,
and of structural analysis. In that year, L. Elsner and R. M. Redheffer [6] studied An for
special cases of k and, two years later, P. Rózsa in [18] originally proved a general formula
for the determinant of An. Independently, the spectrum of a tridiagonal 2-Toeplitz matrix
was also studied by M. J. C. Gover in 1994 [9]. In [7], it is considered the case when k = 3
and, later on, the characteristic polynomial of An was stated, for any k, when analyzing the
invertibility conditions for An based on orthogonal polynomials theory (cf. [8]).
We recall now Rózsa’s solution. Let ∆i1,i2,...,ip be the principal minor of An indexed
by the rows and columns i1, i2, . . . , ip. The determinant of An is given in [18] as
detAn = (
√
b1c1 · · · bkck)q
(
∆1...,rUq(x)+
√
bkckb1c1 · · · brcr√
br+1cr+1 · · · bk−1ck−1
∆r+2,...,k−1 Uq−1(x)
)
with n = qk + r and
x =
∆1,...,k − bkck∆2,...,k−1
2
√
b1c1 · · · bkck
,
assuming that ∆1,...,r = 1 and ∆2,...,r = 0, for r = 0, and with {Un(x)}n⩾0 standing for
the Chebyshev polynomials of the second kind. These polynomials satisfy the three-term
recurrence relation
Un+1(x) = 2xUn(x)− Un−1(x), for all n = 1, 2, . . . , (2.1)
with initial conditions U0(x) = 1 and U1(x) = 2x. We recall that the main explicit formula
for the Chebyshev polynomials of the second kind could be
Un(x) =
sin(n+ 1)θ
sin θ
, with x = cos θ (0 ⩽ θ < π), (2.2)
for all n = 0, 1, 2 . . . . While (2.2) is more common to find in the orthogonal polynomials
theory, there are other explicit representations and relations for Un(x). Among them, the
most frequent are
Un(x) =
(
x+
√
x2 − 1
)n+1 − (x−√x2 − 1)n+1
2
√
x2 − 1
,
an immediate consequence of de Moivre’s formula, and
Un(x) =
⌊n2 ⌋∑
k=0
(−1)k
(
n− k
k
)
(2x)n−2k.
Taking into account the definition of An, we can redefine (1.1) in terms of the determi-
Z. Du et al.: New strong divisibility sequences 125
nant of An, namely,
fn =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
a1 b1
−1
. . . . . .
. . . ak bk
−1 a1 b1
−1
. . . . . .
. . . ak bk
−1 a1 b1
−1
. . . . . .
. . .
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
n×n
. (2.3)
That means that the determinant (2.3) is
fn = (i
k
√
b1 · · · bk)q
(
∆1,...,r Uq(x) +
ir+1
√
bkb1 · · · br
ik−r−1
√
br+1 · · · bk−1
∆r+2,...,k−1 Uq−1(x)
)
,
where
x =
∆1,...,k + bk∆2,...,k−1
ik2
√
b1 · · · bk
.
In particular, if r = k − 1, then
fn = (i
k
√
b1 · · · bk)q∆1,...,k−1 Uq(x), (2.4)
which we will explore to generate new strong divisibility sequences in the next sections.
Before that, we recall a general result relating distinct minors, which can be found for
example in [18].
Lemma 2.1. For any positive integer n and i < j,
∆1,...,j−1∆i+1,...,n − (−1)j−ibi · · · bj−1∆1,...,i−1∆j+1,...,n = ∆1,...,n∆i+1,...,j−1.
In fact, Lemma 2.2 in [20] is a particular case of Lemma 2.1.
3 New divisibility sequences
In [20], the authors asked for conditional (strong) divisibility sequences for r > 2, i.e.,
satisfying (1.1). We start with the weaker condition.
Let us recall several factorization properties for Chebyshev polynomials disclosed
in [17].
Theorem 3.1 ([17]). Let m ⩾ n be two positive integers. Then Um(x) is a multiple of
Un(x) if and only if m = (ℓ+ 1)n+ ℓ, for some nonnegative integer ℓ. More precisely, if
ℓ is even, then
Um(x) = Un(x)
2 ℓ2∑
k=0
Tm−(2k+1)n−2k(x)− 1
 ,
126 Ars Math. Contemp. 22 (2022) #P1.08 / 121–133
and if ℓ is odd, then
Um(x) = 2Un(x)
ℓ−1
2∑
k=0
Tm−(2k+1)n−2k(x).
In Theorem 3.1, {Tn(x)}n⩾0 stands for the Chebyshev polynomial of the first kind.
These polynomials satisfy the same recurrence (2.1), here with initial conditions
T0(x) = 1 and T1(x) = x. An explicit formula for such polynomials is Tn(x) = cosnθ,
with x = cos θ.
The next two results, naturally connected to those in Section 1, can be found in [17].
Theorem 3.2. Let m and n be two nonnegative integers and d = gcd(m,n). Then
gcd(Um−1(x), Un−1(x)) = Ud−1(x).
Corollary 3.3. If m and n are coprime, then gcd(Um−1(x), Un−1(x)) = 1.
The general sequences that we consider are
fn = (±
√
b)n−1Un−1
(
a
±2
√
b
)
,
where a, b are nonzero integers (possibly with b < 0), for n ⩾ 1. In particular, f0 = 0,
f1 = 1 and f2 = a.
It is worth mentioning that the symbol ± can be ignored, that is to say:
fn = (±
√
b)n−1Un−1
(
a
±2
√
b
)
= (
√
b)n−1Un−1
(
a
2
√
b
)
, (3.1)
since the Chebyshev polynomials of the second kind Un(x) have the same parities as n.
We may now state our first main result.
Theorem 3.4. For any integers a and b, {fn} as defined in (3.1) is a divisibility sequence.
Proof. Assume that n | m, say m = sn, where s ⩾ 1. For simplicity, set x = 1
2
√
b
. So
fn =
Un−1(ax)
(2x)n−1
and fm =
Usn−1(ax)
(2x)sn−1
,
which implies that
fm
fn
=
Usn−1(ax)
(2x)(s−1)nUn−1(ax)
.
Set ℓ = s− 1, we have sn− 1 = (ℓ+ 1)(n− 1) + ℓ. From Theorem 3.1, Usn−1(x) is
a multiple of Un−1(x). More precisely, when s is even,
Usn−1(x) = 2Un−1(x)
s−2
2∑
t=0
T(s−2t−1)n(x),
Z. Du et al.: New strong divisibility sequences 127
and when s is odd,
Usn−1(x) = Un−1(x)
2 s−12∑
t=0
T(s−2t−1)n(x)− 1
 .
Therefore
fm
fn
=
Usn−1(ax)
(2x)(s−1)nUn−1(ax)
=
2
(2x)(s−1)n
s−2
2∑
t=0
T(s−2t−1)n(ax)
when s is even, and
fm
fn
=
Usn−1(ax)
(2x)(s−1)nUn−1(ax)
=
2
(2x)(s−1)n
s−1
2∑
t=0
T(s−2t−1)n(ax)−
1
(2x)(s−1)n
when s is odd.
We will prove Usn−1(ax)
(2x)(s−1)nUn−1(ax)
is an integer whether s is even or odd, by involving
with the following two claims.
Claim 1. 2T(s−2t−1)n
(
a
2
)
is an integer, for any 0 ⩽ t ⩽ ⌊ s−12 ⌋.
This claim follows immediately from the recurrence relation about Tn(x) as shown
in (2.1).
Claim 2. (
√
b)(s−1)nT(s−2t−1)n
(
1√
b
)
is an integer, for any 0 ⩽ t ⩽ ⌊ s−12 ⌋.
Observe that among all the terms in T(s−2t−1)n
(
1√
b
)
, the maximum degree of denom-
inator is (
√
b)(s−1)n, which means that all the denominators of T(s−2t−1)n
(
1√
b
)
would be
canceled by (
√
b)(s−1)n. It leads to this claim.
Combining the above claims, it leads to
2
(2x)(s−1)n
⌊ s−12 ⌋∑
t=0
T(s−2t−1)n(ax) = 2(
√
b)(s−1)n
⌊ s−12 ⌋∑
t=0
T(s−2t−1)n
(
a
2
√
b
)
is an integer. When s is even, fn | fm follows now. When s is odd, together with the fact
that 1
(2x)(s−1)n
= (
√
b)(s−1)n is an integer, fn | fm also holds.
4 Strong divisibility sequences
The sequence {fn} defined in (3.1) can have negative terms. Therefore, in our strongly
divisibility definition, we are assuming that gcd(am, an) = |agcd(m,n)|. Since we are
interested in positive conditional recurrence sequences (1.1), all the terms of {fn} will be
considered as positive or, equivalently, a > 0 and a2 − 4b ⩾ 0. Notice that the zeros
of the Chebyshev polynomials of the second kind are in the interval (−1, 1) and, from its
definition, limx→+∞ Un(x) = +∞.
128 Ars Math. Contemp. 22 (2022) #P1.08 / 121–133
In order to provide our characterization to the strong divisibility property of {fn}, let
us state several straightforward relations involving fn, as defined in (3.1). From (2.1), we
have
Un
(
a
2
√
b
)
=
a√
b
Un−1
(
a
2
√
b
)
− Un−2
(
a
2
√
b
)
and
fn = afn−1 − bfn−2. (4.1)
A more general identity can be obtained from (2.1), namely
Us+t (x) = Us (x)Ut (x)− Us−1 (x)Ut−1 (x) ,
and then,
fs+t = fs+1ft − bfsft−1. (4.2)
The next result is an extension of some other results we can find in the literature, as for
example related to the Fibonacci numbers.
Lemma 4.1. If gcd(a, b) = 1, then gcd(fn, fn+1) = 1 for any n ⩾ 1.
Proof. We claim that gcd(fn, b) = 1, for any n ⩾ 1, which can be proved by induction.
From f1 = 1 and f2 = a, this claim holds when n = 1, 2. Assume that gcd(fn−1, b) = 1
and gcd(fn−2, b) = 1. Suppose to the contrary that gcd(fn, b) = s, where s > 1. From
(4.1), s | afn−1. Notice that gcd(s, a) = 1, otherwise it is a contradiction to the hypothesis
that gcd(a, b) = 1. So s | fn−1. However, this is another contradiction to the inductive
hypothesis stating gcd(fn−1, b) = 1.
Now we are ready to show that gcd(fn, fn+1) = 1. Again, from f1 = 1 and f2 = a,
we know that gcd(fn, fn+1) = 1 is true when n = 1. Suppose to the contrary that
gcd(fn−2, fn−1) = 1, for some n ⩾ 3, but gcd(fn−1, fn) = t with t > 1. From
(4.1), t | bfn−2. Note that gcd(t, fn−2) = 1, otherwise we get a contradiction with
gcd(fn−2, fn−1) = 1. Thus, t | b means that t is a common divisor of b and fn, a
contradiction to the above claim that gcd(fn, b) = 1.
The proof is now completed.
We are now able to prove the main result of this section.
Theorem 4.2. The sequence {fn} defined in (3.1) is strongly divisible if and only if
gcd(a, b) = 1.
Proof. The necessity part is easy. Assume that {fn} is a strong divisibility sequence.
Suppose to the contrary that gcd(a, b) ̸= 1. From (4.1), we may obtain the first few values:
f1 = 1, f2 = a, f3 = a2 − b, f4 = a3 − 2ab. Clearly, gcd(f3, f4) ̸= 1 = f1 follows from
gcd(a, b) ̸= 1, which is a contradiction to the strong divisibility property of {fn}.
Now we prove the part of sufficiency. Suppose that gcd(a, b) = 1. Set g = gcd(n,m)
and d = gcd(fn, fm). We would like to show that gcd(fn, fm) = |fgcd(n,m)|, i.e., d =
|fg|, which comes from fg | d and d | fg .
On one hand, from g | n and g | m, we get fg | fn and fg | fm, since {fn} is a
divisibility sequence from Theorem 3.4. Thus, fg | d.
On the other hand, we still need to show that d | fg . Since, g = gcd(n,m), we may
assume that there exist positive integers s, k such that sn = g + km. From (4.2), we have
fsn = fg+km = fgfkm+1 − bfg−1fkm.
Z. Du et al.: New strong divisibility sequences 129
From d | fn, and fn | fsn (since {fn} is a divisibility sequence), we get d | fsn. Simi-
larly, we have d | fkm. Therefore, d | fgfkm+1. Notice that gcd(d, fkm+1) = 1, other-
wise, together with d | fkm, it leads to gcd(fkm, fkm+1) ̸= 1, which is a contraction to
Lemma 4.1. Now, it follows that d | fg .
Combining fg | d and d | fg , we obtain d = |fg|, which reveals the strong divisibility
property of {fn}.
5 Examples
In this final section, from the above results, we provide several examples of new (condi-
tional) strong divisibility sequences.
Setting k = 3, r = 2, we have
x =
∆1,...,k + bk∆2,...,k−1
ik2
√
b1 · · · bk
.
In particular, if r = k − 1, then
fn = (−i
√
b1b2b3)
q(a1a2 + b1)Uq
(
a1a2a3 + a3b1 + a1b2 + a2b3
−i2
√
b1b2b3
)
.
So, if we consider the sequence defined by
fn =

