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  • Laplacian Controllability for Graphs Obtained by Some Standard Products
    Graphs Comb. (IF 0.597) Pub Date : 2020-08-08
    Milica Anđelić, Maurizio Brunetti, Zoran Stanić

    Let \(L_G\) be the Laplacian matrix of a graph G with n vertices, and let \({\mathbf {b}}\) be a binary vector of length n. The pair \((L_G, {\mathbf {b}})\) is said to be controllable (and we also say that G is Laplacian controllable for \({\mathbf {b}}\)) if \(L_G\) has no eigenvector orthogonal to \({\mathbf {b}}\). In this paper we study the Laplacian controllability of joins, Cartesian products

    更新日期:2020-08-09
  • Two-Coloring Triples such that in Each Color Class Every Element is Missed at Least Once
    Graphs Comb. (IF 0.597) Pub Date : 2020-08-06
    Balázs Keszegh

    We give a characterization of finite sets of triples of elements (e.g., positive integers) that can be colored with two colors such that for every element i in each color class there exists a triple which does not contain i. We give a linear (in the number of triples) time algorithm to decide if such a coloring exists and find one if it does. We also consider generalizations of this result and an application

    更新日期:2020-08-06
  • $$P_{3}$$ P 3 -Factors in the Square of a Tree
    Graphs Comb. (IF 0.597) Pub Date : 2020-08-04
    Guowei Dai, Zhiquan Hu

    A spanning subgraph H of a graph G is a \(P_{3}\)-factor ofG if every component of H is a path of order three. The square of a graph G, denoted by \(G^{2}\), is the graph with vertex set V(G) such that two vertices are adjacent in \(G^{2}\) if and only if their distance in G is at most 2. A graph G is subcubic if it has maximum degree at most three. In this paper, we give a sharp necessary condition

    更新日期:2020-08-04
  • An Infinite Family of Sum-Paint Critical Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-08-01
    Thomas Mahoney, Chad Wiley

    Independently, Zhu and Schauz introduced online list coloring in 2009. In each round, a set of vertices allowed to receive a particular color is marked, and the coloring algorithm (Painter) gives a new color to an independent subset of these vertices. A graph G is f-paintable for a function \(f:V(G)\rightarrow \mathbb {N}\) if Painter can produce a proper coloring when the number of times each vertex

    更新日期:2020-08-01
  • Sub-Ramsey Numbers for Matchings
    Graphs Comb. (IF 0.597) Pub Date : 2020-07-24
    Fangfang Wu, Shenggui Zhang, Binlong Li

    Given a graph G and a positive integer k, the sub-Ramsey number sr(G, k) is defined to be the minimum number m such that every \(K_{m}\) whose edges are colored using every color at most k times contains a subgraph isomorphic to G all of whose edges have distinct colors. In this paper, we will concentrate on \(sr(nK_{2},k)\) with \(nK_{2}\) denoting a matching of size n. We first give upper and lower

    更新日期:2020-07-24
  • Group Divisible Designs with Block Size Four and Type $$g^u b^1 (gu/2)^1$$ g u b 1 ( g u / 2 ) 1
    Graphs Comb. (IF 0.597) Pub Date : 2020-07-23
    Anthony D. Forbes

    We discuss group divisible designs with block size four and type \(g^u b^1 (gu/2)^1\), where \(u = 5\), 6 and 7. For integers a and b, we prove the following. (i) A 4-GDD of type \((4a)^5 b^1 (10a)^1\) exists if and only if \(a \ge 1\), \(b \equiv a\) (mod 3) and \(4a \le b \le 10a\). (ii) A 4-GDD of type \((6a+3)^6 b^1 (18a+9)^1\) exists if and only if \(a \ge 0\), \(b \equiv 3\) (mod 6) and \(6a+3

    更新日期:2020-07-23
  • Discrete Geometry on Red and Blue Points in the Plane—A survey
    Graphs Comb. (IF 0.597) Pub Date : 2020-07-16
    Mikio Kano, Jorge Urrutia

    Discrete geometry on colored point sets in the plane has a long history, but this research area has been extensively developed in the last 2 decades. In 2003, a short survey entitled “Discrete geometry on red and blue points in the plane: A survey” was published. Since then, many new and important results have been published, and thus the need of a new and up-to-date survey is evident.

