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  • On existence of Two Classes of Generalized Howell Designs with Block Size Three and Index Two
    Graphs Comb. (IF 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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
  • On the Connectivity and Independence Number of Power Graphs of Groups
    Graphs Comb. (IF 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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 0.488) 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
  • Containment: A Variation of Cops and Robber
    Graphs Comb. (IF 0.488) Pub Date : 2020-02-19
    Danny Crytser, Natasha Komarov, John Mackey

    We consider “Containment”: a variation of the graph pursuit game of Cops and Robber in which cops move from edge to adjacent edge, the robber moves from vertex to adjacent vertex (but cannot move along an edge occupied by a cop), and the cops win by “containing” the robber—that is, by occupying all \(\deg (v)\) of the edges incident with a vertex v while the robber is at v. We develop bounds that relate

    更新日期:2020-02-19
  • The Average Size of Matchings in Graphs
    Graphs Comb. (IF 0.488) Pub Date : 2020-02-08
    Eric O. D. Andriantiana, Valisoa Razanajatovo Misanantenaina, Stephan Wagner

    In this paper, we consider the average size of independent edge sets, also called matchings, in a graph. We characterize the extremal graphs for the average size of matchings in general graphs and trees. In addition, we obtain inequalities between the average size of matchings and the number of matchings as well as the matching energy, which is defined as the sum of the absolute values of the zeros

    更新日期:2020-02-08
  • Degree Lists and Connectedness are 3-Reconstructible for Graphs with At Least Seven Vertices
    Graphs Comb. (IF 0.488) Pub Date : 2020-02-01
    Alexandr V. Kostochka, Mina Nahvi, Douglas B. West, Dara Zirlin

    The k-deck of a graph is the multiset of its subgraphs induced by k vertices. A graph or graph property is l-reconstructible if it is determined by the deck of subgraphs obtained by deleting l vertices. We show that the degree list of an n-vertex graph is 3-reconstructible when \(n\ge 7\), and the threshold on n is sharp. Using this result, we show that when \(n\ge 7\) the \((n-3)\)-deck also determines

    更新日期:2020-02-01
  • Several Extreme Coefficients of the Tutte Polynomial of Graphs
    Graphs Comb. (IF 0.488) Pub Date : 2020-01-10
    Helin Gong, Xian’an Jin, Mengchen Li

    Let \(t_{i,j}\) be the coefficient of \(x^iy^j\) in the Tutte polynomial T(G; x, y) of a connected bridgeless and loopless graph G with order v and size e. It is trivial that \(t_{0,e-v+1}=1\) and \(t_{v-1,0}=1\). In this paper, we obtain expressions for another six extreme coefficients \(t_{i,j}\)’s with \((i,j)=(0,e-v)\),\((0,e-v-1)\),\((v-2,0)\),\((v-3,0)\),\((1,e-v)\) and \((v-2,1)\) in terms of

    更新日期:2020-01-10
  • Forbidding Multiple Copies of Forestable Graphs
    Graphs Comb. (IF 0.488) Pub Date : 2020-01-09
    Neal Bushaw, Nathan Kettle

    The Turán number of a graph H is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. We call a graph Hforestable if it is cyclic, bipartite, and contains a vertex v such that \(H[V\setminus v]\) is a forest. For a forestable graph H, we determine \({\text {ex}}(n,k\cdot H)\) exactly as a function of \({\text {ex}}(n,H)\). This is related to earlier work of

    更新日期:2020-01-09
  • Disjoint Paired-Dominating sets in Cubic Graphs.
    Graphs Comb. Pub Date : null
    Gábor Bacsó,Csilla Bujtás,Casey Tompkins,Zsolt Tuza

    A paired-dominating set of a graph G is a dominating set D with the additional requirement that the induced subgraph G[D] contains a perfect matching. We prove that the vertex set of every claw-free cubic graph can be partitioned into two paired-dominating sets.

    更新日期:2019-11-01
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