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  • The Signature of Chordal Graphs and Cographs
    Graphs Comb. (IF 0.597) Pub Date : 2021-01-21
    Changxiang He, Shujuan Qian, Haiying Shan, Baofeng Wu

    The signature s(G) of a graph G is defined as the difference between its positive inertia index and negative inertia index. In 2013, Ma et al. conjectured that \(-c_3(G)\le s(G) \le c_5(G)\) for an arbitrary simple graph G, where \(c_3(G)\) denotes the number of cycles in G of length 3 modulo 4, \(c_5(G)\) denotes the number of cycles in G of length 1 modulo 4. In this paper, we give some inequalities

    更新日期:2021-01-21
  • Art Gallery Problem with Rook and Queen Vision
    Graphs Comb. (IF 0.597) Pub Date : 2021-01-21
    Hannah Alpert, Érika Roldán

    How many chess rooks or queens does it take to guard all squares of a given polyomino, the union of square tiles from a square grid? This question is a version of the art gallery problem in which the guards can “see” whichever squares the rook or queen attacks. We show that \(\lfloor {\frac{n}{2}} \rfloor \) rooks or \(\lfloor {\frac{n}{3}} \rfloor \) queens are sufficient and sometimes necessary to

    更新日期:2021-01-21
  • On Generalisations of the AVD Conjecture to Digraphs
    Graphs Comb. (IF 0.597) Pub Date : 2021-01-20
    Julien Bensmail, Fionn Mc Inerney

    Given an undirected graph, in the AVD (edge-colouring) Conjecture, the goal is to find a proper edge-colouring with the least number of colours such that every two adjacent vertices are incident to different sets of colours. More precisely, the conjecture says that, a few exceptions apart, every graph G should admit such an edge-colouring with at most \(\Delta (G)+2\) colours. Several aspects of interest

    更新日期:2021-01-20
  • Wiener Indices of Maximal k -Degenerate Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2021-01-09
    Allan Bickle, Zhongyuan Che

    A graph is maximal k-degenerate if each induced subgraph has a vertex of degree at most k and adding any new edge to the graph violates this condition. In this paper, we provide sharp lower and upper bounds on Wiener indices of maximal k-degenerate graphs of order \(n \ge k \ge 1\). A graph is chordal if every induced cycle in the graph is a triangle and chordal maximal k-degenerate graphs of order

    更新日期:2021-01-10
  • Cut-Edges and Regular Factors in Regular Graphs of Odd Degree
    Graphs Comb. (IF 0.597) Pub Date : 2020-10-30
    Alexandr V. Kostochka, André Raspaud, Bjarne Toft, Douglas B. West, Dara Zirlin

    We study 2k-factors in \((2r+1)\)-regular graphs. Hanson, Loten, and Toft proved that every \((2r+1)\)-regular graph with at most 2r cut-edges has a 2-factor. We generalize their result by proving for \(k\le (2r+1)/3\) that every \((2r+1)\)-regular graph with at most \(2r-3(k-1)\) cut-edges has a 2k-factor. Both the restriction on k and the restriction on the number of cut-edges are sharp. We characterize

    更新日期:2021-01-07
  • Fractional Matchings, Component-Factors and Edge-Chromatic Critical Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2021-01-04
    Antje Klopp, Eckhard Steffen

    The first part of the paper studies star-cycle factors of graphs. It characterizes star-cycle factors of a graph G and proves upper bounds for the minimum number of \(K_{1,2}\)-components in a \(\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}\)-factor of a graph G. Furthermore, it shows where these components are located with respect to the Gallai–Edmonds decomposition of G and it characterizes the edges which are

    更新日期:2021-01-05
  • A Size Condition for Diameter Two Orientable Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2021-01-03
    Garner Cochran, Éva Czabarka, Peter Dankelmann, László Székely

    It was conjectured by Koh and Tay [Graphs Combin. 18(4) (2002), 745–756] that for \(n\ge 5\) every simple graph of order n and size at least \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) -n+5\) has an orientation of diameter two. We prove this conjecture and hence determine for every \(n\ge 5\) the minimum value of m such that every graph of order n and size m has an orientation of diameter two

