A quadrangulation on a surface 𝔽 is a fixed embedding of a loopless graph on 𝔽 such that every face is bounded by a closed walk of length 4. Define a “2-vertex deletion" and a “hexagonal contraction" as reductions to transform a quadrangulation G with a vertex of degree at most 3 to a smaller quadrangulation G' so that G' is a minor of G. In this paper, we determine the set of minimal quadrangulations on the torus having vertices of degree at most 3 with respect to those reductions. Moreover, for 4-regular quadrangulations on the torus, we determine minimal ones with respect to the minor relation, and we also do so for simple 4-regular quadrangulations.
A tree t-spanner T of a graph G is a spanning tree with the property that the distance between any pair of nodes in T is at most t times the distance in G. If an edge e in T fails (i.e., is removed from G and T), the tree breaks into two subtrees T_+ and T_- . Let E_X denote the set of edges in G that reconnect T_+ and T_- . Every edge f∈ E_X is a potential swap edge for e that can be used to repair the tree spanner. An edge in E_X that has the largest stretch among all edges in E_X when f is used to reconnect T_+ and T_- is called a critical edge. In this paper, we show that, for every edge e that fails in a tree spanner of an unweighted graph, there is always a set of at most four edges that contains at least one critical edge for every potential swap edge of e. The proof relies on studying the endpoints of a diametrical path in a tree with changing edge weights and may be of independent interest. We also show that there are instances where the smallest critical set has size four.
The first objective of this paper is to characterize all possible parameters of Plotkin-optimal two-homogeneous weight regular projective codes over finite chain rings, as well as their weight distributions. We show the existence of codes with these parameters by constructing an infinite family of two-homogeneous weight codes. The parameters of their Gray images have the same weight distribution as that of the two-weight codes of type SU1 in the sense of Calderbank and Kantor (Bull. Lond. Math. Soc., 18 (1986) 97-122). Further, we also construct three-homogeneous weight regular projective codes over finite chain rings combining with some known results. Finally, we study applications of our constructed codes in secret sharing schemes and graph theory. In particular, infinite families of strongly regular graphs and strongly walk-regular graphs with non-trivial parameters are obtained.
Let G be a graph. For x∈ A and y∈ B , we define d_A(x) and d_B(y) as the vertices’ indegrees. Similarly, we define d_A(y) and d_B(x) as the outdegrees. A partition (A, B) of V(G) is an internal partition of G if ∀ x∈ A , d_A(x)≥ d_B(x) and ∀ y∈ B , d_B(y)≥ d_A(y) . An internal bisection (A, B) of V(G) is an internal partition with | A| =| B| . DeVos (2009) conjectures that for any integer d, every d-regular graph of sufficiently large order n has an internal partition. We proved that there exists a 5-regular graph of order 12 with no internal bisection. We also proved that every k-regular graph with minimum cut less than k has an internal partition where k is an odd integer.
We consider the (up) signless Laplace operator on a simplicial complex X based on the signless differential, which was introduced recently by Kaufman and Oppenheim (2020). We bound the largest eigenvalue of this operator in terms of various combinatorial parameters of the simplicial complex X such as the maximum degree, minimum degree, and average degree of simplexes in X, the dimension and number of simplexes in X, and the diameter of X. Our results also yield algebraic characterizations for the regularity of simplicial complexes.
Let Γ be a graph with vertex set V(Γ ) . A subset C of V(Γ ) is a perfect code of Γ if C is an independent set in Γ such that every vertex in V(Γ )∖ C is adjacent to exactly one vertex in C. A subset T of V(Γ ) is a total perfect code of Γ if every vertex of Γ is adjacent to exactly one vertex in T. Let G be a group with identity element e. The intersection graph of G, denoted by Γ (G) , is the graph whose vertex set consists of all nontrivial proper subgroups of G, and two distinct vertices H and K are adjacent if and only if H∩ K{e} . In this paper, we establish necessary and sufficient conditions for the intersection graphs of finite abelian groups, generalized quaternion groups, and modular groups to have perfect codes and total perfect codes. We characterize dihedral groups and quasi-dihedral groups whose intersection graphs have perfect codes, and prove that the intersection graphs of dihedral groups and quasi-dihedral groups have no total perfect code. Furthermore, we explicitly provide some of the existing perfect codes and total perfect codes in the intersection graphs mentioned above.