    更新日期:2020-07-17
  • Properties of a q -Analogue of Zero Forcing
    Graphs Comb. (IF 0.597) Pub Date : 2020-07-13
    Steve Butler, Craig Erickson, Shaun Fallat, H. Tracy Hall, Brenda Kroschel, Jephian C.-H. Lin, Bryan Shader, Nathan Warnberg, Boting Yang

    Zero forcing is a combinatorial game played on a graph where the goal is to start with all vertices unfilled and to change them to filled at minimal cost. In the original variation of the game there were two options. Namely, to fill any one single vertex at the cost of a single token; or if any currently filled vertex has a unique non-filled neighbor, then the neighbor is filled for free. This paper

    更新日期:2020-07-13
  • Separability of Schur Rings Over Abelian Groups of Odd Order
    Graphs Comb. (IF 0.597) Pub Date : 2020-07-09
    Grigory Ryabov

    An S-ring (a Schur ring) is said to be separable with respect to a class of groups \({\mathcal {K}}\) if every algebraic isomorphism from the S-ring in question to an S-ring over a group from \({\mathcal {K}}\) is induced by a combinatorial isomorphism. A finite group G is said to be separable with respect to \({\mathcal {K}}\) if every S-ring over G is separable with respect to \({\mathcal {K}}\)

    更新日期:2020-07-10
  • Extremal Graphs for Blow-Ups of Keyrings
    Graphs Comb. (IF 0.597) Pub Date : 2020-07-01
    Zhenyu Ni, Liying Kang, Erfang Shan, Hui Zhu

    The blow-up of a graph H is the graph obtained from replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different. Given a graph H and a positive integer n, the extremal number, ex(n, H), is the maximum number of edges in a graph on n vertices that does not contain H as a subgraph. A keyring \(C_s(k)\) is a \((k+s)\)-edge graph obtained from a cycle of

    更新日期:2020-07-01
  • Circular Zero-Sum r -Flows of Regular Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-04-11
    Saieed Akbari, Amir Hossein Ghodrati, Mohammad Ali Nematollahi

    A circular zero-sum flow for a graph G is a function \(f:E(G) \rightarrow {\mathbb {R}}{\setminus }\{0\}\) such that for every vertex v, \(\sum _{e\in E_v}f(e)=0\), where \(E_v\) is the set of all edges incident with v. If for each edge e, \(1\le |f(e)| \le r-1\), where \(r\ge 2\) is a real number, then f is called a circular zero-sum r-flow. Also, if r is a positive integer and for each edge e, f(e)

    更新日期:2020-06-19
  • p -Adic Roots of Chromatic Polynomials
    Graphs Comb. (IF 0.597) Pub Date : 2020-04-27
    Paul Buckingham

    The complex roots of the chromatic polynomial \(P_{G}(x)\) of a graph G have been well studied, but the p-adic roots have received no attention as yet. We consider these roots, specifically the roots in the ring \(\mathbb{Z}_p\) of p-adic integers. We first describe how the existence of p-adic roots is related to the p-divisibility of the number of colourings of a graph—colourings by at most k colours

    更新日期:2020-06-19
  • Packing and Covering Directed Triangles
    Graphs Comb. (IF 0.597) Pub Date : 2020-04-11
    Jessica McDonald, Gregory J. Puleo, Craig Tennenhouse

    We prove that if a directed multigraph D has at most t pairwise arc-disjoint directed triangles, then there exists a set of less than 2t arcs in D which meets all directed triangles in D, except in the trivial case \(t=0\). This answers affirmatively a question of Tuza from 1990.

    更新日期:2020-06-19
  • Degree Sequence Conditions for Maximally Edge-Connected and Super Edge-Connected Hypergraphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-04-11
    Shuang Zhao, Yingzhi Tian, Jixiang Meng

    Let H be a connected hypergraph with minimum degree \(\delta\) and edge-connectivity \(\lambda\). The hypergraph H is maximally edge-connected if \(\lambda = \delta\), and it is super edge-connected or super-\(\lambda\), if every minimum edge-cut consists of edges incident with some vertex. There are several degree sequence conditions, for example, Goldsmith and White (Discrete Math 23: 31–36, 1978)

    更新日期:2020-06-19
  • Harmonic Index t -Designs in the Hamming Scheme for Arbitrary q
    Graphs Comb. (IF 0.597) Pub Date : 2020-04-11
    Makoto Tagami, Ryota Hori

    The notion of T-design in a symmetric association scheme was introduced by Delsarte. A harmonic index t-design is the particular case in which the index set T consists of a single index \(\{t\}\). Zhu et al. studied a harmonic index t-design in the binary Hamming scheme and gave a Fisher type lower bound on the cardinality. Also they defined the notion of a tight harmonic index design using the bound