fn−1 + 3fn−2, ifn ≡ 1 (mod 3),
2fn−1 + fn−2, ifn ≡ 2 (mod 3),
4fn−1 + 2fn−2, if n ≡ 0 (mod 3),
we have
fn = (−i
√
6)q 3Uq
(
20
−i2
√
6
)
.
Now set
gq+1 = (−i
√
6)q Uq
(
20
−i2
√
6
)
,
for q ⩾ 0. The first terms are:
n gn
1 1
2 20
3 406
4 8240
5 167236
6 3394160
7 68886616
8 1398097280
9 28375265296
10 575893889600
Now, we can check, for example, that g3 | g6 or g5 | g10. However,
gcd(g8, g10) = 320.
130 Ars Math. Contemp. 22 (2022) #P1.08 / 121–133
Instead, we take the recurrence relation
fn =

2fn−1 + 3fn−2, if n ≡ 1 (mod 3),
fn−1 + fn−2, if n ≡ 2 (mod 3),
4fn−1 + 2fn−2, if n ≡ 0 (mod 3).
Setting
gq+1 = (−i
√
6)q Uq
(
19
−i2
√
6
)
,
for q ⩾ 0, the first terms are:
n gn
1 1
2 19
3 367
4 7087
5 136855
6 2642767
7 51033703
8 985496959
9 19030644439
10 367495226095
Now we can check, for example, that g4 | g8 or g5 | g10. Moreover,
gcd(g8, g10) = g2 or gcd(g6, g9) = g3,
and, of course,
gcd(g4, g9) = g1.
Let us consider now two more elaborated examples, for k = 4. We start with the
following one
fn =

2fn−1 + 4fn−2, if n ≡ 1 (mod 4),
fn−1 + 3fn−2, if n ≡ 2 (mod 4),
2fn−1 + fn−2, if n ≡ 3 (mod 4),
3fn−1 + fn−2, if n ≡ 0 (mod 4).
Setting
gq+1 = (
√
12)q Uq
(
53
2
√
12
)
,
Z. Du et al.: New strong divisibility sequences 131
for q ⩾ 0, the first terms are:
n gn
1 1
2 53
3 2797
4 147605
5 7789501
6 411072293
7 21693357517
8 1144815080885
9 60414878996701
10 3188250805854533
Straightforward verification shows, for example, that g4 | g8 or g5 | g10. Furthermore,
gcd(g8, g10) = g2 or gcd(g6, g9) = g3,
and, of course,
gcd(g4, g9) = g1.
Finally, we study
fn =