A graph is said to be neighborhood 3-balanced if there exists a vertex labeling with three colors so that each vertex has an equal number of neighbors of each color. We give order constraints on 3-balanced graphs, determine which generalized Petersen and Pappus graphs are 3-balanced, discuss when being 3-balanced is preserved under various graph constructions, give two general characterizations of cubic 3-balanced graphs, and classify cubic 3-balanced graphs of small order.
The present paper forms part of a continuing series in which the author develops an evolutionary geometric theory of the sporadic simple groups. The project was initiated in [12], where Chapter 10 prepared a number of groups and their associated geometries for further detailed study. The geometry associated with the smallest Conway group, Co_3 , was characterised in [13].
A signed graph is a pair of a graph and a mapping from the edge set to {+1,-1} . In 1982, Zaslavsky introduced the notion of a proper coloring of signed graphs as a natural generalization of a proper coloring of unsigned graphs. An odd coloring of a graph is a proper coloring of a graph such that every non-isolated vertex has a color that appears at an odd number of neighbors. This notion was introduced by Petruševski and Škrekovski in 2022, and has been actively studied. As a common generalization of these two concepts, in this paper, we introduce the notion of odd coloring of signed graphs. As an analogy of the Heawood’s map-color problem, for signed graphs embedded in a closed surface, we show that (1) for every closed surface S, every signed graph embedded in S is odd 2H(S)-colorable, and that (2) for every closed surface other than the Klein bottle, there is a signed graph with the odd chromatic number 2H(S)-1 that can be embedded in S, where H(S) denotes the Heawood number of S.
The b-fold indicated L-coloring game on G is played by two players: Ann and Ben, where G is a graph and L is a list assignment of G. In each round, Ann chooses an uncolored vertex v, and Ben colors v with a b-set ϕ (v) from L(v) such that none of the colors in ϕ (v) have been used by its colored neighbors. If all vertices are colored, Ann wins the game. Otherwise, after some rounds, there is an uncolored vertex v with less than b available colors (i.e., colors in its list not used by its colored neighbors), and Ben wins the game. We say G is indicated (L, b)-colorable if Ann has a winning strategy for the b-fold indicated L-coloring game. For a mapping g: V(G) →ℕ , we say G is indicated (g, b)-choosable if G is indicated (L, b)-colorable for every list assignment L of G with |L(v)|≥ g(v) for each vertex v. If g(v)=a for every vertex v, then indicated (g, b)-choosable is called indicated (a, b)-choosable. The indicated choice number ch_i(G) is the least integer k such that G is indicated (k, 1)-choosable (also called indicated k-choosable). The fractional indicated choice number of G is ch_i^f(G)=inf{a/b:G is indicated (a,b)-choosable} . This paper proves that for any finite graph G, ch_i^f(G)=ch_i(G) ; a connected graph G is indicated 2-choosable if and only if its core is K_1 or Θ _2,2,2 or an even cycle; for m ≥ 2 , a graph G is indicated (2m, m)-choosable if and only if G is a tree. A graph G is called indicated k-choosable-critical if G is not indicated (k-1) -choosable, but any proper subgraph is indicated (k-1) -choosable. We give a characterization of indicated 3-choosable critical graphs.
For a graph G, let γ (G) and core(G) denote the cardinality of a minimum dominating set of G, and the intersection of all the minimum dominating sets of G, respectively. In this paper, we prove that if G is a 2K_2 -free graph with γ (G)≥ 3 and without isolated vertices, then v∈ core(G) if and only if γ (G-v)> γ (G); moreover, we give an example to answer an open question proposed by Samodivkin whether there is a connected graph G such that core(G)∅ and γ (G-v)=γ (G) for each vertex v of G. We also prove that for every {claw,Z_2} -free graph G without isolated vertices, v∈ core(G) if and only if γ (G-v)> γ (G) .