    更新日期:2021-01-03
  • On the Connectivity of Enhanced Power Graphs of Finite Groups
    Graphs Comb. (IF 0.597) Pub Date : 2021-01-03
    Sudip Bera, Hiranya Kishore Dey, Sajal Kumar Mukherjee

    This paper deals with the vertex connectivity of enhanced power graphs of finite groups. We classify all abelian groups G such that the vertex connectivity of enhanced power graph of G is 1. We derive an upper bound for the vertex connectivity of the enhanced power graph of any general abelian group G. Also we completely characterize all abelian groups G, such that the proper enhanced power graph is

    更新日期:2021-01-03
  • Measuring Similarity Between Connected Graphs: The Role of Induced Subgraphs and Complementarity Eigenvalues
    Graphs Comb. (IF 0.597) Pub Date : 2021-01-02
    Alberto Seeger, David Sossa

    This work elaborates on the old problem of measuring the degree of similarity, say \({\mathfrak {f}}(G,H)\), between a pair of connected graphs G and H, not necessarily of the same order. The choice of a similarity index \({\mathfrak {f}}\) depends essentially on the graph properties that are considered as important in a given context. As relevant information on a graph, one may consider for instance

    更新日期:2021-01-03
  • On Grundy and b-Chromatic Number of Some Families of Graphs: A Comparative Study
    Graphs Comb. (IF 0.597) Pub Date : 2021-01-01
    Zoya Masih, Manouchehr Zaker

    The Grundy and the b-chromatic number of graphs are two important chromatic parameters. The Grundy number of a graph G, denoted by \(\Gamma (G)\) is the worst case behavior of greedy (First-Fit) coloring procedure for G and the b-chromatic number \(\mathrm{{b}}(G)\) is the maximum number of colors used in any color-dominating coloring of G. Because the nature of these colorings are different they have

    更新日期:2021-01-01
  • Spherical 2-Designs as Stationary Points of Many-Body Systems
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-24
    Makoto Miura, Kazushi Ueda

    We show that spherical 2-designs give stationary points of the classical mechanical system of particles in a Euclidean space interacting by a double-well potential.

    更新日期:2020-11-25
  • Fractional Cross Intersecting Families
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-20
    Rogers Mathew, Ritabrata Ray, Shashank Srivastava

    Let \(\mathcal {A}=\{A_{1},\ldots ,A_{p}\}\) and \(\mathcal {B}=\{B_{1},\ldots ,B_{q}\}\) be two families of subsets of [n] such that for every \(i\in [p]\) and \(j\in [q]\), \(|A_{i}\cap B_{j}|= \frac{c}{d}|B_{j}|\), where \(\frac{c}{d}\in [0,1]\) is an irreducible fraction. We call such families \(\frac{c}{d}\)-cross intersecting families. In this paper, we find a tight upper bound for the product

    更新日期:2020-11-21
  • Embeddings of a Graph into a Surface with Different Weak Chromatic Numbers
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-16
    Kengo Enami, Kenta Noguchi

    A weak coloring of a graph G embedded on a surface is a vertex coloring of G such that no face is monochromatic. The weak chromatic number of G is the minimum number k such that G has a weak k-coloring. Kündgen and Ramamurthi (J Combin Theory Ser B 85, 307–337, 2002) conjectured that for each positive integer k, there is a graph that has two different embeddings on the same surface whose weak chromatic

    更新日期:2020-11-16
  • Anti-Ramsey Problems in Complete Bipartite Graphs for t Edge-Disjoint Rainbow Spanning Trees
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-12
    Yuxing Jia, Mei Lu, Yi Zhang

    Let \(r(K_{p,q},t)\) be the maximum number of colors in an edge-coloring of the complete bipartite graph \(K_{p,q}\) not having t edge-disjoint rainbow spanning trees. We prove that \(r(K_{p,p},1)=p^2-2p+2\) for \(p\ge 4\) and \(r(K_{p,q},1)=pq-2q+1\) for \(p>q\ge 4\). Let \(t\ge 2\). We also show that \(r(K_{p,p},t)=p^2-2p+t+1\) for \(p \ge 2t+\sqrt{3t-3}+4\) and \(r(K_{p,q},t)=pq-2q+t\) for \(p >