    更新日期:2020-06-19
  • Affirmative Solutions on Local Antimagic Chromatic Number
    Graphs Comb. (IF 0.597) Pub Date : 2020-06-16
    Gee-Choon Lau, Ho-Kuen Ng, Wai-Chee Shiu

    An edge labeling of a connected graph \(G = (V, E)\) is said to be local antimagic if it is a bijection \(f:E \rightarrow \{1,\ldots ,|E|\}\) such that for any pair of adjacent vertices x and y, \(f^+(x)\not = f^+(y)\), where the induced vertex label \(f^+(x)= \sum f(e)\), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by \(\chi _{la}(G)\), is the

    更新日期:2020-06-16
  • Multi-color Forcing in Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-06-13
    Chassidy Bozeman, Pamela E. Harris, Neel Jain, Ben Young, Teresa Yu

    Let \(G=(V,E)\) be a finite connected graph along with a coloring of the vertices of G using the colors in a given set X. In this paper, we introduce multi-color forcing, a generalization of zero-forcing on graphs, and give conditions in which the multi-color forcing process terminates regardless of the number of colors used. We give an upper bound on the number of steps required to terminate a forcing

    更新日期:2020-06-13
  • Shortest Directed Networks in the Plane
    Graphs Comb. (IF 0.597) Pub Date : 2020-06-12
    Alastair Maxwell, Konrad J. Swanepoel

    Given a set of sources and a set of sinks as points in the Euclidean plane, a directed network is a directed graph drawn in the plane with a directed path from each source to each sink. Such a network may contain nodes other than the given sources and sinks, called Steiner points. We characterize the local structure of the Steiner points in all shortest-length directed networks in the Euclidean plane

    更新日期:2020-06-12
  • Optimal Domination Polynomials
    Graphs Comb. (IF 0.597) Pub Date : 2020-06-12
    Iain Beaton, Jason I. Brown, Danielle Cox

    Let G be a graph on n vertices and m edges and D(G, x) the domination polynomial of G. In this paper we completely characterize the values of n and m for which optimal graphs exist for domination polynomials. We also show that there does not always exist least optimal graphs for the domination polynomial. Applications to network reliability are highlighted.

    更新日期:2020-06-12
  • Partitioning Claw-Free Subcubic Graphs into Two Dominating Sets
    Graphs Comb. (IF 0.597) Pub Date : 2020-06-08
    Qing Cui

    A dominating set in a graph G is a set \(S\subseteq V(G)\) such that every vertex in \(V(G){\setminus } S\) has at least one neighbor in S. Let G be an arbitrary claw-free graph containing only vertices of degree two or three. In this paper, we prove that the vertex set of G can be partitioned into two dominating sets \(V_1\) and \(V_2\) such that for \(i=1,2\), the subgraph of G induced by \(V_i\)

    更新日期:2020-06-08
  • On Vertex-Disjoint Triangles in Tripartite Graphs and Multigraphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-06-06
    Qingsong Zou, Jiawang Li, Zizheng Ji

    Let G be a tripartite graph with tripartition \((V_{1},V_{2},V_{3})\), where \(\mid V_{1}\mid =\mid V_{2}\mid =\mid V_{3}\mid =k>0\). It is proved that if \(d(x)+d(y)\ge 3k\) for every pair of nonadjacent vertices \(x\in V_{i}, y\in V_{j}\) with \(i\ne j(i,j\in \{1,2,3\})\), then G contains k vertex-disjoint triangles. As a corollary, if \(d(x)\ge \frac{3}{2}k\) for each vertex \(x\in V(G)\), then

    更新日期:2020-06-06
  • Forbidden Subgraphs for a Graph to Have a Hamiltonian Path Square
    Graphs Comb. (IF 0.597) Pub Date : 2020-06-05
    Xiaojing Yang, Liming Xiong

    The square of a graph is obtained by adding an edge between each pair of vertices that are of distance 2 in the original graph. If a graph G has a hamiltonian path (or cycle) square, then the square of the hamiltonian path (or cycle) is called a hamiltonian path (or cycle) square of G. Chen and Shan (Graphs Comb 31:2113–2124, 2015) characterized all forbidden pairs for a 4-connected graph to have a