The Gilbert graph 𝒢_q,n,d , which arises naturally in graph theory and coding theory, is the regular graph on 𝔽_q^n in which two vertices are adjacent if their Hamming distance is less than d, and it is vertex-transitive. We classify all parameters (q, n, d) for which 𝒢_q,n,d is edge-transitive or distance-transitive, and separately classify all parameters for which its complement has these properties. We prove that 𝒢_q,n,d is edge-transitive if and only if it is distance-transitive, and that this occurs precisely when d=2 , (q,d)=(2,3) , or (q,d)=(2,n) . For the complement graphs, we determine all parameters yielding edge- or distance-transitivity using spectral methods based on Krawtchouk polynomials and the structure of the Hamming association scheme. In contrast to the Gilbert graphs, where the parameter sets corresponding to edge- and distance-transitivity coincide, we show that for their complements the set of parameters yielding distance-transitivity is strictly contained in the set yielding edge-transitivity. As an application, we compute the exact values of the Lovász ϑ -function of Gilbert graphs, as well as of their complements, in all cases where either one of them is edge-transitive.
We show that some well-known integer sequences can be represented as sequences of determinant of matrices associated with certain families of threshold graphs. Specifically, let G_n denote a connected threshold graph with n vertices. Define S(G_n):=(s_ij) as the n × n matrix, where s_ii=0 for all i; s_ij=1 if vertices i and j are not adjacent, and -1 if i and j are adjacent. We show that if {μ _j-1}_j ≥ 1 is the sequence of Pell numbers, then μ _j-1=| S(A_j)| , where A_j is the connected antiregular graph on j vertices. We then find a sequence of connected threshold graphs {G_m} such that the sequence of Fibonacci numbers aligns with {| S(G_m)|} . Additionally, we determine a recurrence relation for the sequence { D(A_k)} , where D(A_k) is the distance matrix of the connected antiregular threshold graph with k vertices. Using this relation, we give an explicit formula to compute D(A_k) .
We identify a structural pattern in the construction of known infinite families of trees whose independence polynomials are not log-concave. Using this pattern and properties of polynomial ring ideals, we derive linear recurrences for these polynomials. As a consequence, we prove that the set of non-isolated limit points of their zeros lies on the circle |z+1/3|=1/3 in the complex plane. Building on these recurrences, we also exhibit infinite families of trees whose independence polynomials break log-concavity at one, two, and three consecutive indices, as well as finite families that break log-concavity at four and five consecutive indices. Our approach suggests that arbitrarily many consecutive breaks may be achievable, offering further insight into a question posed by Galvin [D. Galvin, Trees with non log-concave independent set sequences, arXiv:2502.10654v1, 2025].
Mixed covering arrays are generalizations of orthogonal arrays. A set of t vectors {x_1,x_2,… ,x_t} with x_i∈ℤ_g_i^N, 1⩽ i⩽ t, is said to be t-qualitatively independent if for every t-tuple (a_1,a_2,… ,a_t) ∈ℤ_g_1×ℤ_g_2×…×ℤ_g_t, there exists an integer 1⩽ r⩽ N such that (x_1[r],x_2[r],… ,x_t[r])=(a_1,a_2,… ,a_t) . Let H=(V(H),E(H)) be a weighted hypergraph with V(H)={v_1,v_2,… ,v_k} and weights w(v_i)=g_i, 1⩽ i ⩽ k . A mixed covering array on H, denoted by MCA (N;H, ∏ _i=1^kg_i), is an N× k array such that column i corresponds to vertex v_i∈ V(H) with weight g_i ; the entries in column i are from ℤ_g_i ; if e={v_1,v_2,… ,v_t}∈ E(H), the columns correspond to vertices v_1, v_2, … , v_t are t-qualitatively independent. In this paper, we introduce some basic hypergraph operations. As their applications, we provide constructions for optimal mixed covering arrays on some 3-uniform or 4-uniform hypergraphs.