    更新日期:2020-11-13
  • Sharp Upper Bounds on the k -Independence Number in Graphs with Given Minimum and Maximum Degree
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-11
    Suil O, Yongtang Shi, Zhenyu Taoqiu

    The k-independence number of a graph G is the maximum size of a set of vertices at pairwise distance greater than k. In this paper, for each positive integer k, we prove sharp upper bounds for the k-independence number in an n-vertex connected graph with given minimum and maximum degree.

    更新日期:2020-11-12
  • Acyclic Coloring of Graphs with Maximum Degree 7
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-11
    Juan Wang, Lianying Miao, Wenyao Song, Yunlong Liu

    The acyclic chromatic number a(G) of a graph G is the minimum number of colors such that G has a proper vertex coloring and no bichromatic cycles. For a graph G with maximum degree \(\Delta \), Grünbaum (1973) conjectured \(a(G)\le \Delta +1\). Up to now, the conjecture has only been shown for \(\Delta \le 4\). In this paper, it is proved that \(a(G)\le 12\) for \(\Delta =7\), thus improving the result

    更新日期:2020-11-12
  • On $$\ell $$ ℓ -Distance-Balanced Product Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-09
    Janja Jerebic, Sandi Klavžar, Gregor Rus

    A graph G is \(\ell \)-distance-balanced if for each pair of vertices x and y at a distance \(\ell \) in G, the number of vertices closer to x than to y is equal to the number of vertices closer to y than to x. A complete characterization of \(\ell \)-distance-balanced corona products is given and characterization of lexicographic products for \(\ell \ge 3\), thus complementing known results for \(\ell

    更新日期:2020-11-09
  • Strict Neighbor-Distinguishing Index of Subcubic Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-09
    Jing Gu, Weifan Wang, Yiqiao Wang, Ying Wang

    A proper edge coloring of a graph G is strict neighbor-distinguishing if for any two adjacent vertices u and v, the set of colors used on the edges incident to u and the set of colors used on the edges incident to v are not included with each other. The strict neighbor-distinguishing index of G is the minimum number \(\chi '_\mathrm{snd}(G)\) of colors in a strict neighbor-distinguishing edge coloring

    更新日期:2020-11-09
  • Equitable Clique-Coloring in Claw-Free Graphs with Maximum Degree at Most 4
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-09
    Zuosong Liang, Yanxia Dong, Yancai Zhao, Huiyu Xing

    A clique of a graph G is a set of pairwise adjacent vertices of G. A clique-coloring of G is an assignment of colors to the vertices of G in such a way that no inclusion-wise maximal clique of size at least two of G is monochromatic. An equitable clique-coloring of G is a clique-coloring such that any two color classes differ in size by at most one. Bacsó and Tuza proved that connected claw-free graphs

    更新日期:2020-11-09
  • Proportional Choosability of Complete Bipartite Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-08
    Jeffrey A. Mudrock, Jade Hewitt, Paul Shin, Collin Smith

    Proportional choosability is a list analogue of equitable coloring that was introduced in 2019. The smallest k for which a graph G is proportionally k-choosable is the proportional choice number of G, and it is denoted \(\chi _{pc}(G)\). In the first ever paper on proportional choosability, it was shown that when \(2 \le n \le m\), \(\max \{ n + 1, 1 + \lceil m / 2 \rceil \} \le \chi _{pc}(K_{n,m})

    更新日期:2020-11-09
  • Decompositions of 6-Regular Bipartite Graphs into Paths of Length Six
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-07
    Yanan Chu, Genghua Fan, Chuixiang Zhou