    更新日期:2020-06-05
  • On existence of Two Classes of Generalized Howell Designs with Block Size Three and Index Two
    Graphs Comb. (IF 0.597) Pub Date : 2020-06-05
    Jing Shi, Jinhua Wang

    Let \(t, k, {\lambda }, s\) and v be nonnegative integers, and let X be a set of v symbols. A generalized Howell design, denoted t-GHD\(_k (s, v; {\lambda })\), is an \(s \times s\) array, each cell of which is either empty or contains a k-set of symbols from X, called a block, such that: (i) each symbol appears exactly once in each row and in each column (i.e. each row and column is a resolution of

    更新日期:2020-06-05
  • A Note on Combinatorial Proofs for Extended Touchard’s and Extended Koshy’s Identities
    Graphs Comb. (IF 0.597) Pub Date : 2020-06-04
    Roberta R. Zhou, Kun Yan, Yihui He, Xinxin Xu

    Touchard’s and Koshy’s identities are beautiful identities about Catalan numbers. It is worth noting that combinatorial interpretations for extended Touchard’s identity and extended Koshy’s identity can intuitively reflect the equations. In this paper, we give a new combinatorial proof for the extended Touchard’s identity by means of Dyck Paths. The principle of inclusion–exclusion (sieve method) is

    更新日期:2020-06-04
  • Commuting Involution Graphs for 4-Dimensional Projective Symplectic Groups
    Graphs Comb. (IF 0.597) Pub Date : 2020-06-04
    Alistaire Everett, Peter Rowley

    For a group G and X a subset of G the commuting graph of G on X, denoted by \(\mathcal {C}(G,X)\), is the graph whose vertex set is X with \(x,y\in X\) joined by an edge if \(x\ne y\) and x and y commute. If the elements in X are involutions, then \(\mathcal {C}(G,X)\) is called a commuting involution graph. This paper studies \(\mathcal {C}(G,X)\) when G is a 4-dimensional projective symplectic group

    更新日期:2020-06-04
  • Neighbor Sum Distinguishing Total Choosability of Cubic Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-06-02
    Donghan Zhang, You Lu, Shenggui Zhang

    Let \(G=(V, E)\) be a graph and \({\mathbb {R}}\) be the set of real numbers. For a k-list total assignment L of G that assigns to each member \(z\in V\cup E\) a set \(L_{z}\) of k real numbers, a neighbor sum distinguishing (NSD) total L-coloring of G is a mapping \(\phi :V\cup E \rightarrow {\mathbb {R}}\) such that every member \(z\in V\cup E\) receives a color of \(L_z\), every pair of adjacent

    更新日期:2020-06-02
  • Gallai–Ramsey Numbers for a Class of Graphs with Five Vertices
    Graphs Comb. (IF 0.597) Pub Date : 2020-06-02
    Xihe Li, Ligong Wang

    Given two graphs G and H, the k-colored Gallai–Ramsey number \(gr_k(G : H)\) is defined to be the minimum integer n such that every k-coloring of the complete graph on n vertices contains either a rainbow copy of G or a monochromatic copy of H. In this paper, we consider \(gr_k(K_3 : H)\), where H is a connected graph with five vertices and at most six edges. There are in total thirteen graphs in this

    更新日期:2020-06-02
  • Bivariate Order Polynomials
    Graphs Comb. (IF 0.597) Pub Date : 2020-06-02
    Matthias Beck, Maryam Farahmand, Gina Karunaratne, Sandra Zuniga Ruiz

    Motivated by Dohmen–Pönitz–Tittmann’s bivariate chromatic polynomial \(\chi _G(x,y)\), which counts all x-colorings of a graph G such that adjacent vertices get different colors if they are \(\le y\), we introduce a bivarate version of Stanley’s order polynomial, which counts order preserving maps from a given poset to a chain. Our results include decomposition formulas in terms of linear extensions

    更新日期:2020-06-02
  • Reconfiguring 10-Colourings of Planar Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-30
    Carl Feghali

    Let \(k \ge 1\) be an integer. The reconfiguration graph \(R_k(G)\) of the k-colourings of a graph G has as vertex set the set of all possible k-colourings of G and two colourings are adjacent if they differ on exactly one vertex. A conjecture of Cereceda from 2007 asserts that for every integer \(\ell \ge k + 2\) and k-degenerate graph G on n vertices, \(R_{\ell }(G)\) has diameter \(O(n^2)\). The

    更新日期:2020-05-30
  • On Cycles that Alternate Through Selected Sets of Vertices
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-30
    Colton Magnant, Yaping Mao