2fn−1 + 4fn−2, if n ≡ 1 (mod 4),
fn−1 + 2fn−2, if n ≡ 2 (mod 4),
2fn−1 + fn−2, if n ≡ 3 (mod 4),
3fn−1 + fn−2, if n ≡ 0 (mod 4).
Setting
gq+1 = (
√
8)q Uq
(
46
2
√
8
)
,
for q ⩾ 0, the first terms are:
n gn
1 1
2 46
3 2108
4 96600
5 4426736
6 202857056
7 9296010688
8 425993635200
9 19521339133696
10 894573651068416
Now we can check, for instance, that g4 | g8 or g5 | g10. However,
gcd(g8, g10) = 2944.
132 Ars Math. Contemp. 22 (2022) #P1.08 / 121–133
ORCID iDs
Zhibin Du https://orcid.org/0000-0001-5795-3580
Darko Dimitrov https://orcid.org/0000-0002-1648-9600
Carlos M. da Fonseca https://orcid.org/0000-0001-7742-4416
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P1.09 / 135–147
https://doi.org/10.26493/1855-3974.2257.6de
(Also available at http://amc-journal.eu)
Maximal order group actions on Riemann
surfaces
Jay Zimmerman * , Coy L. May
Towson University, 7800 York Road, Towson, USA
Received 20 February 2020, accepted 2 July 2021, published online 13 May 2022
Abstract
A natural problem is to determine, for each value of the integer g ≥ 2, the largest order
of a group that acts on a Riemann surface of genus g. Let N(g) (respectively M(g)) be the
largest order of a group of automorphisms of a Riemann surface of genus g ≥ 2 preserving
the orientation (respectively possibly reversing the orientation) of the surface.
The basic inequalities comparing N(g) and M(g) are N(g) ≤ M(g) ≤ 2N(g). There
are well-known families of extended Hurwitz groups that provide an infinite number of
integers g satisfying M(g) = 2N(g). It is also easy to see that there are solvable groups
which provide an infinite number of such examples.
We prove that, perhaps surprisingly, there are an infinite number of integers g such that
N(g) = M(g). Specifically, if p is a prime satisfying p ≡ 1 (mod 6) and g = 3p + 1 or
g = 2p+1, there is a group of order 24(g− 1) that acts on a surface of genus g preserving
the orientation of the surface. For all such values of g larger than a fixed constant, there are
no groups with order larger than 24(g − 1) that act on a surface of genus g.
Keywords: Riemann surface, genus, group action, NEC group, strong symmetric genus.
Math. Subj. Class. (2020): 57M60, 20F38, 20H10
1 Introduction
A finite group G can be represented as a group of automorphisms of a compact Riemann
surface. In most of the classical work, the group actions were required to preserve the
orientation of the Riemann surface. It is also possible to allow the group actions to reverse
the orientation of the surfaces.
Among the most interesting group actions for a particular value of the genus g are those
such that the orders of the groups are “large” relative to the genus g. A natural problem,
*Corresponding author.
E-mail addresses: jzimmerman@towson.edu (Jay Zimmerman), cmay@towson.edu (Coy L. May)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
136 Ars Math. Contemp. 22 (2022) #P1.09 / 135–147
then, is to determine, for each value of the integer g ≥ 2, the largest order of a group that
acts on a Riemann surface of genus g.
Let N(g) (respectively M(g)) be the largest order of a group of automorphisms of a
Riemann surface of genus g ≥ 2 preserving the orientation (respectively possibly reversing
the orientation) of the surface. Now suppose the group G acts on the Riemann surface X
of genus g ≥ 2 (possibly reversing the orientation of X). Let G+ be the subgroup of G
consisting of the orientation preserving automorphisms. Then |G+| ≤ N(g) and
|G| ≤ 2|G+| ≤ 2N(g). (1.1)
Consequently, we obtain the basic inequalities comparing N(g) and M(g).
N(g) ≤ M(g) ≤ 2N(g). (1.2)
The classical upper bound of Hurwitz shows that, for all g ≥ 2,
N(g) ≤ 84(g − 1) and M(g) ≤ 168(g − 1). (1.3)
A group G of order 84(g − 1) is called a Hurwitz group if it acts on a surface of genus g
preserving orientation. If the Hurwitz group has an extension G∗ of order 2|G| that acts on
the same surface, then G∗ is an extended Hurwitz group. If g is a genus for which there is
an extended Hurwitz group, then N(g) = 84(g − 1) and M(g) = 2N(g). These groups
have generated considerable interest; see especially [5] but also [3, 4] and [22]. There are
known infinite families of extended Hurwitz groups. For example, Conder showed that all
symmetric groups Σn for n > 167 are extended Hurwitz groups [8, p. 75]. Consequently,
the bounds in (1.3) and the upper bound for M(g) in (1.2) are attained for infinitely many
g.
On the other hand, the general lower bound for N(g) is
N(g) ≥ 8(g + 1) (1.4)
for all g ≥ 2. Further, this lower bound is the best possible, that is, there are infinitely many
g such that N(g) = 8(g + 1). These results were established independently by Accola [1]
and Maclachlan [13].
The corresponding lower bound for M(g) is easy to establish.
Theorem 1.1. For all integers g ≥ 2, M(g) ≥ 16(g + 1). Further, there are infinitely
many g such that M(g) = 16(g + 1).
The family of groups used by Accola [1] and Maclachlan [13] to establish the re-
sults about the lower bound (1.4) can be extended following the approach of Singerman
[22, p. 22]. This yields, for each g ≥ 2, the construction of a group of order 16(g+ 1) that
acts on a Riemann surface of genus g so that M(g) ≥ 16(g + 1).
These groups are another family of groups such that M(g) = 2N(g) for infinitely
many genera g. Indeed, intuitively, one expects M(g) to “often” be equal to 2N(g). But
it is certainly possible that M(g) < 2N(g). For example, the two smallest values of
g for which M(g) < 2N(g) are 17 and 20 with N(17) = 1344, M(17) = 1536 and
N(20) = 228, M(20) = 336. Also, the classification of orientably regular maps of genus
p+ 1 [9] and the Belolipetsky-Jones group of order 12p for prime p [2, p. 382] shows that
M(g) < 2N(g) for infinitely g.
J. Zimmerman and C. L. May: Maximal order group actions on Riemann surfaces 137
However, for some values of g, N(g) = M(g). The two smallest values of g satisfying
N(g) = M(g) are 27 and 28 with N(27) = M(27) = 624 and N(28) = M(28) = 1296.
Surprisingly, this equality holds for infinitely many g. Our main result is the following.
Theorem 1.2. There are infinitely many g such that M(g) = N(g).
Specifically, if p is a prime satisfying p ≡ 1 (mod 6) and g = 3p+ 1, there is a group
of order 24(g−1) that acts on a surface of genus g preserving the orientation of the surface.
For all such values of g = 3p + 1 larger than a fixed constant, there are no groups with
order larger than 24(g − 1) that act on a surface of genus g (including those that reverse
orientation). Similar results hold if p ≡ 1 (mod 6) and g = 2p+ 1.
Here we acknowledge our debt to the data on large group actions on surfaces of low
genus calculated by Conder [6]. This data was quite helpful in conjecturing Theorem 5.6
and its corollary Theorem 1.2.
We would also like to express our sincere gratitude to the referee for numerous helpful
comments. These led to significant improvements in the first three sections.
2 Background results
Much of the following background information is taken from [18]; also see
[10, Section 2]. Let the finite group G act on the (compact) Riemann surface X of genus
g ≥ 2. Then represent X = U/K, where K is a Fuchsian surface group and obtain
an NEC group Γ and a homomorphism ϕ : Γ → G onto G such that K = kernel ϕ.
Associated with the NEC group Γ are its signature and canonical presentation.
Further, the non-euclidean area µ(Γ) of a fundamental region for Γ can be calculated
directly from its signature. Here see [21, p.235], where µ(Γ) is given in terms of the
topological genus of the quotient surface U/Γ and the periods and link periods of Γ. Then
the genus of the surface X on which G acts is given by
g = 1 + |G| · µ(Γ)/4π. (2.1)
The simpler, classical case is that G acts on X preserving orientation. This is the case
if and only if Γ is a Fuchsian group and G is generated by elements ai, bi for 1 ≤ i ≤ h
and xj of order mj for 1 ≤ j ≤ k with relation x1 · · ·xk[a1, b1] · · · [ah, bh] = 1. Then the
application of (2.1) yields the classical Riemann-Hurwitz equation
2g − 2 = |G|
2h− 2 + k∑
j=1
(
1− 1
mj
) . (2.2)
The group G acts reversing the orientation of X in case Γ is a proper NEC group. Then
it is necessary to check that the surface group K does not contain orientation-reversing
elements, or equivalently, the image ϕ(Γ+) has index two in G [20, Theorem 1, p. 52].
If this condition holds, then we will say that G has a particular partial presentation with
the Singerman subgroup condition. The Riemann-Hurwitz equation in this case is more
complicated and is in [10, p. 274], for instance. In this case, though, |G| = 2|G+| and
(2.2) can be employed to calculate the relationship between the genus g and |G|.
In connection with group actions on surfaces, there are two natural parameters associ-
ated with each finite group. The symmetric genus σ(G) of the group G is the minimum
138 Ars Math. Contemp. 22 (2022) #P1.09 / 135–147
genus of any Riemann surface on which G acts faithfully (possibly reversing orientation).
The strong symmetric genus σ0(G) of G is the minimum genus of any Riemann surface on
which G acts faithfully preserving orientation.
Next we quickly survey the NEC groups with relatively small non-euclidean area. We
use the notation of [18]. First, an (ℓ,m, n) triangle group is a Fuchsian group Λ with
signature
(0;+; [ℓ,m, n]; {}), where 1/ℓ+ 1/m+ 1/n < 1.
If the group G is a quotient of Λ by a surface group, then G has a presentation of the form
Xℓ = Y m = (XY )n = 1. (2.3)
We will say that G has partial presentation T (ℓ,m, n).
There are two types of NEC groups with a triangle group as canonical Fuchsian sub-
group. A full (or extended) (ℓ,m, n) triangle group is an NEC group Γ with signature
(0;+; [ ]; {(ℓ,m, n)}), where 1/ℓ+ 1/m+ 1/n < 1.
If G is a quotient of Γ (by a surface group), then G has a presentation of the form
A2 = B2 = C2 = (AB)ℓ = (BC)m = (CA)n = 1, (2.4)
and, further, the subgroup generated by AB and BC (the image of Γ+) has index 2. The
partial presentation (2.4) will be denoted FT (ℓ,m, n).
A hybrid (m;n) triangle group is an NEC group Γ with signature
(0;+; [m]; {(n)}), where 2/m+ 1/n < 1.
The canonical Fuchsian subgroup Γ+ is a (m,m, n) triangle group. If G is a quotient of Γ,
then G has a presentation of the form
C2 = Xm = [C,X]n = 1, (2.5)
and the subgroup generated by X and CXC has index 2. This partial presentation will be
denoted HT (m;n).
An (ℓ,m, n, t) quadrilateral group is a Fuchsian group Λ with signature
(0;+; [ℓ,m, n, t]; { }), where 1/ℓ+ 1/m+ 1/n+ 1/t < 2.
A quotient group G of Λ has a presentation of the form
Xℓ = Y m = Zn = (XY Z)t = 1 (2.6)
We will denote this partial presentation Q(ℓ,m, n, t). If a group has presentation (2.6) with
ℓ, m, n, t all equal to 2, then the group acts on a torus.
Suppose G is a group that acts on a Riemann surface X of genus g ≥ 2, where X is
represented X = U/K and G = Γ/K. Particularly important here is the case in which
|G| > 24(g − 1), and we will say G is a large group of automorphisms of X . There
is, of course, a corresponding restriction on the non-euclidean area of the NEC group Γ
and the types of partial presentations that Γ can have. The area restriction is µ(Γ)/2π <
1/12, which is fairly limiting. A careful check of the signatures gives the following. This
result appears in [18, Theorem 2] and also [10, p. 275]. Here we have added the specific
Riemann-Hurwitz equation for each case. For example, if G has the partial presentation
FT (2, 4, s), then µ(Γ)/2π = (s − 4)/8s. Then using equation (2.2) gives 16(g − 1) =
|G|(s− 4)/s.
J. Zimmerman and C. L. May: Maximal order group actions on Riemann surfaces 139
Theorem A. Let G be a group that acts on a Riemann surface of genus g ≥ 2. Then
|G| > 24(g − 1) if and only if G has a partial presentation (with the relations fulfilled)
of type T (2, 4, 5) or T (2, 3, s), where 7 ≤ s ≤ 11, or one of the following types with the
Singerman subgroup condition satisfied. The application of the Riemann-Hurwitz equation
is included for each case.
1. FT (2, 3, s), 24(g − 1) = |G|(s− 6)/s where s ≥ 7,
2. FT (2, 4, s), 16(g − 1) = |G|(s− 4)/s where 5 ≤ s ≤ 11,
3. FT (2, 5, s), 20(g − 1) = |G|(3s− 10)/s where 5 ≤ s ≤ 7,
4. FT (3, 3, s), 12(g − 1) = |G|(s− 3)/s where 4 ≤ s ≤ 5,
5. HT (3; 4), 48(g − 1) = |G|,
6. HT (3; 5), 30(g − 1) = |G|,
7. HT (5; 2), 40(g − 1) = |G|.
3 Basic lower bound for M(g)
We begin by constructing the family of groups that provides the lower bound in Theo-
rem 1.1.
Fix the integer m ≥ 3, and let Lm be the group defined by the presentation
x2 = y4 = z2m = xyz = 1, (z2)x = z−2. (3.1)
It is easy to see that Lm is an extension of the cyclic group Zm by the dihedral group D4 and
consequently |Lm| = 8m. Then the group Lm has partial presentation T (2, 4, 2m). Then
a calculation using (2.1) shows that Lm acts on a Riemann surface X of genus g = m− 1