    Let T be a tree with m edges. It was conjectured that every m-regular bipartite graph can be decomposed into edge-disjoint copies of T. In this paper, we prove that every 6-regular bipartite graph can be decomposed into edge-disjoint paths with 6 edges. As a consequence, every 6-regular bipartite graph on n vertices can be decomposed into \(\frac{n}{2}\) paths, which is related to the well-known Gallai’s

    更新日期:2020-11-09
  • Questions on Color-Critical Subgraphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-07
    Nicholas Newman, Matthew Noble

    In our work, we define a k-tuple of positive integers \((x_1, \ldots , x_k)\) to be a \(\chi \)-sequence if there exists a k-chromatic graph G such that for each \(i \in \{1, \ldots , k\}\), the order of a minimum i-chromatic subgraph of G is equal to \(x_i\). Denote by \(\mathcal {X}_k\) the set of all \(\chi \)-sequences of length k. A very difficult question is to determine, for a given \((x_1,

    更新日期:2020-11-09
  • Ramsey and Gallai-Ramsey Numbers for Two Classes of Unicyclic Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-06
    Zhao Wang, Yaping Mao, Colton Magnant, Jinyu Zou

    Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge coloring of \(K_n\) contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we consider two classes of unicyclic graphs, the star with an extra edge and the path with a triangle at one end. We provide the 2-color Ramsey

    更新日期:2020-11-09
  • An Injective Version of the 1-2-3 Conjecture
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-06
    Julien Bensmail, Bi Li, Binlong Li

    In this work, we introduce and study a new graph labelling problem standing as a combination of the 1-2-3 Conjecture and injective colouring of graphs, which also finds connections with the notion of graph irregularity. In this problem, the goal, given a graph G, is to label the edges of G so that every two vertices having a common neighbour get incident to different sums of labels. We are interested

    更新日期:2020-11-06
  • Upper Bounds on the k -Tuple (Roman) Domination Number of a Graph
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-03
    Michael A. Henning, Nader Jafari Rad

    Rautenbach and Volkmann (Appl Math Lett 20:98–102, 2007), gave an upper bound for the k-tuple domination number of a graph. Rad (J Combin Math Comb Comput, 2019, in press) presented an improvement of the above bound using the Caro-Wei Theorem. In this paper, using the well-known Brooks’ Theorem for vertex coloring and vertex covers, we improve the above bounds on the k-tuple domination number under

    更新日期:2020-11-04
  • Graphs with Diameter 2 and Large Total Domination Number
    Graphs Comb. (IF 0.597) Pub Date : 2020-11-02
    Artūras Dubickas

    In this paper we show that for each sufficiently large n there exist graphs G of order n and diameter 2 whose total domination number \(\gamma _t(G)\) is greater than \(\sqrt{(3n\log n)/8}-\sqrt{n}\). On the other hand, it is shown that the total domination number of a graph of order \(n \geqslant 3\) and diameter 2 is always less than \(\sqrt{(n\log n)/2}+\sqrt{n/2}\).

    更新日期:2020-11-02
  • Counting Locally Flat-Foldable Origami Configurations Via 3-Coloring Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-10-26
    Alvin Chiu, William Hoganson, Thomas C. Hull, Sylvia Wu

    Origami, where two-dimensional sheets are folded into complex structures, is rich with combinatorial and geometric structure, most of which remains to be fully understood. In this paper we consider flat origami, where the sheet of material is folded into a two-dimensional object, and consider the mountain (convex) and valley (concave) creases that result, called a MV assignment of the crease pattern

    更新日期:2020-10-26
  • Strongly Spanning Trailable Graphs with Small Circumference and Hamilton-Connected Claw-Free Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-10-12
    Xia Liu, Liming Xiong, Hong-Jian Lai

    A graph G is strongly spanning trailable if for any \(e_1=u_1v_1, e_2=u_2v_2\in E(G)\) (possibly \(e_1=e_2\)), \(G(e_1, e_2)\), which is obtained from G by replacing \(e_1\) by a path \(u_1v_{e_1}v_1\) and by replacing \(e_2\) by a path \(u_2v_{e_2}v_2\), has a spanning \((v_{e_1}, v_{e_2})\)-trail. A graph G is Hamilton-connected if there is a spanning path between any two vertices of V(G). In this