    Given two disjoint sets of k vertices each in a graph G, it was recently asked whether \(\delta (G) \ge n/2\) and \(\kappa (G) \ge 2k\) would be a sufficient assumption to guarantee the existence of a Hamiltonian cycle of G which alternates visiting vertices of the two selected sets. We answer this question in the affirmative when the graph is large and we also provide a generalization to more sets

    更新日期:2020-05-30
  • Which Graphs Occur as $$\gamma $$γ -Graphs?
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-29
    Matt DeVos, Adam Dyck, Jonathan Jedwab, Samuel Simon

    The \(\gamma \)-graph of a graph G is the graph whose vertices are labelled by the minimum dominating sets of G, in which two vertices are adjacent when their corresponding minimum dominating sets (each of size \(\gamma (G)\)) intersect in a set of size \(\gamma (G)-1\). We extend the notion of a \(\gamma \)-graph from distance-1-domination to distance-d-domination, and ask which graphs H occur as

    更新日期:2020-05-29
  • Reducing Vizing’s 2-Factor Conjecture to Meredith Extension of Critical Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-27
    Xiaodong Chen, Qing Ji, Mingda Liu

    A simple graph G is called \(\varDelta\)-critical if \(\chi '(G) =\varDelta (G) +1\) and \(\chi '(H) \le \varDelta (G)\) for every proper subgraph H of G, where \(\varDelta (G)\) and \(\chi '(G)\) are the maximum degree and the chromatic index of G, respectively. Vizing in 1965 conjectured that any \(\varDelta\)-critical graph contains a 2-factor, which is commonly referred to as Vizing’s 2-factor

    更新日期:2020-05-27
  • Chromatic Number, Induced Cycles, and Non-separating Cycles
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-27
    Hanbaek Lyu

    We study two parameters obtained from the Euler characteristic by replacing the number of faces with that of induced and induced non-separating cycles. By establishing monotonicity of such parameters under certain homomorphism and edge contraction, we obtain new upper bounds on the chromatic number in terms of the number of induced cycles and the Hadwiger number in terms of the number of induced non-separating

    更新日期:2020-05-27
  • A Relation Between Schröder Paths and Motzkin Paths
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-24
    Lin Yang, Sheng-Liang Yang

    A small q-Schröder path of semilength n is a lattice path from (0, 0) to (2n, 0) using up steps \(U = (1, 1)\), horizontal steps \(H = (2, 0)\), and down steps \(D = (1,-1)\) such that it stays weakly above the x-axis, has no horizontal steps on the x-axis, and each horizontal step comes in q colors. In this paper, we provide a bijection between the set of small q-Schröder paths of semilength \(n+1\)

    更新日期:2020-05-24
  • Decomposing Degenerate Graphs into Locally Irregular Subgraphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-23
    Julien Bensmail, François Dross, Nicolas Nisse

    A (undirected) graph is locally irregular if no two of its adjacent vertices have the same degree. A decomposition of a graph G into k locally irregular subgraphs is a partition \(E_1,\dots ,E_k\) of E(G) into k parts each of which induces a locally irregular subgraph. Not all graphs decompose into locally irregular subgraphs; however, it was conjectured that, whenever a graph does, it should admit

    更新日期:2020-05-23
  • Maximum Cuts in $$\mathscr {H}$$H -Free Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-23
    Huawen Ma

    For a graph G, let f(G) be the maximum number of edges in a cut of G. For a positive integer m and a set of graphs \(\mathscr {H}\), let \(f(m,\mathscr {H})\) be the minimum possible cardinality of f(G), as G ranges over all graphs on m edges which contain no graphs in \(\mathscr {H}\). Suppose that \(r\ge 8\) is an even integer and \(\mathscr {H}=\{C_{5},C_{6},\ldots ,C_{r-1}\}\). In this paper, we

    更新日期:2020-05-23
  • Burning the Plane
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-22
    Anthony Bonato, Karen Gunderson, Amy Shaw

    Graph burning is a discrete-time process on graphs, where vertices are sequentially burned, and burned vertices cause their neighbours to burn over time. We consider extremal properties of this process in the new setting where the underlying graph is also changing at each time-step. The main focus is on the possible densities of burning vertices when the sequence of underlying graphs are growing grids

    更新日期:2020-05-22
  • On Mixed Graphs Whose Hermitian Spectral Radii are at Most 2
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-19
    Bo-Jun Yuan, Yi Wang, Shi-Cai Gong, Yun Qiao