preserving the orientation of the surface. The group Lm has order 8(g + 1). This fam-
ily of groups is certainly not new. The family Lm was used, independently, to estab-
lish the lower bound 8(g + 1) by both Accola and Maclachlan; here see [1, p. 400] and
[13, Theorem 4, p. 266]. The construction of this family also appears in [2, p. 384].
Next we construct an extension of the group Lm by Z2, following the approach in
[15, p. 128]. To Lm adjoin an element t of order 2 that transforms the elements of Lm
according to the automorphism
α(x) = x−1, α(y) = y−1. (3.2)
Then the extension L∗m has presentation
t2 = x2 = y4 = z2m = xyz = (tx)2 = (ty)2 = 1, (z2)x = z−2. (3.3)
The extension L∗m of Lm has order 2|Lm| and has partial presentation FT (2, 4, 2m). Thus
the group L∗m is a group of order 16(g + 1) that acts on the surface X of genus g. This
extended family was described by Singerman in [22, p. 24]. Now it is easy to prove
Theorem 1.1.
Proof. Fix g ≥ 2, and set m = g + 1. Then M(g) ≥ |L∗m| = 16(g + 1). Also, there are
infinitely many values of g such that N(g) = 8(g + 1); here see [1, Theorem 4, p. 407]
or [13, Theorem 5, p. 272]. Then for such a value of g, M(g) ≤ 2N(g) = 16(g + 1)
using the basic inequality (1.2). Hence there are infinitely many values of g such that
M(g) = 16(g + 1).
140 Ars Math. Contemp. 22 (2022) #P1.09 / 135–147
Before proving Theorem 1.2, we establish an interesting result about the family of
groups Lm. We have seen that Lm acts preserving orientation on a Riemann surface of
genus g = m− 1. In fact, this value is the strong symmetric genus of the group Lm.
First, we get rid of a redundant generator in the definition of Lm and obtain the presen-
tation
x2 = z2m = (zx)4 = 1, (z2)x = z−2. (3.4)
Theorem 3.1. σ0(Lm) = m− 1.
Proof. Let Lm have generators x and z and relations (3.4) and be generated by u and v.
Define N = ⟨z2, (zx)2⟩. The element (zx)2 is in the center of Lm and since conjugation
by x inverts z2, N is a normal subgroup of Lm. Since Lm/N ∼= Z2 × Z2, uN , vN and
uvN are the same as the cosets xN , zN and zxN in some order. All elements of Lm in
the set xN have order 2 and all elements in the set zxN have order 4. The elements of the
set zN are of the form zk or zk(zx)2 and have order 2m/d, where d = gcd(k, 2m).
Let u be an element from xN and v an element from zxN . The product is contained
in zN . So uv = (xz)2zt = ((xz)2z)t or uv = zt, where t is odd. Next, suppose the
product uv has order smaller than 2m. So gcd(t,m) = d > 2. Let M = ⟨(xz)2, zt⟩.
Since xzx = (xz)2z−1, M is a normal subgroup of Lm of order 4m/d. It follows that
⟨u, v⟩ = ⟨u, uv⟩ ⊆ ⟨u,M⟩ and since |⟨u,M⟩| = 8m/d ̸= 8m, the elements u and v do
not generate Lm. Therefore, the product of two generators, u of order 2 and v of order 4,
must have order 2m.
Hence Lm has presentation T (2, 4, 2m), and the corresponding triangle group is the
only one that maps faithfully onto Lm.
Suppose that Lm has partial presentation Q(2, 2, 2, 2) and acts on a torus. Let Lm be
generated by involutions s, t, u and v satisfying stuv = 1. Let N = ⟨(zx)2, z2⟩. All
elements of order 2 are either contained in the coset xN or are in the normal subgroup
V = ⟨(zx)2, zm⟩ of Lm. Since there are no elements of order 2 in the coset zxN , an even
number of s, t, u or v must be from xN . If none of the generators are from xN , then
⟨s, t, u, v⟩ ⊆ ⟨z,N⟩ ̸= Lm. If all four generators are in xN , then ⟨s, t, u, v⟩ ⊆ ⟨x,N⟩ ≠
Lm. Suppose that only two of the generators are from xN , say u and v. Then ⟨s, t, u, v⟩ ⊆
⟨x, V ⟩ ̸= Lm and again we get a contradiction. It also follows that σ0(Lm) ̸= 1 for all
m > 2.
Suppose that Lm has partial presentation Q(2, 2, 2, 3). Let Lm be generated by involu-
tions s, t, u and the element v of order 3. The element v must be contained in ⟨z2⟩ ⊆ N .
Since there are no elements of order 2 in the coset zxN and (sN)(tN)(uN) = (1N), we
can’t have one of the cosets be xN and another be zN , since then the third would be in
zxN . If one or more of s, t and u are in xN , then ⟨s, t, u, v⟩ ⊆ ⟨x,N⟩ ≠ Lm. If one
or more of s, t and u are in zN , then ⟨s, t, u, v⟩ ⊆ ⟨z,N⟩ ≠ Lm. So Lm does not have
presentation Γ(2, 2, 2, 3). No other Fuchsian group has small enough non-euclidean area
and the proof is complete.
Theorem 3.1 shows that there is at least one group with strong symmetric genus g for all
g, which is the main result of [16]. One interesting thing here is that the well-known groups
of Theorem 3.1 provide an alternate proof of [16, Theorem 1], which was established using
groups of the form Zk ×Dn.
Theorem 3.1 also has a consequence for the function that counts the number of groups
of each genus. Using direct products and dicyclic groups, it was shown that there are at
J. Zimmerman and C. L. May: Maximal order group actions on Riemann surfaces 141
least four groups of strong symmetric genus g for all g ≥ 0 [14, Theorem 1]. It is not hard
to see that the group Lg+1 is another group of genus g, and we have the following.
Theorem 3.2. If g is a non-negative integer, then there are at least 5 groups of strong
symmetric genus g.
We remark here that these families Lm and L∗m are groups that act on the torus. These
groups are in in Proulx classes (g) and (k) respectively; for the associated partial presenta-
tions, see [12, pp. 291,292]. The orientation preserving subgroup of the action of L∗m on
the torus is not Lm, even though Lm is the orientation preserving subgroup of the action of
L∗m on the surface of genus m − 1. Consequently, the two families Lm and L∗m are of no
help in filling the symmetric genus spectrum. The groups Zk ×Dn used in [16] to fill all
the gaps in the strong symmetric genus spectrum are also groups that act on the torus and
have symmetric genus one.
4 A family of 24(g − 1) automorphisms
Our main task here is to show there are infinitely many values of g such that M(g) = N(g).
This result was something of a surprise, to us at least, and it is not easy to prove.
We start with the construction of another family of groups. Let p be a prime satisfying
p ≡ 1 (mod 6) and m an integer satisfying m3 ≡ 1 (mod p) and not congruent to 1
(mod p). Define the groups Jp by the presentation
x3 = u3 = v2 = zp = (uv)4 = [x, u] = [x, v] = [z, u] = 1, (4.1)
zx = zm, zv = z−1.
It is easy to see that Jp is the semidirect product of the cyclic group Zp by the group
Z3 ×Σ4, namely Zp ×ϕ (Z3 ×Σ4) where ϕ is a homomorphism mapping x into z → zm,
u into z → z and v into z → z−1.
Theorem 4.1. The group Jp has partial presentation T (2, 3, 12) and hence acts on a sur-
face of genus 1 + 3p.
Proof. We will use the presentation (4.1). First, o(v) = 2, o(ux) = 3 and o(vux) = 12.
In addition, ⟨v, ux⟩ = ⟨x, u, v⟩ ∼= Z3 × Σ4.
Define r = vz and w = z−1ux. Clearly, o(r) = 2. It is easy to verify that z−kx =
xz−km. Therefore, w3 = (z−1x)3 = z−(m
2+m+1) = 1, since m2 +m+ 1 ≡ 0 (mod p).
It follows that o(w) = 3 and o(rw) = 12.
Next, we need to show that Jp = ⟨r, w⟩. First, [r, w] = z−(m+1)[v, u]. Next, we show
that [r, w]3 = z−3(m+1). If m ≡ −1 (mod p), then m3 ≡ −1 (mod p) and this is false.
Therefore, z ∈ ⟨[r, w]⟩ and it follows that ⟨r, w⟩ = ⟨z, x, u, v⟩ = Jp. Thus Jp has partial
presentation T (2, 3, 12).
It is not difficult to see that, in fact, σ0(Jp) = 1 + 3p. An obvious consequence of
Theorem 4.1 is the following.
Theorem 4.2. Let p be a prime such that p ≡ 1 (mod 6), and let g = 3p + 1. Then the
group Jp is a group of order 24(g − 1) that acts on a surface of genus g preserving the
orientation of the surface. Consequently, for any such g,
M(g) ≥ N(g) ≥ 24(g − 1). (4.2)
142 Ars Math. Contemp. 22 (2022) #P1.09 / 135–147
There are, of course, infinitely many such g. We will show that, for most of these values
of g, M(g) = N(g) = 24(g − 1), establishing Theorem 1.2.
5 Large groups of automorphisms
Assume p is a prime such that p ≡ 1 (mod 6), and let g = 3p + 1. Then Theorem 4.2
shows that there is a group of order 24(g − 1) that acts on a surface of genus g preserving
the orientation of the surface, and inequality (4.2) holds. The hard part of the proof of
Theorem 1.2 is to show that, for most of these values of g, there are no large groups of
automorphisms, that is, no groups with order larger than 24(g− 1). We use Theorem A. In
this section we do not assume that p ≡ 1 (mod 6). However, in the proof it is necessary to
assume that that the prime p is not small. This will enable us to apply the following useful
result of Accola [1, Lemma 5, p. 402].
Accola’s Lemma. Let G be a non-abelian image of the triangle group T (2, 3, λ) of order
µλ. Then λ ≤ µ2.
Let X be a Riemann surface of genus g, and suppose that G were a large group of auto-
morphisms of X . Then |G| > 24(g − 1) = 72p, and G has one of the partial presentations
in Theorem A. We show that, in fact, G cannot have any of these partial presentations.
While it is necessary to consider each presentation, we describe the overall outline of the
argument but omit some details. In addition, to apply Accola’s Lemma, it is necessary to
assume that the prime p is not small, and we assume that p > (36)2.
Lemma 5.1. If the prime p > (36)2, then p divides |G| but p2 does not.
Proof. Suppose first that G has any of the partial presentations in Theorem A except
FT (2, 3, s). In these cases, the Riemann-Hurwitz formulas in Theorem A give |G| in terms
of the parameter s, and for the values of s that can occur, |G| is a multiple of p but p2 does
not divide |G| (for large p). For example, suppose G has partial presentation FT (2, 5, s),
where s is 5, 6 or 7. Then if s is 5 or 6, then G is 120p or 90p, respectively, and p2 does
not divide |G| if p > 5. If s = 7, then |G| is not an integer.
Suppose now that G has partial presentation FT (2, 3, s) where s ≥ 7. In this case,
|G| = 72ps/(s − 6) so that 72ps = |G|(s − 6). First, for small s, 7 ≤ s ≤ 12, s ̸= 11,
|G| is a multiple of p but p2 does not divide |G| (for large p). For example, if s = 8,
|G| = 288p. If s = 11, |G| is not an integer.
Assume then that G has partial presentation FT (2, 3, s) where s > 12, the hard case.
Now by Euclid’s Lemma, either p divides |G| or p divides (s− 6).
Assume that p divides (s − 6) and write s − 6 = mp for some integer m ≥ 1. Now
s = mp + 6 > p > (36)2 (by assumption). But on the other hand, |G| = 72ps/mp =
72s/m. Then |G+| = 36s/m. The group of orientation preserving automorphisms G+
is a T (2, 3, s) group of order cs, where c = 36/m ≤ 36. Now by Accola’s Lemma,
p < s ≤ c2 ≤ (36)2, an obvious contradiction. Thus, if G is a FT (2, 3, s) group (and
p > (36)2), then p divides |G|.
Finally, we have |G|/p = 72s/(s − 6). With s > 12, s/(s − 6) < 2 so that
|G|/p = 72s/(s− 6) < 144. Hence, p2 does not divide |G| for large p.
Lemma 5.2. The Sylow p-subgroup Sp ∼= Zp of G is normal in G.
J. Zimmerman and C. L. May: Maximal order group actions on Riemann surfaces 143
Proof. A review of the calculations in the previous proof shows that in each case that is
arithmetically possible, |G| = cp for some constant c < p for large p. The constant c
depends on the presentation, of course. Now, obviously, |Sp| = p. Also, the number np of
Sylow p-subgroups of G is ≡ 1 (mod p) and is a divisor of |G|. Then np ≥ (p + 1) is
clearly not possible. Hence Sp is normal in G.
Now let Sp act on X with Y = X/Sp the quotient space, γ the genus of Y and
π : X → Y the quotient map.
Lemma 5.3. The quotient map π is unramified, and the quotient space Y = X/Sp has
genus γ = 4. Further, the quotient group Q = G/Sp is a large group of automorphisms
of Y .
Proof. Let τ be the number of branch points of π. Then the Riemann-Hurwitz formula
gives
2(g − 1)/p = 2(γ − 1) + τ(p− 1)/p. (5.1)
Then 2(g − 1) = 2p(γ − 1) + τ(p − 1) and we have g − 1 = 3p. Now τ(p − 1) =
6p− 2p(γ − 1) = 2p(4− γ). Since τ(p− 1) ≥ 0, 4 ≥ γ. If γ = 4, then τ = 0. Assume
γ < 4. Then p− 1 divides p(8− 2γ). Since p− 1 and p are relatively prime, p− 1 divides
(8 − 2γ) so that p − 1 ≤ 8 − 2γ ≤ 8. Now p ≤ 9 contradicting the assumption that p is
large. Thus γ = 4 and the number of branch points τ = 0, that is, the quotient map π is
unramified.
Now the quotient group Q acts on the surface Y of genus 4. Since G is a large group
of automorphisms of X , |G| > 24(g − 1) = 72p. Then |Q| = |G|/p > 72 = 24(4 − 1)
and Q is a large group of automorphisms of Y .
The large group actions on Riemann surfaces of genus 4 have been classified, and these
are presented in Table 1. These group actions were considered in determining the groups
of symmetric genus 4; here see [18, pp. 4089,4090] and [10, p. 285]. With a single
exception, these actions correspond to groups of reflexible regular maps. A description
of the connection between groups of regular maps and large groups of automorphisms of
Riemann surfaces is in [17, p. 24]. The regular maps of genus 4 were first classified by
Garbe [11, p. 53]. These maps also appear in [7, Table 1]. In Table 1, we give the group