    更新日期:2020-10-12
  • Classification of Partially Metric Q-Polynomial Association Schemes with $$m_{ 1} = 4$$ m 1 = 4
    Graphs Comb. (IF 0.597) Pub Date : 2020-10-06
    Da Zhao

    We classify the Q-polynomial association schemes with \(m_{1} = 4\) which are partially metric with respect to the nearest neighbourhood relation. An association scheme is partially metric with respect to a relation \(R_1\) if the scheme graph of \(R_2\) is exactly the distance-2 graph of the scheme graph of \(R_1\) under a certain ordering of the relations.

    更新日期:2020-10-07
  • Correction to: Gallai–Ramsey Numbers for a Class of Graphs with Five Vertices
    Graphs Comb. (IF 0.597) Pub Date : 2020-10-03
    Xihe Li, Ligong Wang

    The following items are the errors that should be corrected.

    更新日期:2020-10-04
  • Rainbow Antistrong Connection in Tournaments
    Graphs Comb. (IF 0.597) Pub Date : 2020-09-28
    Yumei Hu, Yarong Wei

    An arc-coloured digraph is rainbow antistrong connected if any two distinct vertices u, v are connected by both a forward antidirected (u, v)-trail and a forward antidirected (v, u)-trail which do not use two arcs with the same colour. The rainbow antistrong connection number of a digraph D is the minimum number of colours needed to make the digraph rainbow antistrong connected, denoted by \(\overset{\rightarrow

    更新日期:2020-09-28
  • On Well-Dominated Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-09-25
    Sarah E. Anderson, Kirsti Kuenzel, Douglas F. Rall

    A graph is well-dominated if all of its minimal dominating sets have the same cardinality. It is proved that there are exactly eleven connected, well-dominated, triangle-free graphs whose domination number is at most 3. We prove that at least one of the factors is well-dominated if the Cartesian product of two graphs is well-dominated. In addition, we show that the Cartesian product of two connected

    更新日期:2020-09-25
  • Encoding Labelled p -Riordan Graphs by Words and Pattern-Avoiding Permutations
    Graphs Comb. (IF 0.597) Pub Date : 2020-09-21
    Kittitat Iamthong, Ji-Hwan Jung, Sergey Kitaev

    The notion of a p-Riordan graph generalizes that of a Riordan graph, which, in turn, generalizes the notions of a Pascal graph and a Toeplitz graph. In this paper we introduce the notion of a p-Riordan word, and show how to encode p-Riordan graphs by p-Riordan words. For special important cases of Riordan graphs (the case \(p=2\)) and oriented Riordan graphs (the case \(p=3\)) we provide alternative

    更新日期:2020-09-21
  • On Degree Sum Conditions and Vertex-Disjoint Chorded Cycles
    Graphs Comb. (IF 0.597) Pub Date : 2020-09-21
    Bradley Elliott, Ronald J. Gould, Kazuhide Hirohata

    In this paper, we consider a general degree sum condition sufficient to imply the existence of k vertex-disjoint chorded cycles in a graph G. Let \(\sigma _t(G)\) be the minimum degree sum of t independent vertices of G. We prove that if G is a graph of sufficiently large order and \(\sigma _t(G)\ge 3kt-t+1\) with \(k\ge 1\), then G contains k vertex-disjoint chorded cycles. We also show that the degree

    更新日期:2020-09-21
  • Spiders and their Kin: An Investigation of Stanley’s Chromatic Symmetric Function for Spiders and Related Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-09-19
    Angèle M. Foley, Joshua Kazdan, Larissa Kröll, Sofía Martínez Alberga, Oleksii Melnyk, Alexander Tenenbaum

    We study the chromatic symmetric functions of graph classes related to spiders, namely generalized spider graphs (line graphs of spiders), and what we call horseshoe crab graphs. We show that no two generalized spiders have the same chromatic symmetric function, thereby extending the work of Martin, Morin and Wagner. Additionally, we establish Query Text="Please confirm if the inserted city name in