    A mixed graph is a graph with undirected and directed edges. Guo and Mohar in 2017 determined all mixed graphs whose Hermitian spectral radii are less than 2. In this paper, we give a sufficient condition which can make Hermitian spectral radius of a connected mixed graph strictly decreasing when an edge or a vertex is deleted, and characterize all mixed graphs with Hermitian spectral radii at most

    更新日期:2020-05-19
  • Consecutive Detecting Arrays for Interaction Faults
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-18
    Ce Shi, Ling Jiang, Aiyuan Tao

    The concept of detecting arrays was developed to locate and detect interaction faults arising between the factors in a component-based system during software testing. In this paper, we propose a family of consecutive detecting arrays (CDAs) in which the interactions between factors are considered to be ordered. CDAs can be used to generate test suites for locating and detecting interaction faults between

    更新日期:2020-05-18
  • Stirling Pairs of Permutations
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-15
    Richard A. Brualdi

    Stirling permutations are permutations \(\pi \) of the multiset \(\{1,1,2,2,\ldots ,n,n\}\) in which those integers between the two occurrences of an integer are greater than it. We identify a permutation \(\pi \) of \(\{1,1,2,2,\ldots ,n,n\}\) as a pair of permutations \((\pi _1,\pi _2)\) which we call a Stirling pair. We characterize Stirling pairs using the weak Bruhat order and the notion of a

    更新日期:2020-05-15
  • Richardson’s theorem in quasi-transitive and pre-transitive digraphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-14
    Hortensia Galeana-Sánchez, Rocío Sánchez-López

    A subset N of V(D) is said to be a kernel if it satisfies the following two properties: (1) for any two different vertices x and y in N there is no arc between them, and (2) for each vertex u in V(D)\(\setminus N\) there exists v in N such that (u,v) \(\in\) A(D). If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. In Galeana-Sánchez and Rojas-Monroy (Discrete Math

    更新日期:2020-05-14
  • Comments on “On Integral Graphs Which Belong to the Class $$\overline{\alpha K_{a,b}}$$αKa,b¯ ” [Graphs Comb. 19, 527–532 (2003)]
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-11
    Milan Pokorný, Dragan Stevanović

    Lepović gave a characterization of integral graphs of the form \(\overline{\alpha K_{a,b}}\) in “On Integral Graphs Which Belong to the Class \(\overline{\alpha K_{a,b}}\)” [Graphs Comb. 19, 527–532 (2003)]. However, for majority of the graphs \(\overline{\alpha K_{a,b}}\) declared to be integral in this result, their spectrum also includes non-integer eigenvalues \(-1\pm \sqrt{ab}\). Here we correct

    更新日期:2020-05-11
  • A Generalization of Some Results on List Coloring and DP-Coloring
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-09
    Keaitsuda Maneeruk Nakprasit, Kittikorn Nakprasit

    Let G be a graph and let \(f_i, i \in \{1,\ldots ,s\},\) be a function from V(G) to the set of nonnegative integers. In Sittitrai and Nakprasit (Analogue of DP-coloring on variable degeneracy and its applications, 2020), the concept of DP-F-coloring, a generalization of DP-coloring and variable degeneracy, was introduced. We use DP-F-coloring to define DPG-[k, t]-colorable graphs and modify the proofs

    更新日期:2020-05-09
  • Repetition of Spectral Radiuses Among Connected Induced Subgraphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-05
    Alberto Seeger

    A connected graph G is spectrally non-redundant if the spectral radiuses of the connected induced subgraphs of G are all different. As observed by Fernandes, Júdice, and Trevisan (2017), for such graphs the number of connected induced subgraphs is equal to the number of complementarity eigenvalues. This note is an attempt at quantifying the so-called spectral redundancy phenomenon. Such a phenomenon

    更新日期:2020-05-05
  • Gallai–Ramsey Numbers for Rainbow Paths
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-02
    Xihe Li, Pierre Besse, Colton Magnant, Ligong Wang, Noah Watts

    Given graphs G and H and a positive integer k, the Gallai–Ramsey number, denoted by \(gr_{k}(G : H)\) is defined to be the minimum integer n such that every coloring of \(K_{n}\) using at most k colors will contain either a rainbow copy of G or a monochromatic copy of H. We consider this question in the cases where \(G \in \{P_{4}, P_{5}\}\). In the case where \(G = P_{4}\), we completely solve the

    更新日期:2020-05-02
  • Unimodality of Principal Eigenvector and its Applications
    Graphs Comb. (IF 0.597) Pub Date : 2020-05-02
    Jie Xue, Ruifang Liu, Jinlong Shu

    In this paper, we consider the unimodality of the principal eigenvector of graphs. A unimodal lemma of the principal eigenvector on internal paths is obtained. This unimodal lemma is used to establish a cycle version of Li–Feng transformation with respect to the spectral radius. Another application of unimodal lemma is to determine the unicyclic graph with minimal spectral radius.