number in the MAGMA small groups library. Map symbols are from [7].
Table 1: Large Group Actions on Surfaces of Genus 4.
Group Order Library Partial Map G/G′
Number Presentation Symbol
Σ3 × Σ4 144 183 FT (2, 3, 12) R4.1 (Z2)2
Z2 × Σ5 240 189 FT (2, 4, 5) R4.2 (Z2)2
Σ5 120 34 T (2, 4, 5) R4.2 Z2
144 186 FT(2,4,6) R4.3 (Z2)3
D4 ×D5 80 39 FT(2,4,10) R4.4 (Z2)3
Z2 ×A5 120 35 FT(2,5,5) R4.6 Z2
Σ5 120 34 HT (5; 2) Z2
The group G is an extension of Sp ∼= Zp by Q. Since |Q| is relatively prime to p, the
group G is a semidirect product, by the Schur-Zassenhaus Lemma.
144 Ars Math. Contemp. 22 (2022) #P1.09 / 135–147
Lemma 5.4. G ∼= Zp ×ϕ Q.
The following is important here. The proof is an exercise using the definition of semidi-
rect product.
Lemma 5.5. Let H be the semidirect product K ×θ Q, and let L = kernel(θ). Then L is
normal in the big group H .
Theorem 5.6. Let p be a prime such that p > (36)2. There are no large groups of auto-
morphisms that act on a surface of genus g = 3p+ 1.
Proof. For each of the possibilities for Q, we show that G cannot have the relevant partial
presentation.
First suppose there is a group G of order 144p with partial presentation FT (2, 3, 12).
In particular, G is generated by involutions. Then G ∼= Zp ×ϕ Q, where Q ∼= Σ3 × Σ4.
Let L = kernel(ϕ). Since ϕ : Q → Aut(Zp) ∼= Zp−1, Q/L is cyclic. It follows that Q′ ⊂
L ⊂ Q. Now a calculation shows that the commutator quotient group Q/Q′ ∼= (Z2)2.
Hence L must have index 1 or 2 in Q, and L is normal in G by Lemma 5.5. If L = Q, then
G ∼= Zp × Q. Then G is obviously not generated by involutions, since Zp is not. Hence
[Q : L] = 2 and the quotient group G/L has order 2p so that G/L is isomorphic to either
Z2p or the dihedral group Dp. Since Z2p is not generated by involutions, we must have
G/L ∼= Dp. But Dp is not a quotient of a FT (2, 3, 12) group (the product of reflections
in Dp has order p or 1). Thus there is no group of order 144p with partial presentation
FT (2, 3, 12).
Essentially the same proof (using the same notation) shows that there are no groups of
order 80p with presentation FT (2, 4, 10) and also none of order 144p with presentation
FT (2, 4, 6). The only difference in each of the cases is that the commutator quotient group
Q/Q′ ∼= (Z2)3. But it still follows that L has index 1 or 2 in Q.
The proof is very similar but even easier in case there were a group G of order 120p
with presentation FT (2, 5, 5). Then G ∼= Zp ×ϕ Q, where Q ∼= Z2 × A5. Now Q′ ∼= A5
so that either L = Q or L = Q′. Again, as in the previous cases, L has index 1 or 2 in Q,
and it follows in the same way that this case is not possible either.
Now suppose there were such a group G with order 120p with partial presentation
T (2, 4, 5). Then G is generated by two elements of orders 2 and 5. Then G ∼= Zp ×ϕ Q,
where Q ∼= Σ5. Then Q′ ∼= A5, and thus either L = Q or L = Q′. If L = Q, then
G ∼= Zp ×Q. Then G is obviously not a quotient of a T (2, 4, 5) group, since Zp is not (Zp
has no elements of order 2). Hence L = Q′ and the quotient group G/L has order 2p so
that G/L is isomorphic to either Z2p or the dihedral group Dp. Neither group is a quotient
of T (2, 4, 5) groups; neither group has an element of order 5. Thus there is no group of
order 120p with partial presentation T (2, 4, 5).
Further, there is no group G of order 240p with partial presentation FT (2, 4, 5). If
there were such a group G acting on a surface X of genus 3p + 1, then the group G+ of
orientation preserving automorphisms would be a T (2, 4, 5) group acting on X . But we
have just seen that this is not possible. Hence there is no FT (2, 4, 5) group of order 240p.
Finally, assume that there were a group G of order 120p with partial presentation
HT (5; 2). Then G is generated by two elements of orders 2 and 5, and G ∼= Zp ×ϕ Q,
where Q ∼= Σ5. Now Q′ ∼= A5, and either L = Q or L = Q′. If L = Q, then G ∼= Zp×Q.
Then G is obviously not a quotient of a HT (5; 2) group, since Zp is not (Zp has no ele-
ments of order 2). Hence L = Q′ and the quotient group G/L has order 2p so that G/L is
J. Zimmerman and C. L. May: Maximal order group actions on Riemann surfaces 145
isomorphic to either Z2p or the dihedral group Dp. Neither of these groups is a quotient of
a HT (5; 2) group; neither group has an element of order 5. Thus there is no group of order
120p with partial presentation HT (5; 2).
In summary, none of the partial presentations listed in Table 1 are possible, and there is
no large group action on a surface of genus g = 3p+ 1 for large p > (36)2.
Combining Theorems 4.2 and 5.6 gives the following.
Theorem 5.7. Let p be a prime such that p ≡ 1 (mod 6) and p > (36)2, and let
g = 3p+ 1. Then for any such g,
M(g) = N(g) = 24(g − 1). (5.2)
Applying Dirichlet’s Theorem about the number of primes in an arithmetic sequence
establishes Theorem 1.2.
6 Another family of 24(g − 1) automorphisms
There is another interesting family of groups that can be used to determine an infinite
sequence of odd values of g such that M(g) = N(g). This provides an alternate proof of
Theorem 1.2, and it may be established using arguments similar to those in the two previous
sections. But there is no improvement to Theorem 1.2, of course, and technically the proof
is somewhat harder. We describe this family of groups but only comment very briefly on
the arguments in this case.
Let p be a prime satisfying p ≡ 1 (mod 6) and m an integer satisfying m3 ≡ 1
(mod p) and not congruent to 1 (mod p). Define the groups Kp by the presentation
u3 = v2 = (uv)3(u−1v)3 = zp = 1, zu = zm, zv = z−1. (6.1)
It is easy to see that Kp is the semidirect product of the cyclic group Zp by the group
P48. The group P48 has order 48 and contains SL(2, 3) as a subgroup; a presentation is in
[15, p. 116]. It is one of the groups of symmetric genus 2 [15, Theorem 4]. The group Kp
has partial presentation T (2, 3, 12) and acts on a surface of genus 1 + 2p. This gives the
following analog of Theorem 4.2.
Theorem 6.1. Let p be a prime such that p ≡ 1 (mod 6), and let g = 2p + 1. Then the
group Kp is a group of order 24(g − 1) that acts on a surface of genus g preserving the
orientation of the surface. Consequently, for any such g,
M(g) ≥ N(g) ≥ 24(g − 1). (6.2)
Using the approach (and notation) of Section 6, it is possible to show that there are no
large groups of automorphisms for most of these values of g. The analog of Lemma 5.1
holds; it is necessary to assume that p > (24)2 to apply Accola’s result. Then the analog
of Lemma 5.2 is easy to establish. The result corresponding to Lemma 5.3 holds with a
similar proof. There is an important difference here, though. The quotient space Y has
genus γ = 3. It is necessary, then, to consider the large group actions on Riemann surfaces
of genus 3. These actions have been classified; see [18, p. 4089] and [10, p. 285]. The
regular maps of genus 3 were classified by Sherk [19]; also see [7, Table 1]. There are 10
large group actions in all. Eight of these are map groups, but there are also two groups of
96 to consider, a FT (3, 3, 4) group and a HT (3; 4) group. The analogs of Lemmas 5.4
and 5.5 continue to hold, as does the following companion to Theorem 5.6.
146 Ars Math. Contemp. 22 (2022) #P1.09 / 135–147
Theorem 6.2. Let p be a prime such that p > (24)2. There are no large groups of auto-
morphisms that act on a surface of genus g = 2p+ 1.
It is necessary to consider the ten possibilities for the quotient group Q. In nine of the
cases, as in the proof of Theorem 5.6, an argument using the commutator quotient group
suffices; in these cases, Q/Q′ is isomorphic to 1, Z2, (Z2)2, or (Z2)3. The exceptional
case is the one in which G is a group of order 96p with partial presentation HT (3; 4). In
this case, Q/Q′ ∼= Z6; this case can be handled by considering the group G+. Then it
is not hard to show that there is no T (3, 3, 4) group of order 48p and hence no HT (3; 4)
group of order 96p.
Combining Theorems 6.1 and 6.2 gives the following.
Theorem 6.3. Let p be a prime such that p ≡ 1 (mod 6) and p > (24)2, and let
g = 2p+ 1. Then for any such g,
M(g) = N(g) = 24(g − 1). (6.3)
Finally, it is worth noting that there are genera in which N(g) = M(g) but the genus g
does not have either the form 2p + 1 or the form 3p + 1 for a prime p. Two examples are
genus 28 and genus 37.
ORCID iDs
Jay Zimmerman https://orcid.org/0000-0003-2749-0057
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P1.10 / 149–161
https://doi.org/10.26493/1855-3974.2329.97a
(Also available at http://amc-journal.eu)
Signed graphs with two eigenvalues and vertex
degree five*
Zoran Stanić
Faculty of Mathematics, University of Belgrade,
Studentski trg 16, 11 000 Belgrade, Serbia
Received 6 May 2020, accepted 13 July 2021, published online 17 May 2022
Abstract
It is known that a signed graph with exactly 2 eigenvalues must be regular, and all those
whose vertex degree does not exceed 4 are known. In this paper we characterize all signed
graphs with 2 eigenvalues and vertex degree 5. We also determine all signed graphs with 2
eigenvalues and 12 or 13 vertices, which is a natural step since those with a fewer number
of vertices are known.
Keywords: Regular signed graph, adjacency matrix, weighing matrix, bipartite double.
Math. Subj. Class. (2020): 05C22, 05C50
1 Introduction
A signed graph Ġ is a pair (G, σ), where G = (V,E) is a simple graph, called the underly-
ing graph, and σ : E −→ {1,−1} is the signature. The number of vertices of Ġ is denoted
by n. The edge set of Ġ is composed of subsets of positive and negative edges. Two ver-
tices are positive (resp. negative) neighbours if they are joined by a positive (resp. negative)
edge.
The adjacency matrix AĠ of Ġ is obtained from the adjacency matrix of its underlying
graph by reversing the sign of all 1s which correspond to negative edges. The eigenvalues
of Ġ are identified as the eigenvalues of AĠ, and they form the spectrum of Ġ.
We interpret a graph as a signed graph with all the edges being positive and, where
no confusion arises, we write ‘Ġ has k eigenvalues’ to mean that Ġ has exactly k distinct
eigenvalues.
Signed graphs with 2 eigenvalues have been investigated in [5, 7, 8, 11] and some re-
lated references. They are known to be regular, moreover every connected signed graph
*Research is partially supported by the Serbian Ministry of Education, Science and Technological Develop-
ment via the Faculty of Mathematics, University of Belgrade.
E-mail address: zstanic@math.rs (Zoran Stanić)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
150 Ars Math. Contemp. 22 (2022) #P1.10 / 149–161
with 2 eigenvalues is strongly regular in the sense of [10]. All signed graphs with 2 eigen-
values and (vertex) degree at most 4 are explicitly determined in [7, 11], and they can also
be deduced from the results reported in [8]. In particular, there is an infinite family of those
with degree 4. In [8] this family is determined in terms of cyclotomic matrices, and in
[11] the same family is determined in terms of signed line graphs. Signed graphs with 2
eigenvalues and at most 11 vertices are reported in [11]. Those whose least eigenvalue is
greater than −2 and those which are signed line graphs are also known and can be found
in the same reference. There are also some sporadic results related to other classes of
signed graphs with 2 eigenvalues [7, 10, 11]. Lastly, the Seidel matrix of a simple graph
G can be seen as the adjacency matrix of the complete signed graph whose negative edges
correspond to the edges of G. Accordingly, many results of [2, 3] concerning graphs with
exactly 2 eigenvalues of the Seidel matrix can be interpreted in the context of signed graphs.
Moreover, in this paper we use a similar approach.
Since all signed graphs with 2 eigenvalues and degree at most 4 are known, the next
natural step is to consider those with degree 5. In Section 3 we characterize all of them.
Moreover, we explicitly determine all except those that belong to some of the two particular
infinite families, which remain undetermined but well characterized by certain structural
properties.
In Section 4 we determine all signed graphs with 2 eigenvalues and 12 or 13 vertices,
which is an extension of the aforementioned result on those with at most 11 vertices.
We start with a preparatory section in which we give some terminology, notation and
known results. The paper is concluded by the Appendix that contains certain inequivalent
weighing matrices which are frequently used in this paper.
2 Preliminaries
We say that a signed graph is connected, regular or bipartite if the same holds for its un-
derlying graph. The negation −Ġ is obtained by reversing the sign of every edge of Ġ.
The degree of a vertex is equal to the number of edges incident with it. In particular, the
negative degree is the number of negative edges incident with it.