    更新日期:2020-09-20
  • The Largest Coefficient of the Highest Root and the Second Smallest Exponent
    Graphs Comb. (IF 0.597) Pub Date : 2020-09-18
    Tan Nhat Tran

    There are many different ways that the exponents of Weyl groups of irreducible root systems have been defined and put into practice. One of the most classical and algebraic definitions of the exponents is related to the eigenvalues of Coxeter elements. While the coefficients of the highest root when expressed as a linear combination of simple roots are combinatorial objects in nature, there are several

    更新日期:2020-09-20
  • Bipartite Independent Number and Hamilton-Biconnectedness of Bipartite Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-09-15
    Binlong Li

    Let G be a balanced bipartite graph with bipartite sets X, Y. We say that G is Hamilton-biconnected if there is a Hamilton path connecting any vertex in X and any vertex in Y. We define the bipartite independent number \(\alpha ^o_B(G)\) to be the maximum integer \(\alpha \) such that for any integer partition \(\alpha =s+t\), G has an independent set formed by s vertices in X and t vertices in Y.

    更新日期:2020-09-16
  • Unit Ball Graphs on Geodesic Spaces
    Graphs Comb. (IF 0.597) Pub Date : 2020-09-15
    Masamichi Kuroda, Shuhei Tsujie

    Consider finitely many points in a geodesic space. If the distance of two points is less than a fixed threshold, then we regard these two points as “near”. Connecting near points with edges, we obtain a simple graph on the points, which is called a unit ball graph. If the space is the real line, then it is known as a unit interval graph. Unit ball graphs on a geodesic space describe geometric characteristics

    更新日期:2020-09-15
  • On Colorful Edge Triples in Edge-Colored Complete Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-09-09
    Gábor Simonyi

    An edge-coloring of the complete graph \(K_n\) we call F-caring if it leaves no F-subgraph of \(K_n\) monochromatic and at the same time every subset of |V(F)| vertices contains in it at least one completely multicolored version of F. For the first two meaningful cases, when \(F=K_{1,3}\) and \(F=P_4\) we determine for infinitely many n the minimum number of colors needed for an F-caring edge-coloring

    更新日期:2020-09-10
  • On 0-Rotatable Graceful Caterpillars
    Graphs Comb. (IF 0.597) Pub Date : 2020-09-03
    Atílio G. Luiz, C. N. Campos, R. Bruce Richter

    An injection \(f :V(T) \rightarrow \{0,\ldots ,|E(T)|\}\) of a tree T is a graceful labelling if \(\{|f(u)-f(v)| :uv \in E(T)\}=\{1,\ldots ,|E(T)|\}\). Tree T is 0-rotatable if, for any \(v \in V(T)\), there exists a graceful labelling f of T such that \(f(v)=0\). In this work, the following families of caterpillars are proved to be 0-rotatable: caterpillars with a perfect matching; caterpillars obtained

    更新日期:2020-09-03
  • Supereulerian Graphs with Constraints on the Matching Number and Minimum Degree
    Graphs Comb. (IF 0.597) Pub Date : 2020-09-03
    Mansour J. Algefari, Hong-Jian Lai

    A graph is supereulerian if it has a spanning eulerian subgraph. We show that a connected simple graph G with \(n = |V(G)| \ge 2\) and \(\delta (G) \ge \alpha '(G)\) is supereulerian if and only if \(G \ne K_{1,n-1}\) if n is even or \(G \ne K_{2, n-2}\) if n is odd. Consequently, every connected simple graph G with \(\delta (G) \ge \alpha '(G)\) has a hamiltonian line graph.

    更新日期:2020-09-03
  • Ear-Slicing for Matchings in Hypergraphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-09-02
    András Sebő

    We study when a given edge of a factor-critical graph is contained in a matching avoiding exactly one, pregiven vertex of the graph. We then apply the results to always partition the vertex-set of a 3-regular, 3-uniform hypergraph into at most one triangle (hyperedge of size 3) and edges (subsets of size 2 of hyperedges), corresponding to the intuition, and providing new insight to triangle and edge

    更新日期:2020-09-03
  • Signed Circuit Cover of Bridgeless Signed Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-08-27
    Mengmeng Xie, Chuixiang Zhou

    Let \((G, \sigma )\) be a 2-edge-connected flow-admissible signed graph. In this paper, we prove that \((G,\sigma )\) has a signed circuit cover with length at most 3|E(G)|.

