On the Pancyclicity of Lexicographic Products

We prove that if G and H are graphs containing at least one edge each, then their lexicographic product G[H] is weakly pancyclic, i. e., it contains a cycle of every length between the length of a shortest cycle and that of a longest one. This supports some conjectures on locally connected graphs and on product graphs. We obtain an analogous result on even cycles in products G[H] that are bipartite. We also investigate toughness conditions on G implying that G[H] is hamiltonian (and hence pancyclic). Supported by the project LN00A056 of the Czech Ministry of Education     

Lambda-fold 2-perfect 6-cycle systems in equipartite graphs

A 6-cycle system of a graph G is an edge-disjoint decomposition of G into 6-cycles. Graphs G, for which necessary and sufficient conditions for existence of a 6-cycle system have been found, include complete graphs and complete equipartite graphs. A 6-cycle system of G is said to be 2-perfect if the graph formed by joining all vertices distance 2 apart in the 6-cycles is again an edge-disjoint decomposition of G, this time into 3-cycles, since the distance 2 graph in any 6-cycle is a pair of disjoint 3-cycles.Necessary and sufficient conditions for existence of 2-perfect 6-cycle systems of both complete graphs and complete equipartite graphs are known, and also of λ-fold complete graphs. In this paper, we complete the problem, giving necessary and sufficient conditions for existence of λ-fold 2-perfect 6-cycle systems of complete equipartite graphs.     

Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k

A graph G is (k,0)‐colorable if its vertices can be partitioned into subsets V₁ and V₂ such that in G[V₁] every vertex has degree at most k, while G[V₂] is edgeless. For every integer k?0, we prove that every graph with the maximum average degree smaller than (3k+4)/(k+2) is (k,0)‐colorable. In particular, it follows that every planar graph with girth at least 7 is (8, 0)‐colorable. On the other hand, we construct planar graphs with girth 6 that are not (k,0)‐colorable for arbitrarily large k. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:83–93, 2010     

Super Restricted Edge Connectivity of Regular Graphs

An edge cut of a connected graph is called restricted if it separates this graph into components each having order at least 2; a graph G is super restricted edge connected if G−S contains an isolated edge for every minimum restricted edge cut S of G. It is proved in this paper that k-regular connected graph G is super restricted edge connected if k > |V(G)|/2+1. The lower bound on k is exemplified to be sharp to some extent. With this observation, we determined the number of edge cuts of size at most 2k−2 of these graphs. Supported by NNSF of China (10271105); Ministry of Science and Technology of Fujian (2003J036); Education Ministry of Fujian (JA03147)     

A characterization of weakly four‐connected graphs

A graph G = (V, E) is called weakly four‐connected if G is 4‐edge‐connected and G ? x is 2‐edge‐connected for all x ∈ V. We give sufficient conditions for the existence of ‘splittable’ vertices of degree four in weakly four‐connected graphs. By using these results we prove that every minimally weakly four‐connected graph on at least four vertices contains at least three ‘splittable’ vertices of degree four, which gives rise to an inductive construction of weakly four‐connected graphs. Our results can also be applied in the problem of finding 2‐connected orientations of graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 217–229, 2006     

Factor-Critical Graphs with Given Number of Maximum Matchings

A connected graph G is said to be factor-critical if G − ν has a perfect matching for every vertex ν of G. In this paper, the factor-critical graphs G with |V(G)| maximum matchings and with |V(G)| + 1 ones are characterized, respectively. From this, some special bicritical graphs are characterized. This work is supported by the Ph.D. Programs Foundation of Ministry of Education of China (No.20070574006) and the NNSF(10201019) of China.     

Signed 2-independence in graphs

A function f : V→{−1,1} defined on the vertices of a graph G=(V,E) is a signed 2-independence function if the sum of its function values over any closed neighbourhood is at most one. That is, for every vV, f(N[v])1, where N[v] consists of v and every vertex adjacent to v. The weight of a signed 2-independence function is f(V)=∑f(v), over all vertices vV. The signed 2-independence number of a graph G, denoted α_s²(G), equals the maximum weight of a signed 2-independence function of G. In this paper, we establish upper bounds for α_s²(G) in terms of the order and size of the graph, and we characterize the graphs attaining these bounds. For a tree T, upper and lower bounds for α_s²(T) are established and the extremal graphs characterized. It is shown that α_s²(G) can be arbitrarily large negative even for a cubic graph G.     

Vertex decompositions of sparse graphs into an independent vertex set and a subgraph of maximum degree at most 1

A graph G is (1, 0)-colorable if its vertex set can be partitioned into subsets V ₁ and V ₀ so that in G[V ₁] every vertex has degree at most 1, while G[V ₀] is edgeless. We prove that every graph with maximum average degree at most $\tfrac{{12}} {5} $\tfrac{{12}} {5} is (1, 0)-colorable. In particular, every planar graph with girth at least 12 is (1, 0)-colorable. On the other hand, we construct graphs with the maximum average degree arbitrarily close (from above) to $\tfrac{{12}} {5} $\tfrac{{12}} {5} which are not (1, 0)-colorable.     