We say that signed graphs Ġ1 and Ġ2 are switching isomorphic if there is a monomial
(0, 1,−1)-matrix P such that AĠ2 = P
−1AĠ1P . (We recall that P
−1 = P ⊺.) In this case
we write Ġ1 ∼= Ġ2. Switching isomorphic signed graphs share the same spectrum and in
many considerations they are identified.
The product Ġ1 × Ġ2 of Ġ1 and Ġ2 is the signed graph with the vertex set
V (Ġ1) × V (Ġ2) in which two vertices (u1, u2) and (v1, v2) are adjacent if and only if
ui and vi are adjacent in Gi, for 1 ≤ i ≤ 2. The sign of an edge of the product is the prod-
uct of the signs of the corresponding edges of Ġ1 and Ġ2. The adjacency matrix AĠ1×Ġ2
is equal to the Kronecker product AĠ1 ⊗ AĠ2 . We recall that the Kronecker product is
not symmetric, but the resulting matrices are permutation equivalent, which means that the
corresponding signed graphs are switching isomorphic. In particular, Ġ × K2 is called
the bipartite double (or the bipartite double cover) of Ġ and denoted by bd(Ġ). The bi-
partite double is always bipartite, and it is connected if and only if Ġ is connected and
non-bipartite. If λ1, λ2, . . . , λn are the eigenvalues of Ġ (with possible repetitions), then
the eigenvalues of bd(Ġ) are ±λ1,±λ2, . . . ,±λn.
A weighing matrix N of order n and weight r is an n× n (0, 1,−1)-matrix satisfying
N⊺N = rI . We say that two rows of a weighing matrix intersect in k places if their non-
Z. Stanić: Signed graphs with two eigenvalues and vertex degree five 151
zero entries match in exactly k positions. Two weighing matrices M and N are said to be
equivalent if there are (0, 1,−1)-monomial matrices P,Q such that N = PMQ.
Results reported in the following two sections rely on the classification of weighing
matrices of weight 5 and weighing matrices of order 12 obtained by Harada and Mune-
masa [6]. The first classification is given in the following theorem, while the corresponding
matrices can be found in [4, 6]. To make the paper more self-contained we list them in the
Appendix. The notation is transferred from the mentioned references.
Theorem 2.1 ([6]). Any weighing matrix of weight 5 is equivalent to a matrix which is a
direct sum of an arbitrary number of matrices W (6, 5), W (8, 5), W12,5, W14,5, D(16, 5),
E(4ti + 2, 5), F (4tj + 4, 5), where ti, tj ≥ 2.
Every matrix of the previous theorem determines a bipartite signed graph with the ad-
jacency matrix (
O N⊺
N O
)
,
where N is the matrix in question. Throughout the paper we denote the corresponding
signed graphs by Ẇ (6, 5), Ẇ (8, 5), Ẇ12,5, Ẇ14,5, Ḋ(16, 5), Ė(4t + 2, 5), Ḟ (4t + 4, 5).
In other words, a dot indicates that we are dealing with the signed graph. The number of
vertices in each of these signed graphs is twice of the first parameter.
3 Connected signed graphs with 2 eigenvalues and degree 5
Here is the main result of this section.
Theorem 3.1. A connected signed graph with degree 5 has 2 eigenvalues if and only if it
is switching isomorphic to
(i) K6, −K6,
(ii) one of the bipartite signed graphs Ẇ (6, 5), Ẇ (8, 5), Ẇ12,5, Ẇ14,5, Ḋ(16, 5),
Ė(4t+ 2, 5), Ḟ (4t+ 4, 5),
(iii) one of the non-bipartite signed graphs illustrated in Figure 1 or a non-bipartite
signed graph Ġ such that either bd(Ġ) ∼= Ė(4t+ 2, 5) or bd(Ġ) ∼= Ḟ (4t+ 4, 5),
where, wherever it occurs, the parameter t satisfies t ≥ 2.
Every signed graph of (ii) or (iii) has the eigenvalues ±
√
5.
Evidently, a disconnected signed graph with 2 eigenvalues and degree 5 is a disjoint
union of connected ones. The proof of the previous theorem is based on the subsequent
lemmas. In the first one we consider the case in which the eigenvalues are asymmetric (i.e.,
they are not equal in absolute value).
Lemma 3.2. Every connected signed graph with degree 5 and 2 asymmetric eigenvalues
is switching isomorphic to K6 or −K6.
Proof. By considering the minimal polynomial we deduce that the eigenvalues must be
integral and the negation of their product is equal to the degree, i.e., 5. This gives just 2
possibilities which further produce the desired solutions.
152 Ars Math. Contemp. 22 (2022) #P1.10 / 149–161
(a) (b)
(c)
Figure 1: Signed graphs of Theorem 3.1(iii). In this and the forthcoming figures, negative
edges are dashed.
In what follows we assume that the eigenvalues are symmetric. In this case they are
±
√
5, so our task is to consider all signed graphs with these particular eigenvalues. We first
consider the bipartite case.
Lemma 3.3. Every connected bipartite signed graph with eigenvalues ±
√
5 is switching
isomorphic to either Ẇ (6, 5), Ẇ (8, 5), Ẇ12,5, Ẇ14,5, Ḋ(16, 5), Ė(4t+2, 5) or Ḟ (4t+4, 5)
(for t ≥ 2).
Proof. Let
AĠ =
(
O M⊺
M O
)
be the adjacency matrix of a signed graph under consideration. It follows that M is a
square matrix satisfying M⊺M = 5I , i.e., M is a weighing matrix of weight 5. Since Ġ
is connected, M is equivalent to one of the seven matrices listed in the formulation of this
lemma. If we denote this matrix by N , we have M = PNQ for some (0, 1,−1)-monomial
Z. Stanić: Signed graphs with two eigenvalues and vertex degree five 153
matrices P and Q, which implies
AĠ =
(
Q⊺ O
O P
)(
O N⊺
N O
)(
Q O
O P ⊺
)
=
(
Q O
O P ⊺
)−1 (
O N⊺
N O
)(
Q O
O P ⊺
)
,
and the result follows.
The essential part of the proof of Theorem 3.1 refers to the non-bipartite case. Observe
that every connected non-bipartite signed graph Ġ with eigenvalues ±
√
5 is extracted from
a decomposition of a bipartite one (with the same eigenvalues) having the form Ġ × K2,
i.e., which is a bipartite double of Ġ. We remark that not every bipartite signed graph is a
bipartite double, and a decomposition does not need to be unique (in the sense that it may
produce switching non-isomorphic signed graphs). We also have bd(Ġ) ∼= bd(−Ġ). In
what follows we consider possible decompositions of signed graphs of Lemma 3.3. For
this purpose we need the following simple but very useful result. It helps us to determine
the common neighbourhood of the particular vertices of Ġ.
Lemma 3.4. If W1 and W2 are the colour classes of bd(Ġ), then there are bijections
f1 : V (Ġ) −→ W1 and f2 : V (Ġ) −→ W2 such that u, v are adjacent in Ġ if and only if
f1(u), f2(v) and f2(u), f1(v) are adjacent in bd(Ġ).
Proof. The result follows by definition of the Kronecker product.
In the following two lemmas we consider Ẇ (6, 5) and Ẇ (8, 5). Despite the corre-
sponding results can be obtained by computer search, we give theoretical proofs. They
illustrate a technique which is used in the forthcoming considerations.
Lemma 3.5. bd(Ġ) ∼= Ẇ (6, 5) holds if and only if Ġ is switching isomorphic to the signed
graph illustrated in Figure 1(a).
Proof. Since every two rows of W (6, 5) intersect in 4 places, we conclude (by Lemma 3.4)
that Ġ is a complete signed graph with 6 vertices. Since its eigenvalues are ±
√
5, its adja-
cency matrix is equivalent to W (6, 5), and (up to switching) there is the unique possibility,
W (6, 5) itself, which leads us to the signed graph of Figure 1(a).
The opposite implication is immediate.
Considering the minimal polynomial, we get that if Ġ has the eigenvalues ±
√
5, then
for every pair of its vertices we have
w2(u, v) =
{
5 if u = v
0 if u ̸= v,
(3.1)
where w2(u, v) denotes the difference between the number of 2-walks between u and v
which traverse edges of the same sign and the number of 2-walks between the same vertices
which traverse edges of different sign.
We proceed with Ẇ (8, 5).
Lemma 3.6. bd(Ġ) ∼= Ẇ (8, 5) holds if and only if Ġ is switching isomorphic to the signed
graph illustrated in Figure 1(b).
154 Ars Math. Contemp. 22 (2022) #P1.10 / 149–161
Proof. Since the first (resp. last) four rows of W (8, 5) intersect each other in the first
(resp. last) 4 places, from Lemma 3.4, we conclude that the vertices of Ġ are partitioned
into two sets of equal size in such a way that each vertex of the first set is adjacent to each
vertex of the second. We claim that, up to switching, there is a unique distribution of the
edges between the vertices of the mentioned sets as given in Figure 1(b), where the vertices
of the first set are drawn in left part. Indeed, without loss of generality, we may assume
that Ġ contains a vertex whose negative degree is 0 (the top-left vertex in the figure), and
then every remaining vertex in its set has two positive and two negative neighbours in the
other set, which together with (3.1) gives the desired edge distribution. It remains to insert
the remaining 4 edges, 2 in each set. Again, on the basis of (3.1) we arrive at the unique
possibility, which gives Ġ.
The uniqueness (up to switching) follows by the way of construction, and the opposite
implication is immediate.
Observe that, according to (3.1), every pair of vertices of Ġ has an even number of
common neighbours. Consequently, every pair of rows of Abd(Ġ) intersects in an even
number of places. Since the degree is 5, they intersect in 0, 2 or 4 places. In what follows,
we consider one particular case. We say that a signed graph is triangle-free if it does not
contain a triangle as a subgraph.
Lemma 3.7. If every pair of rows of Abd(Ġ) intersects in either 0 or 2 places, then Ġ is
triangle-free.
Proof. Assume that Ġ is not triangle-free. Then, every edge of Ġ that belongs to some
triangle, in fact, belongs to exactly 2 triangles. Indeed, by (3.1) applied to the vertices
incident with such an edge, we conclude that it must belong to an even number of triangles
which, by Lemma 3.4 and the assumption of this lemma, makes this number equal to 2.
If so, then Ġ contains a tetrahedron, or an octahedron, or an icosahedron as an induced
subgraph. Moreover, the octahedron is eliminated immediately since it contains a pair
of vertices with 4 common neighbours, which would imply the existence of two rows in
Abd(Ġ) that intersect in 4 places.
First let Ġ contain a tetrahedron, and let its vertices be denoted by a, b, c, d. Assume that
the negative degree of a is 0. Under the assumption of this lemma, a, b have no common
neighbours outside the fixed tetrahedron, and thus w2(a, b) = 0 yields σ(bc) = −σ(bd).
Similarly, w2(a, c) = 0 gives σ(cb) = −σ(cd), which implies σ(bd) = σ(cd). But from
w2(a, d) = 0, we get σ(bd) = −σ(cd), which contradicts the previous equality.
If Ġ contains an icosahedron, then it also contains (as an induced subgraph) a pentagon
and an additional vertex adjacent to all the vertices of the pentagon. By assuming that the
negative degree of the additional vertex is 0, we arrive at a contradiction in a very similar
way as before.
We now eliminate Ẇ12,5 and Ẇ14,5 (as candidates for bd(Ġ)) and consider Ḋ(16, 5).
Lemma 3.8. There is no signed graph Ġ such that bd(Ġ) ∼= Ẇ12,5 or bd(Ġ) ∼= Ẇ14,5.
Proof. Assume by way of contradiction that either bd(Ġ) ∼= Ẇ12,5 or bd(Ġ) ∼= Ẇ14,5. In
both cases Abd(Ġ) satisfies the assumption of Lemma 3.7, and thus Ġ is triangle-free, and
of course the same holds for its underlying graph G.
Z. Stanić: Signed graphs with two eigenvalues and vertex degree five 155
The spectrum of the underlying graph of Ẇ12,5 is [±5, (±
√
5)6, (±1)5]. Since the non-
integral algebraic conjugates are equal in multiplicity, there is just one candidate for the
spectrum of G: [5, (±
√
5)3, (−1)5]. According to [12], there is exactly one graph with this
spectrum (known as the icosahedron), but it contains a triangle – a contradiction.
Similarly, the spectrum of the underlying graph of Ẇ14,5 is [±5,±3, (±(1 +
√
2))6,
(±(1 −
√
2))6]. There is just one candidate for the spectrum of G that passes the numer-
ical condition tr(AG) = 0: [5, 3, 1 ±
√
2, (−1 ±
√
2)5]. But since in this case we have
tr(A3G) ̸= 0, we deduce that G contains a triangle – a contradiction.
Lemma 3.9. bd(Ġ) ∼= Ḋ(16, 5) holds if and only if Ġ is switching isomorphic to the
signed graph illustrated in Figure 1(c).
Proof. Since Abd(Ġ) satisfies the assumption of Lemma 3.7, we get that G is triangle-free.
The spectrum of the underlying graph of Ḋ(16, 5) is [±5, (±3)5, (±1)10], which together
with tr(AG) = 0, tr(A3G) = 0, leads to the unique possibility for the spectrum of G:
[5, 110, (−3)5]. There is exactly one graph with this spectrum known as the Clebsch graph
(the underlying graph of the signed graph illustrated in the figure).
In what follows we consider the signature σ defined on E(G) which would produce the
desired Ġ. From this point we use a descriptive terminology and refer the reader to follow
the corresponding figure. As in the previous proofs, without loss of generality, we assume
that (at least) one vertex is not incident with negative edges; let this be the central vertex of
the figure, denoted here by a. Observe that (in the figure) a is surrounded by the 5 vertices
which form a pentagram. Denote the set of these vertices by P and the set of the remaining
vertices at distance 2 from a by Q. Now, we have the following:
• Every neighbour of a has 2 neighbours in P and exactly one of them is a positive