    更新日期:2020-05-02
  • Flag-Transitive Block Designs and Finite Simple Exceptional Groups of Lie Type
    Graphs Comb. (IF 0.597) Pub Date : 2020-04-09
    Seyed Hassan Alavi

    In this article, we study 2-designs with \(\gcd (r,\lambda )=1\) admitting a flag-transitive almost simple automorphism group with socle a finite simple exceptional group of Lie type. We obtain four infinite families of such designs and provide some examples in each of these families.

    更新日期:2020-04-09
  • Regular and Vertex-Transitive Kähler Graphs Having Commutative Principal and Auxiliary Adjacency Operators
    Graphs Comb. (IF 0.597) Pub Date : 2020-04-06
    Toshiaki Adachi, Guanyuan Chen

    A Kähler graph is a compound of two graphs having a common set of vertices and is a discrete model of a Riemannian manifold equipped with magnetic fields. In this paper we study selfadjointness of adjacency operators of Kähler graphs and express their zeta functions in terms of eigenvalues of their principal and auxiliary adjacency operators when they are commutative. Also, we construct finite vertex-transitive

    更新日期:2020-04-06
  • Hamiltonian Cycle Properties in k -Extendable Non-bipartite Graphs with High Connectivity
    Graphs Comb. (IF 0.597) Pub Date : 2020-04-03
    Zhiyong Gan, Dingjun Lou, Yanping Xu

    Let G be a graph, \(\nu \) the order of G and k a positive integer such that \(k\le (\nu -2)/2\). Then G is said to be k-extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G. A graph G is Hamiltonian if it contains a Hamiltonian cycle. A graph G is Hamiltonian-connected if, for any two of its vertices, it contains a spanning path joining the two

    更新日期:2020-04-03
  • On Near Polygons All Whose Hexes are Dual Polar Spaces
    Graphs Comb. (IF 0.597) Pub Date : 2020-04-02
    Bart De Bruyn

    One of the most fundamental results in the theory of regular near polygons is the result that every regular near 2d-gon, \(d \ge 3\), whose parameters \(s,t,t_i\), \(i \in \{ 0,1,\ldots ,d \}\), satisfy \(s,t_2 \ge 2\) and \(t_3=t_2^2+t_2\) is a dual polar space. The proof of that theorem heavily relies on Tits’ theory of buildings, in particular on Tits’ strong results on covering of chamber systems

    更新日期:2020-04-02
  • On the Connectivity and Independence Number of Power Graphs of Groups
    Graphs Comb. (IF 0.597) Pub Date : 2020-03-31
    Peter J. Cameron, Sayyed Heidar Jafari

    Let G be a group. The power graph of G is a graph with vertex set G in which two distinct elements x, y are adjacent if one of them is a power of the other. We characterize all groups whose power graphs have finite independence number, show that they have clique cover number equal to their independence number, and calculate this number. The proper power graph is the induced subgraph of the power graph

    更新日期:2020-03-31
  • Every 4-Equivalenced Association Scheme is Frobenius
    Graphs Comb. (IF 0.597) Pub Date : 2020-03-28
    Bora Moon

    For a positive integer k, we say that an association scheme \((\varOmega ,S)\) is k-equivalenced if each non-diagonal element of S has valency k. An association scheme \((\varOmega ,S)\) is called Frobenius when the set S is equal to the set of orbitals of a Frobenius group G on a finite set \(\varOmega \). It is known that every k-equivalenced association scheme is Frobenius when k=2, 3. In this paper

    更新日期:2020-03-28
  • Partially Broken Orientations of Eulerian Plane Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-03-27
    Gen Kawatani, Yusuke Suzuki

    It is well-known that every Eulerian plane graph G is face 2-colorable and admits an orientation which is an assignment of a direction to each edge of G such that incoming edges and outgoing edges appear alternately around any \(v \in V(G)\); we say that such a vertex v has the alternate property, and that such an orientation is good. In this paper, we discuss orientations given to Eulerian plane graphs