    更新日期:2020-08-27
  • On Specific Factors in Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-08-25
    Csilla Bujtás, Stanislav Jendrol’, Zsolt Tuza

    It is well known that if \(G = (V, E)\) is a connected multigraph and \(X\subset V\) is a subset of even order, then G contains a spanning forest H such that each vertex from X has an odd degree in H and all the other vertices have an even degree in H. This spanning forest may have isolated vertices. If this is not allowed in H, then the situation is much more complicated. In this paper, we study this

    更新日期:2020-08-26
  • Correction to: Properties of a q -Analogue of Zero Forcing
    Graphs Comb. (IF 0.597) Pub Date : 2020-08-24
    Steve Butler, Craig Erickson, Shaun Fallat, H. Tracy Hall, Brenda Kroschel, Jephian C.-H. Lin, Bryan Shader, Nathan Warnberg, Boting Yang

    The LaTeX control sequence \deg is interpreted differently in the authors’ and the publisher’s LaTeX setting

    更新日期:2020-08-24
  • Generalized Power Sum and Newton-Girard Identities
    Graphs Comb. (IF 0.597) Pub Date : 2020-08-17
    Sudip Bera, Sajal Kumar Mukherjee

    In this article we prove an algebraic identity which significantly generalizes the formula for sum of powers of consecutive integers involving Stirling numbers of the second kind. Also we have obtained a generalization of Newton-Girard power sum identity.

    更新日期:2020-08-17
  • Spanning Triangle-Trees and Flows of Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-08-13
    Jiaao Li, Xueliang Li, Meiling Wang

    In this paper we study the flow properties of graphs containing a spanning triangle-tree. Our main results provide a structure characterization of graphs with a spanning triangle-tree admitting a nowhere-zero 3-flow. All these graphs without nowhere-zero 3-flows are constructed from \(K_4\) by a so-called bull-growth operation. This generalizes a result of Fan et al. in 2008 on triangularly-connected

    更新日期:2020-08-14
  • On q -Series and Split Lattice Paths
    Graphs Comb. (IF 0.597) Pub Date : 2020-08-13
    Megha Goyal

    In this paper a natural question which arise to study the graphical aspect of split \((n+t)\)-color partitions, is answered by introducing a new class of lattice paths, called split lattice paths. A direct bijection between split \((n+t)\)-color partitions and split lattice paths is proved. This new combinatorial object is applied to give new combinatorial interpretations of two basic functions of

    更新日期:2020-08-14
  • Uniform Length Dominating Sequence Graphs
    Graphs Comb. (IF 0.597) Pub Date : 2020-08-12
    Aysel Erey

    A sequence of vertices \((v_1,\, \dots , \,v_k)\) of a graph G is called a dominating closed neighborhood sequence if \(\{v_1,\, \dots , \,v_k\}\) is a dominating set of G and \(N[v_i]\nsubseteq \cup _{j=1}^{i-1} N[v_j]\) for every i. A graph G is said to be \(k-\)uniform if all dominating closed neighborhood sequences in the graph have equal length k. Brešar et al. (Discrete Math 336:22–36, 2014)

    更新日期:2020-08-12
  • The Triangle Groups (2, 4, 5) and (2, 5, 5) are not Systolic
    Graphs Comb. (IF 0.597) Pub Date : 2020-08-09
    Annette Karrer, Petra Schwer, Koen Struyve

    In this paper we provide new examples of hyperbolic but nonsystolic groups by showing that the triangle groups (2, 4, 5) and (2, 5, 5) are not systolic. Along the way we prove some results about subsets of systolic complexes stable under involutions.

    更新日期:2020-08-10
  • 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
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