• 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
• 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
• 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
• 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 • 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 • 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• Graphs Comb. (IF 0.597) Pub Date : 2020-08-01

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
• 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
• 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 • 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 • 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 • 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
• 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
• Graphs Comb. (IF 0.597) Pub Date : 2020-06-28
Kieran Clancy, Michael Haythorpe, Alex Newcombe, Ed Pegg

We show that no cubic graphs of order 26 have crossing number larger than 9, which proves a conjecture of Ed Pegg Jr and Geoffrey Exoo that the smallest cubic graphs with crossing number 11 have 28 vertices. This result is achieved by first eliminating all girth 3 graphs from consideration, and then using the recently developed QuickCross heuristic to find drawings with few crossings for each remaining

更新日期：2020-06-28
• Graphs Comb. (IF 0.597) Pub Date : 2020-06-28
Muhuo Liu, Yang Wu, Hong-Jian Lai

There have been researches on sufficient spectral conditions for Hamiltonian properties and path-coverable properties of graphs. Utilizing the Bondy–Chvátal closure, we provide a unified approach to study sufficient graph eigenvalue conditions for these properties and sharpen several former spectral Hamiltonian results on balanced bipartite graphs and complementary graphs.

更新日期：2020-06-28
• Graphs Comb. (IF 0.597) Pub Date : 2020-06-27
Rogers Mathew, Tapas Kumar Mishra

In this note, we give a simple, counting based proof of Fisher’s Inequality that does not use any tools from linear algebra.

更新日期：2020-06-27
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• Graphs Comb. (IF 0.597) Pub Date : 2020-06-12

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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
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