PARTITION A GRAPH WITH SMALL DIAMETER INTO TWO INDUCED MATCHINGS

§1　IntroductionGraphs considered in this paper are finite and simple.For a graph G,its vertex setandedge set are denoted by V(G) and E(G) ,respectively.If vertices u and v are connected inG,the distance between u and v,denoted by d G(u,v) ,is the length ofa shortest(u,v) -pathin G.The diameter of a connected graph G is the maximum distance between two verticesof G.For X V(G) ,the neighbor set NG(X) of X is defined byNG(X) ={ y∈V(G) \X:there is x∈X such thatxy∈E(G) } .NG({ x} )…     

A Construction of Uniquely C4-free colourable Graphs

Abstract

An F-free colouring of a graph G is a partition {V₁,V₂,…,V_n} of the vertex set V(G) of G such that F is not an induced subgraph of G[V_i] for each i. A graph is uniquely F-free colourable if any two .F-free colourings induce the same partition of V(G). We give a constructive proof that uniquely C₄-free colourable graphs exist.     

Near-proper vertex 2-colorings of sparse graphs

A graph G is (2, 1)-colorable if its vertices can be partitioned into subsets V ₁ and V ₂ such that each component in G[V ₁] contains at most two vertices while G[V ₂] is edgeless. We prove that every graph with maximum average degree mad(G) < 7/3 is (2, 1)-colorable. It follows that every planar graph with girth at least 14 is (2, 1)-colorable. We also construct a planar graph G _n with mad (G _n ) = (18n − 2)/(7n − 1) that is not (2, 1)-colorable.     

Tight products and graph expansion

In this article, we study a new product of graphs called tight product. A graph H is said to be a tight product of two (undirected multi) graphs G₁ and G₂, if V(H) = V(G₁) × V(G₂) and both projection maps V(H)→V(G₁) and V(H)→V(G₂) are covering maps. It is not a priori clear when two given graphs have a tight product (in fact, it is NP‐hard to decide). We investigate the conditions under which this is possible. This perspective yields a new characterization of class‐1 (2k+ 1)‐regular graphs. We also obtain a new model of random d‐regular graphs whose second eigenvalue is almost surely at most O(d^3/4). This construction resembles random graph lifts, but requires fewer random bits. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Two characterizations of chain partitioned probe graphs

Chain graphs are exactly bipartite graphs without induced 2K ₂ (a graph with four vertices and two disjoint edges). A graph G=(V,E) with a given independent set S⊆V (a set of pairwise non-adjacent vertices) is said to be a chain partitioned probe graph if G can be extended to a chain graph by adding edges between certain vertices in S. In this note we give two characterizations for chain partitioned probe graphs. The first one describes chain partitioned probe graphs by six forbidden subgraphs. The second one characterizes these graphs via a certain “enhanced graph”: G is a chain partitioned probe graph if and only if the enhanced graph G ^* is a chain graph. This is analogous to a result on interval (respectively, chordal, threshold, trivially perfect) partitioned probe graphs, and gives an O(m ²)-time recognition algorithm for chain partitioned probe graphs.     

Induced Decompositions of Graphs

We consider those graphs G that admit decompositions into copies of a fixed graph F, each copy being an induced subgraph of G. We are interested in finding the extremal graphs with this property, that is, those graphs G on n vertices with the maximum possible number of edges. We discuss the cases where F is a complete equipartite graph, a cycle, a star, or a graph on at most four vertices.     

A decomposition for strongly perfect graphs

A graph G is strongly perfect if every induced subgraph H of G contains a stable set that meets all the maximal cliques of H. We present a graph decomposition that preserves strong perfection: more precisely, a stitch decomposition of a graph G = (V, E) is a partition of V into nonempty disjoint subsets V₁, V₂ such that in every P₄ with vertices in both V_iapos;s, each of the three edges has an endpoint in V₁ and the other in V₂. We give a good characterization of graphs that admit a stitch decomposition and establish several results concerning the stitch decomposition of strongly perfect graphs.     