neighbour (otherwise, w2(a, b) ̸= 0 for at least one b ∈ P );
• Every neighbour of a has 2 neighbours in Q and exactly one of them is a positive
neighbour (for a similar reason);
• For b, c being neighbours of a such that b, c have a common neighbour d in P , we
have σ(bd) = −σ(cd). The same holds if d ∈ Q.
Using these conditions we arrive at a unique (up to switching) signature for the edges
incident with neighbours of a. Since w2(b, c) = 0 for b, c ∈ P , we get that the edges
between the vertices of P have the same sign. If they are positive (as in the figure), then
all the remaining (undecided) edges are negative (due to (3.1) applied to non-neighbours of
a), and we get the desired Ġ. If the mentioned edges are taken to be negative, we arrive at
a switching isomorphic signed graph.
The opposite implication is immediate.
We now consider the families of signed graphs illustrated in Figure 2. We first explain
their structure. The signed graph of Figure 2(a) has 4t + 2 (t ≥ 2) vertices, and the
vertices distinct from a, b are partitioned into the isomorphic blocks with 4 vertices. For
example, the vertices c, d, e, f belong to one block. There are no negative edges between
two blocks. The signed graph of Figure 2(b) has 4t + 4 (t ≥ 2) vertices, and is obtained
from the previous one by deleting the 4 edges between two blocks, then inserting two
vertices adjacent by a positive edge along with the edges between them and each of the 4
vertices of degree 3 of the corresponding blocks in such a way that the negative edges are
156 Ars Math. Contemp. 22 (2022) #P1.10 / 149–161
(a) (b)
aa
g
h
d
fe
c
bb
Figure 2: Signed graphs of Lemma 3.10.
just the two edges between one of the new vertices and the block vertices adjacent by a
positive edge. If (in the figure) a, b are transferred from the previous signed graph, then the
new vertices are g, h.
Lemma 3.10. If Ġ is switching isomorphic to the signed graph illustrated in Figure 2(a)
(resp. Figure 2(b)), then bd(Ġ) ∼= Ė(4t+ 2, 5) (resp. bd(Ġ) ∼= Ḟ (4t+ 4, 5)).
Proof. In both cases, Ġ is regular of degree 5 and w2(u, v) = 0 holds for every pair
of its distinct vertices, which is verified easily. Therefore, A2
Ġ
= 5I , which means that
the eigenvalues of Ġ are ±
√
5. Since, obviously, Ġ is non-bipartite, its bipartite double
is connected and has the same eigenvalues. In other words, bd(Ġ) is one of the bipartite
signed graphs listed in Lemma 3.3. Now, Ẇ (6, 5), Ẇ (8, 5) are eliminated since bd(Ġ) has
at least 20 vertices, Ẇ12,5, Ẇ14,5 are eliminated by Lemma 3.8, and Ḋ(16, 5) is eliminated
since, contrary to bd(Ġ), it does not contain a pair of vertices with 4 common neighbours.
Therefore, either bd(Ġ) ∼= Ė(4t+2, 5) or bd(Ġ) ∼= Ḟ (4t+4, 5). Comparing the numbers
of vertices, we deduce that the first possibility occurs precisely if Ġ is the first signed graph
of the figure.
We prove the main result of this section.
Proof of Theorem 3.1. If the eigenvalues are asymmetric, then Lemma 3.2 leads to (i). If
the eigenvalues are symmetric, from Lemma 3.3 we get (ii), while from Lemmas 3.5, 3.6,
3.8, 3.9 we get (iii).
Lemma 3.10 shows that the presence of Ė(4t+2, 5) and Ḟ (4t+4, 5) in the formulation
of Theorem 3.1(iii) is essential. This lemma leads to the question on how we get the signed
graphs of Figure 2, and the answer is simple: We arrive at them by following a simple
reasoning based on (3.1) and Lemma 3.4. To determine all the signed graphs that can be
extracted from decompositions of Ė(4t+ 2, 5) and Ḟ (4t+ 4, 5), one should proceed with
the similar reasoning, which in fact becomes complicated especially under the assumption
Z. Stanić: Signed graphs with two eigenvalues and vertex degree five 157
that there exist non-adjacent vertices with 4 common neighbours. An example is given in
the next section.
4 Signed graphs with 2 eigenvalues and 12 or 13 vertices
We have mentioned in Section 1 that signed graphs with 2 eigenvalues and at most 11 ver-
tices are known. Those with at most 10 vertices are obtained by computer search reported
in [11]. According to the same reference, there are exactly 2 such signed graphs with 11
vertices up to switching: K11 and its negation.
It occurs that, for n = 12, a computer search which considers all regular signed graphs
takes a long time, and so this task requires the application of more sophisticated methods. In
what follows we report the results obtained on the basis of a theoretical and computational
search on connected signed graphs with 2 eigenvalues and 12 vertices. Let Ġ denote such
a signed graph and let λ, µ be its eigenvalues. Recall that vertex degree of Ġ is r = −λµ.
Assume first that the eigenvalues λ, µ are asymmetric (and then they must be integral).
Considering r, we easily eliminate the possibility |λ|, |µ| ≥ 3. If |µ| = 1, we arrive at K12
and −K12. If µ = −2, by taking into account tr(AĠ) = 0 and tr(A2Ġ) = rn, we arrive
at the unique possible spectrum [44, (−2)8]. Now, since the least eigenvalue is not less
than −2, the corresponding signed graph is either the signed line graph (in the sense of a
definition given in [1, 11]) or the so-called exceptional signed graph. Moreover, it cannot be
exceptional since the multiplicity of −2 in every exceptional signed graph is n−6, or n−7,
or n− 8, as confirmed in [11]. On the other hand, all signed line graphs with 2 eigenvalues
are determined in the same reference, and accordingly Ġ is switching isomorphic to the
line graph of a signed multigraph obtained by inserting a negative edge between every pair
of vertices of the complete graph K4. For all details (including definition of the signed
line graph) we refer the reader to the corresponding reference, but we also illustrate Ġ in
Figure 3(a). Of course, case µ = 2 produces its negation, and we are done.
Assume now that λ, µ are symmetric. Since the case r ≤ 4 is resolved for every n,
we may also assume that r ≥ 5. For r = 5, we immediately get Ẇ (6, 5) and the signed
graph of Figure 2(b) (obtained for n = 12). We redraw this signed graph in Figure 3(b).
To conclude the list we consider other Ġ’s satisfying bd(Ġ) ∼= Ḟ (12, 5). This can be
performed either by hand or by a brute force, i.e., a computer search. In either way, we
arrive at another solution illustrated in Figure 3(c).
We proceed with r ≥ 6. According to [6], there are 8 inequivalent weighing matrices
of order 12 and weight 6, 3 of weight 7, 7 of weight 8, 4 of weight 9, 5 of weight 10 and
1 of weight 11. They determine all connected bipartite signed graphs with 24 vertices, 2
symmetric eigenvalues and degree r, where 6 ≤ r ≤ 11. Using the method exploited in the
previous section, we consider possible decompositions of each of them. In each case we
start from the given weighing matrix (the data on them can be found in [6]), then compute
the spectrum of the underlying graph of the corresponding bipartite signed graph, and then
compute the spectrum of a putative graph which can be extracted from it.
In this way we arrive at the data of Table 1. So, the 8 weighing matrices of weight 6 pro-
duce the underlying bipartite graphs with 4 distinct spectra (listed in the second column),
and each of these spectra gives spectra (listed in the third column) of the underlying graph
G of a putative signed graph Ġ. The remaining weights are considered in the same way,
and all the possible spectra of the third column satisfy tr(AG) = 0, tr(A2G) = 12r. Note
that in one case for weight 8, there is no spectrum that obeys these numerical conditions.
158 Ars Math. Contemp. 22 (2022) #P1.10 / 149–161
(a) (b)
(c) (d)
Figure 3: Signed graphs of Theorem 4.1.
The table omits the data for weight 11, since this case is resolved directly. Namely, if the
corresponding bipartite signed graph is a bipartite double of a signed graph Ġ, then Ġ is
the complete signed graph with 12 vertices and eigenvalues ±
√
11, but such a signed graph
does not exist, which can be confirmed by examining a list of Seidel’s strong graphs – an
appropriate reference is [9].
It remains to consider the spectra of the third column. (Observe that they are enumer-
ated by 1–15). First, there is no graph with spectrum 4, 5, 9, 11 or 14, which can easily
be confirmed by inspecting regular graphs with 12 vertices. There are 2 graphs with spec-
trum 7, and each contains pairs of vertices with 5 common neighbours which implies the
non-existence of Ġ in these cases. There is exactly 1 graph with spectrum 1, 10, 12, 13 or
15, and the non-existence of Ġ is confirmed either by hand (as these graphs have a high
level of symmetry along with a large number of vertices with the same neighbourhood) or
by computer search. There are exactly 4 graphs with spectrum 3, 1 graph with spectrum
6 and 1 graph with spectrum 8. The non-existence of Ġ is confirmed by computer search.
Finally, there is exactly 1 graph with spectrum 2. By computer search we find exactly 1
(up to switching) Ġ – it is illustrated in Figure 3(d); with consistent vertex labellings its
Z. Stanić: Signed graphs with two eigenvalues and vertex degree five 159
weight spectrum of G×K2 spectrum of G
6
±6, (±3.24)3, (±1.24)3, 010 1. 6, 1.243, 05, (−3.24)3
±6,±4, (±2)5, 010 2. 6, 4, 0
5(−2)5
3. 6, 22, 05, (−2)3,−4
±6,±3.86,±2.83, (±2)3,±1.04, 010 4. 6, 3.86, 0
5,−1.04, (−2)3,−2.83
5. 6, 2.83, 1.04, 05, (−2)3,−3.86
±6,±3.46, (±2)3, 012 6. 6, 3.46, 06, (−2)3,−3.46
7 ±7, (±3)
3, (±1)8 7. 7, 1
5, (−1)3, (−3)3
8. 7, 3, 12, (−1)6, (−3)2
±7, (±2.24)6, (±1)5 9. 7, 2.243,−2.243, (−1)5
8
±8, (±4)2, 018 10. 8, 09, (−4)2
±8,±3.46, (±2)2, 014 11. 8, 3.46, 07, (−2)2,−3.46
±8,±4, (±2)4, 012 12. 8, 2, 06, (−2)3,−4
±8,±4, (±2.83)2, 016
9 ±9, (±3)2, (±1)9 13. 9, 1
3, (−1)6, (−3)2
14. 9, 3, (−1)9,−3
10 ±10, (±2)5, 012 15. 10, 06, (−2)5
Table 1: Data for the search on connected non-bipartite signed graphs with 2 symmetric
eigenvalues and 12 vertices.
adjacency matrix is (
W (6, 5) I
I −W (6, 5)
)
.
The results are summarized in the following theorem.
Theorem 4.1. A connected signed graph with 12 vertices and degree at least 5 has 2
eigenvalues if and only if it is switching isomorphic to either K12, −K12, Ẇ (6, 5), one of
signed graphs illustrated in Figure 3 or the negation of the signed graph of Figure 3(a).
Finally, we quickly resolve the case n = 13.
Theorem 4.2. Every signed graph with 2 eigenvalues and 13 vertices is switching isomor-
phic to K13 or −K13.
Proof. Since the number of vertices is odd, the eigenvalues λ, µ must be asymmetric,
and then the proof is similar to the proof of Lemma 3.2 or to above consideration re-
lated to the asymmetric case for n = 12. Accordingly, for |µ| = 1 we arrive at the
desired signed graphs. Up to negation, the remaining possibilities are µ = −2, λ ∈
{3, 4, 5, 6} and (µ, λ) = (−3, 4), and both are eliminated by the conditions tr(AĠ) = 0,
tr(A3
Ġ
) = 13r.
ORCID iDs
Zoran Stanić https://orcid.org/0000-0002-4949-4203
160 Ars Math. Contemp. 22 (2022) #P1.10 / 149–161
References
[1] F. Belardo and S. K. Simić, On the Laplacian coefficients of signed graphs, Linear Algebra
Appl. 475 (2015), 94–113, doi:10.1016/j.laa.2015.02.007.
[2] A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, volume 18 of
Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1989, doi:
10.1007/978-3-642-74341-2.
[3] P. J. Cameron and J. H. van Lint, Designs, Graphs, Codes and Their Links, volume 22 of
London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1991,
doi:10.1017/cbo9780511623714.
[4] H. C. Chan, C. A. Rodger and J. Seberry, On inequivalent weighing matrices, Ars Comb. 21
(1986), 299–333.
[5] E. Ghasemian and G. H. Fath-Tabar, On signed graphs with two distinct eigenvalues, Filomat
31 (2017), 6393–6400, doi:10.2298/fil1720393g.
[6] M. Harada and A. Munemasa, On the classification of weighing matrices and self-orthogonal
codes, J. Comb. Des. 20 (2012), 40–57, doi:10.1002/jcd.20295.
[7] Y. Hou, Z. Tang and D. Wang, On signed graphs with just two distinct adjacency eigenvalues,
Discrete Math. 342 (2019), 111615, 8, doi:10.1016/j.disc.2019.111615.
[8] J. McKee and C. Smyth, Integer symmetric matrices having all their eigenvalues in the interval
[−2, 2], J. Algebra 317 (2007), 260–290, doi:10.1016/j.jalgebra.2007.05.019.
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10.1016/j.dam.2019.06.017.
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124627, 9, doi:10.1016/j.amc.2019.124627.
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(1998), 233–257, doi:10.1016/s0012-365x(98)00085-5.
Z. Stanić: Signed graphs with two eigenvalues and vertex degree five 161
A Appendix
Here we list the inequivalent weighing matrices of weight 5 mentioned in Theorem 2.1. The
first parameter denotes the order, and the second one denotes the weight. In the schematic
representations we write + for 1, − for −1 and dot for 0. For E(4t + 2, 5), F (4t + 4, 5)
we have t ≥ 2 and every unspecified entry is 0.