    更新日期:2020-03-27
  • Exact Values of Multicolor Ramsey Numbers $$R_l(C_{\le l+1})$$Rl(C≤l+1)
    Graphs Comb. (IF 0.597) Pub Date : 2020-03-19
    Weiguo Zhu, Yongqi Sun, Yali Wu, Hanshuo Zhang

    Let \(ex(n, C_{\le m})\) denote the maximum size of a graph of order n and girth at least \(m+1\), and \(EX(n, C_{\le m})\) be the set of all graphs of girth at least \(m+1\) and size \(ex(n, C_{\le m})\). The Ramsey number \(R_l(C_{\le m})\) is the smallest n such that every \(K_n\), whose edges are in l colors, must contain a monochromatic cycle of length k for some \(3\le k\le m\). In this paper

    更新日期:2020-03-19
  • Isolation of Cycles
    Graphs Comb. (IF 0.597) Pub Date : 2020-03-12
    Peter Borg

    For any graph G, let \(\iota _{\mathrm{c}}(G)\) denote the size of a smallest set D of vertices of G such that the graph obtained from G by deleting the closed neighbourhood of D contains no cycle. We prove that if G is a connected n-vertex graph that is not a triangle, then \(\iota _{\mathrm{c}}(G) \le n/4\). We also show that the bound is sharp. Consequently, this settles a problem of Caro and Hansberg

    更新日期:2020-03-12
  • 2-Edge-Colored Chromatic Number of Grids is at Most 9
    Graphs Comb. (IF 0.597) Pub Date : 2020-03-11
    Janusz Dybizbański

    A 2-edge-colored graph is a pair \((G, \sigma )\) where G is a graph, and \(\sigma :E(G)\rightarrow \{\text {'}+\text {'},\text {'}-\text {'}\}\) is a function which marks all edges with signs. A 2-edge-colored coloring of the 2-edge-colored graph \((G, \sigma )\) is a homomorphism into a 2-edge-colored graph \((H, \delta )\). The 2-edge-colored chromatic number of the 2-edge-colored graph \((G, \sigma

    更新日期:2020-03-11
  • Extremal Graphs for Odd-Ballooning of Paths and Cycles
    Graphs Comb. (IF 0.597) Pub Date : 2020-03-09
    Hui Zhu, Liying Kang, Erfang Shan

    The odd-ballooning of a graph F is the graph obtained from F by replacing each edge in F by an odd cycle of length between 3 and \(q\ (q\ge 3)\) where the new vertices of the odd cycles are all different. Given a forbidden graph H and a positive integer n, the extremal number, ex(n, H), is the maximum number of edges in a graph on n vertices that does not contain H as a subgraph. Erdös et al. and Hou

    更新日期:2020-03-09
  • The Chromatic Number of Graphs with No Induced Subdivision of $$K_4$$K4
    Graphs Comb. (IF 0.597) Pub Date : 2020-02-27
    Guantao Chen, Yuan Chen, Qing Cui, Xing Feng, Qinghai Liu

    In 2012, Lévêque, Maffray and Trotignon conjectured that if a graph does not contain an induced subdivision of \(K_4\), then it is 4-colorable. Recently, Le showed that every such graph is 24-colorable. In this paper, we improve the upper bound to 8.

    更新日期:2020-02-27
  • Some Multi-Color Ramsey Numbers on Stars versus Path, Cycle or Wheel
    Graphs Comb. (IF 0.597) Pub Date : 2020-02-22
    Longqin Wang

    For given simple graphs \(H_1,H_2,\ldots ,H_t\), the Ramsey number \(R(H_1,H_2,\ldots ,H_t)\), which is often called multi-color Ramsey number, is the smallest integer n such that for an arbitrary decomposition \(\{G_i\}_{i=1}^t\) of the complete graph \(K_n\), there is at least one \(G_i\) has a subgraph isomorphic to \(H_i\). Let \(m,n_1,n_2,\ldots , n_t\) be positive integers and \(\Sigma =\sum

    更新日期:2020-02-22
  • An Explicit Formula Enumerating the Finely Bounded Trees Using Young Tableaux
    Graphs Comb. (IF 0.597) Pub Date : 2020-02-20
    Ayaka Ishikawa

    Most of the tree enumeration formulas are generating functions or recurrence formulas. In this article, we show the explicit formula for the number of unlabeled rooted trees with a certain condition. The formula is described in terms of Young tableaux.

    更新日期:2020-02-20
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