A classification of graph capacity functions

Given two graphs G = (V(G), E(G)) and H = (V(H), E(H)), the sum of G and H, G + H, is the disjoint union of G and H. The product of G and H, G × H, is the graph with the vertex set V(G × H) that is the Cartesian product of V(G) and V(H), and two vertices (g₁, h₁), (g₂, h₂) are adjacent if and only if [g₁, g₂] (ELEMENT) E(G) and [h₁, h₂] (ELEMENT) E(H). Let G denote the set of all graphs. Given a graph G, the G-matching function, γ_G, assigns any graph H (ELEMENT) G to the maximum integer k such that kG is a subgraph of H. The graph capacity function for G, P_G: G → (RFRAKTUR), is defined as P_G(H) = lim_n→z[γ_G(Hⁿ)]^1/n, where Hⁿ denotes the n-fold product of H × H × … × H. Different graphs G may have different graph capacity functions, all of which are increasing. In this paper, we classify all graphs whose capacity functions are additive, multiplicative, and increasing; all graphs whose capacity functions are pseudo-additive, pseudo-multiplicative, and increasing; and all graphs whose capacity functions fall under neither of the above cases. © 1996 John Wiley & Sons, Inc.     

A generalization of perfect graphs—i-perfect graphs

Let i be a positive integer. We generalize the chromatic number _X(G) of G and the clique number o(G) of G as follows: The i-chromatic number of G, denoted by _X(G), is the least number k for which G has a vertex partition V₁, V₂,…, V_k such that the clique number of the subgraph induced by each V_j, 1 ≤ j ≤ k, is at most i. The i-clique number, denoted by o_i(G), is the i-chromatic number of a largest clique in G, which equals [o(G/i]. Clearly _X1(G) = _X(G) and o₁(G) = o(G). An induced subgraph G′ of G is an i-transversal iff o(G′) = i and o(G − G′) = o(G) − i. We generalize the notion of perfect graphs as follows: (1) A graph G is i-perfect iff _Xi(H) = o_i(H) for every induced subgraph H of G. (2) A graph G is perfectly i-transversable iff either o(G) ≤ i or every induced subgraph H of G with o(H) > i contains an i-transversal of H. We study the relationships among i-perfect graphs and perfectly i-transversable graphs. In particular, we show that 1-perfect graphs and perfectly 1-transversable graphs both coincide with perfect graphs, and that perfectly i-transversable graphs form a strict subset of i-perfect graphs for every i ≥ 2. We also show that all planar graphs are i-perfect for every i ≥ 2 and perfectly i-transversable for every i ≥ 3; the latter implies a new proof that planar graphs satisfy the strong perfect graph conjecture. We prove that line graphs of all triangle-free graphs are 2-perfect. Furthermore, we prove for each i greater than or equal to2, that the recognition of i-perfect graphs and the recognition of perfectly i-transversable graphs are intractable and not likely to be in co-NP. We also discuss several issues related to the strong perfect graph conjecture. © 1996 John Wiley & Sons, Inc.     

Maximal and minimal entry in the principal eigenvector for the distance matrix of a graph

Let G=(V,E) be a simple, connected and undirected graph with vertex set V(G) and edge set E(G). Also let D(G) be the distance matrix of a graph G (Jane?i? et al., 2007) [13]. Here we obtain Nordhaus–Gaddum-type result for the spectral radius of distance matrix of a graph.A sharp upper bound on the maximal entry in the principal eigenvector of an adjacency matrix and signless Laplacian matrix of a simple, connected and undirected graph are investigated in Das (2009) [4] and Papendieck and Recht (2000) [15]. Generally, an upper bound on the maximal entry in the principal eigenvector of a symmetric nonnegative matrix with zero diagonal entries and without zero diagonal entries are investigated in Zhao and Hong (2002) [21] and Das (2009) [4], respectively. In this paper, we obtain an upper bound on minimal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs. Moreover, we present the lower and upper bounds on maximal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs.     

The Prism Over the Middle-levels Graph is Hamiltonian

Let B _k be the bipartite graph defined by the subsets of {1,…,2k + 1} of size k and k + 1. We prove that the prism over B _k is hamiltonian. We also show that B _k has a closed spanning 2-trail. Supported by project 1M0021620808 and Research Plan MSM 4977751301 of the Czech Ministry of Education.     

Colored graphs without colorful cycles

A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation mon ≡ m + n − 2, the omitted lengths of colorful cycles in a colored graph form a monoid isomorphic to a submonoid of the natural numbers which contains all integers past some point. We prove that several but not all such monoids are realized. We then characterize exact Gallai graphs, i.e., graphs in which every triangle has edges of exactly two colors. We show that these are precisely the graphs which can be iteratively built up from three simple colored graphs, having 2, 4, and 5 vertices, respectively. We then characterize in two different ways the monochromes, i.e., the connected components of maximal monochromatic subgraphs, of exact Gallai graphs. The first characterization is in terms of their reduced form, a notion which hinges on the important idea of a full homomorphism. The second characterization is by means of a homomorphism duality. The first author would like to express his thanks for support by project LN 00A056 of the Ministry of Education of the Czech Republic. The second author would like to express his thanks for support by project LN 00A056 of the Ministry of Education of the Czech Republic, by the NSERC of Canada and by the Gudder Trust of the University of Denver.