For a simple undirected graph G with complement G , the central graph C(G) is constructed by adding a path of length two edges between all pairs of non-adjacent vertices in G . In this work, utilizing the notion of central graphs, we provide a novel classification scheme for all simple undirected graphs based upon the existence of a minimum cardinality vertex cover that concomitantly serves as a dominating set in the complement. In particular, letting γ (H) be the domination number and τ (H) be the vertex cover number for an arbitrary graph H, we show that either γ (C(G))=τ (G) or γ (C(G))=τ (G)+1 . Here, with one well-characterized set of exceptions, γ (C(G))=τ (G) holds if and only if some vertex cover for G is a dominating set for G . In addition, we explicitly characterize the domination number of the central graph for a variety of graph classes, and show that it is NP-hard both to compute γ (C(G)) and to decide whether γ (C(G))=τ (G) or γ (C(G))=τ (G)+1 . Finally, we establish that it is NP-hard to decide if some open neighborhood in a graph is a minimum cardinality vertex cover, and discuss the implications of this result for a generalization of the art gallery visibility problem.
Let G=(V,E,c) be an edge-colored graph, where c:E→ℕ is an edge coloring of G. Let d_G^c(v) be the color degree of v, which is the number of distinct colors on incident edges of v. Let δ ^c(G)=min{d_G^c(v) | v∈ V(G)} be the minimum color degree of G (with respect to c). Let s, t be two integers with s≥ t≥ 2 and G be an edge-colored graph with δ ^c(G)≥ s+t+1 . Fujita, Li and Wang in 2019 conjectured that G admits a partition (S, T) such that δ ^c(G[S])≥ s and δ ^c(G[T])≥ t . Here G[U] denotes the subgraph of G induced by the vertex set U. Let E^i={e∈ E(G) | c(e)=i} be the color class for color i, let G^i=(V(G),E^i) be the corresponding subgraph. A bipartite graph K_1,3 is called a claw, and a graph is claw-free if it does not contain a claw as an induced subgraph. We say an edge-colored graph G is a (monochromatic claw)-free graph if for each i∈ℕ , G^i is claw-free. In this paper, we first show that a (monochromatic claw)-free graph G admits a partition (S, T) such that δ ^c(G[S])≥ s and δ ^c(G[T])≥ t if δ ^c(G)≥ 2s+t+1 . A monochromatic path of length 2 is referred to as a monochromatic 2-path. Let G be an edge-colored graph in which no end-vertex of a monochromatic 2-path lies on another monochromatic 2-path of a different color. Then, we show that such an edge-colored graph G admits a partition (S, T) such that δ ^c(G[S])≥ s and δ ^c(G[T])≥ t if δ ^c(G)≥ s+t+1 .
The (b, c)-Motzkin paths are paths that start from the origin and end on the x-axis, not going below the x-axis, using up steps U=(1,1) , level steps L=(1,0) and down steps D=(1,-1) where each level step L can be one of b possible colors and each down step D be one of c possible colors. The free (b, c)-Motzkin paths are defined similarly but with no restriction of staying above the x-axis. The generalized central trinomial coefficient T_n(b,c) counts the number of free (b, c)-Motzkin paths from (0, 0) to (n, 0), the generalized sub-central trinomial coefficient T_n,1(b,c) counts the number of partial free (b, c)-Motzkin paths from (0, 0) to (n, 1), and the generalized Motzkin number M_n(b,c) counts the number of (b, c)-Motzkin paths from (0, 0) to (n, 0). The numbers M_n(b,c) , T_n(b,c) , T_n(b,c)+ T_n,1(b,c) , and T_n(b,c)- T_n,1(b,c) are collectively referred to as the generalized Motzkin family. In particular, when b=c=1 we have the classical Motzkin family defined by Barcucci, Pinzani and Sprugnoli [4]. Using Riordan arrays, we investigate their mutual relationships and connections to the generalized trinomial coefficients.
The notion of word-representable graphs is a generalization of comparability graphs, in which graphs are represented by words. The complexity of word-representation of a word-representable graph is captured through representation number, whereas the corresponding concept is the permutation-representation number for comparability graphs. The graphs with the (permutation-)representation number at most two were characterized in the literature. While certain examples in the class of graphs with the (permutation-)representation number three are known, no characterization for these classes are available. In this work, we prove that the representation number of melon graphs is at most three. Further, we characterize the class of melon graphs restricted to comparability graphs and show that their permutation-representation number is also at most three. Moreover, this work characterizes the word-representable line graphs of melon graphs and establish that their representation number is at most three.