· +++++
+ · +−−+
++ · +−−
+−+ · +−
+−−+ · +
++−−+ ·

W (6,5)

+++++ · · ·
++−− · + · ·
+−−+ · · + ·
+−+− · · · +
+ · · · −−−−
· + · · −−++
· · + · −++−
· · · +−+−+

W (8,5)

+++++ · · · · · · ·
+− · · · +++ · · · ·
+ · − · · − · · ++ · ·
+ · · − · · − · − · + ·
+ · · · − · · − · −− ·
· +− · · · · + · − · +
· + · − · + · · · +− ·
· + · · − · + · · · +−
· · +− · −+ · · · · +
· · + · − · −++ · · ·
· · · +− · · · −+ · +
· · · · · + · −+ · ++

W12,5

+++++ · · · · · · · · ·
+− · · · +++ · · · · · ·
+ · − · · − · · ++ · · · ·
+ · · − · · − · − · + · · ·
+ · · · − · · − · −− · · ·
· +− · · + · · · · · ++ ·
· + · − · · · + · · − · − ·
· + · · − · + · · · +− · ·
· · +− · · · · + · · · +−
· · + · − · · · · + · + · +
· · · +− · −+ · · · · · −
· · · · · +− · + · · − · +
· · · · · + · − · + · · −−
· · · · · · · · +−++− ·

W14,5

+++++ · · · · · · · · · · ·
+− · · · +++ · · · · · · · ·
+ · − · · − · · ++ · · · · · ·
+ · · − · · − · − · + · · · · ·
+ · · · − · · − · −− · · · · ·
· +− · · + · · · · · ++ · · ·
· + · − · · + · · · · − · + · ·
· + · · − · · + · · · · −− · ·
· · +− · · · · + · · + · · + ·
· · + · − · · · · + · · + · − ·
· · · +− · · · · · + · · ++ ·
· · · · · +− · + · · − · · · +
· · · · · + · − · + · · − · · −
· · · · · · +− · · + · · − · +
· · · · · · · · +−+ · · · −−
· · · · · · · · · · · +−+−+

D(16,5)

+++++ ·
++−− · +
+− · · · · +++ ·
+− · · · · −− · +
+ · · · −− · · −−
· + · · −− · · ++
+− · ·
+− · ·
+ · −+
· +−+
+++ ·
−− · +
· · −−
· · ++
+− · · + · −+
+− · · · −+−
+ · −+−+ · ·
· +−++− · ·

E(4t+2,5)

+++++ ·
++−− · +
+− · · · · +++ ·
+− · · · · −− · +
+ · · · −− · · −−
· + · · −− · · ++
+− · ·
+− · ·
+ · −+
· +−+
+++ ·
−− · +
· · −−
· · ++
+− · · +− · · + ·
+− · · −+ · · · +
+ · −+ · · · · −−
· +−+ · · · · ++
+ · −+−+
· +−++−

F (4t+4,5)

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Maribor Graph Theory Conference (MGTC 2022)
Maribor, Slovenia, 11–15 September 2022
https://conferences.matheo.si/event/37/
The conference aims to bring together researchers from all areas of graph theory and related
topics with an emphasis on the topics studied by Maribor graph theorists. These topics are
largely influenced by the work of Sandi Klavžar whose 60th birthday will be celebrated
during the event. In addition, we will devote one of the sessions to Wilfried Imrich, an
honorary doctor of the University of Maribor.
Keynote speakers:
• László Babai (University of Chicago, USA)
• Marthe Bonamy (Université de Bordeaux, France)
• Michael Henning (University of Johannesburg, South Africa)
• Kolja Knauer (Universitat de Barcelona, Spain)
• Douglas Rall (Furman University, USA)
• Mariusz Woźniak (AGH University of Science and Technology, Poland)
• Xuding Zhu (Zhejiang Normal University, China)
Special invited speakers:
• Paul Dorbec (Université de Caen Normandy, France)
• Ismael G. Yero (Universidad de Cadiz, Spain)
• Andreas M. Hinz (LMU München, Germany)
• Martyn Mulder (Erasmus University Rotterdam, Netherlands)
• Balázs Patkós (Alfréd Rényi Institute of Mathematics, Hungary)
Contributed talks are welcome!
Venue: University of Maribor, Faculty of Natural Sciences and Mathematics. The faculty
is located on the edge of the city of Maribor not far from the Drava river. The city center is
just a half an hour walk from the conference venue.
Organized by: University of Maribor, Faculty of Natural Sciences and Mathematics in
collaboration with the Institute of Mathematics, Physics and Mechanics, and NM Klub.
Program and organizing committee:
Boštjan Brešar (chair), Tanja Dravec, Marko Jakovac
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