Polarity of chordal graphs

Polar graphs are a common generalization of bipartite, cobipartite, and split graphs. They are defined by the existence of a certain partition of vertices, which is NP-complete to decide for general graphs. It has been recently proved that for cographs, the existence of such a partition can be characterized by finitely many forbidden subgraphs, and hence tested in polynomial time. In this paper we address the question of polarity of chordal graphs, arguing that this is in essence a question of colourability, and hence chordal graphs are a natural restriction. We observe that there is no finite forbidden subgraph characterization of polarity in chordal graphs; nevertheless we present a polynomial time algorithm for polarity of chordal graphs. We focus on a special case of polarity (called monopolarity) which turns out to be the central concept for our algorithms. For the case of monopolar graphs, we illustrate the structure of all minimal obstructions; it turns out that they can all be described by a certain graph grammar, permitting our monopolarity algorithm to be cast as a certifying algorithm.     

Claw‐Free Graphs,Skeletal Graphs,and a Stronger Conjecture on ω, Δ, and χ

The second author's (B.A.R.) ω, Δ, χ conjecture proposes that every graph satisfies . In this article, we prove that the conjecture holds for all claw‐free graphs. Our approach uses the structure theorem of Chudnovsky and Seymour. Along the way, we discuss a stronger local conjecture, and prove that it holds for claw‐free graphs with a three‐colorable complement. To prove our results, we introduce a very useful χ‐preserving reduction on homogeneous pairs of cliques, and thus restrict our view to so‐called skeletal graphs.     

The Degree‐Diameter Problem for Claw‐Free Graphs and Hypergraphs

We study the degree‐diameter problem for claw‐free graphs and 2‐regular hypergraphs. Let be the largest order of a claw‐free graph of maximum degree Δ and diameter D. We show that , where , for any D and any even . So for claw‐free graphs, the well‐known Moore bound can be strengthened considerably. We further show that for with (mod 4). We also give an upper bound on the order of ‐free graphs of given maximum degree and diameter for . We prove similar results for the hypergraph version of the degree‐diameter problem. The hypergraph Moore bound states that the order of a hypergraph of maximum degree Δ, rank k, and diameter D is at most . For 2‐regular hypergraph of rank and any diameter D, we improve this bound to , where . Our construction of claw‐free graphs of diameter 2 yields a similar result for hypergraphs of diameter 2, degree 2, and any even rank .     

A Closure for 1‐Hamilton‐Connectedness in Claw‐Free Graphs

A graph G is 1‐Hamilton‐connected if is Hamilton‐connected for every vertex . In the article, we introduce a closure concept for 1‐Hamilton‐connectedness in claw‐free graphs. If is a (new) closure of a claw‐free graph G, then is 1‐Hamilton‐connected if and only if G is 1‐Hamilton‐connected, is the line graph of a multigraph, and for some , is the line graph of a multigraph with at most two triangles or at most one double edge. As applications, we prove that Thomassen's Conjecture (every 4‐connected line graph is hamiltonian) is equivalent to the statement that every 4‐connected claw‐free graph is 1‐Hamilton‐connected, and we present results showing that every 5‐connected claw‐free graph with minimum degree at least 6 is 1‐Hamilton‐connected and that every 4‐connected claw‐free and hourglass‐free graph is 1‐Hamilton‐connected.     

Large Induced Forests in Graphs

In this article, we prove three theorems. The first is that every connected graph of order n and size m has an induced forest of order at least with equality if and only if such a graph is obtained from a tree by expanding every vertex to a clique of order either 4 or 5. This improves the previous lower bound of Alon–Kahn–Seymour for , and implies that such a graph has an induced forest of order at least for . This latter result relates to the conjecture of Albertson and Berman that every planar graph of order n has an induced forest of order at least . The second is that every connected triangle‐free graph of order n and size m has an induced forest of order at least . This bound is sharp by the cube and the Wagner graph. It also improves the previous lower bound of Alon–Mubayi–Thomas for , and implies that such a graph has an induced forest of order at least for . This latter result relates to the conjecture of Akiyama and Watanabe that every bipartite planar graph of order n has an induced forest of order at least . The third is that every connected planar graph of order n and size m with girth at least 5 has an induced forest of order at least with equality if and only if such a graph is obtained from a tree by expanding every vertex to one of five specific graphs. This implies that such a graph has an induced forest of order at least , where was conjectured to be the best lower bound by Kowalik, Lu?ar, and ?krekovski.     

On stability of Hamilton‐connectedness under the 2‐closure in claw‐free graphs

It is easy to characterize chordal graphs by every k‐cycle having at least f(k) = k ? 3 chords. I prove new, analogous characterizations of the house‐hole‐domino‐free graphs using f(k) = 2?(k ? 3)/2?, and of the graphs whose blocks are trivially perfect using f(k) = 2k ? 7. These three functions f(k) are optimum in that each class contains graphs in which every k‐cycle has exactly f(k) chords. The functions 3?(k ? 3)/3? and 3k ? 11 also characterize related graph classes, but without being optimum. I consider several other graph classes and their optimum functions, and what happens when k‐cycles are replaced with k‐paths. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:137‐147, 2011     

Cycle Extensions in BIBD Block‐Intersection Graphs

A cycle C in a graph G is extendable if there is some other cycle in G that contains each vertex of C plus one additional vertex. A graph is cycle extendable if every non‐Hamilton cycle in the graph is extendable. A balanced incomplete block design, BIBD, consists of a set V of v elements and a block set of k‐subsets of V such that each 2‐subset of V occurs in exactly λ of the blocks of . The block‐intersection graph of a design is the graph having as its vertex set and such that two vertices of are adjacent if and only if their corresponding blocks have nonempty intersection. In this paper, we prove that the block‐intersection graph of any BIBD is cycle extendable. Furthermore, we present a polynomial time algorithm for constructing cycles of all possible lengths in a block‐intersection graph.     

Equimatchable Regular Graphs

A graph is called equimatchable if all of its maximal matchings have the same size. Kawarabayashi, Plummer, and Saito showed that the only connected equimatchable 3‐regular graphs are K₄ and K_{3, 3}. We extend this result by showing that for an odd positive integer r, if G is a connected equimatchable r‐regular graph, then . Also it is proved that for an even r, a connected triangle‐free equimatchable r‐regular graph is isomorphic to one of the graphs C₅, C₇, and .     

A Polynomial Invariant of Graphs On Orientable Surfaces

We consider cyclic graphs, that is, graphs with cyclic ordersat the vertices, corresponding to 2-cell embeddings of graphsinto orientable surfaces, or combinatorial maps. We constructa three variable polynomial invariant of these objects, thecyclic graph polynomial, which has many of the useful propertiesof the Tutte polynomial. Although the cyclic graph polynomialgeneralizes the Tutte polynomial, its definition is very different,and it depends on the embedding in an essential way. 2000 MathematicalSubject Classification: 05C10.     

关于双圈图的谱半径

如果G是连通的并且G的边数是n 1,那么n阶图G叫做双圈图,设B(n)是所有的阶为n的双圈图构成的集合,本文给出了B(n)(n(?)9)中前三大的邻接谱半径以及它们对应的图.     

Maximal Induced Matchings in Triangle‐Free Graphs

An induced matching in a graph is a set of edges whose endpoints induce a 1‐regular subgraph. It is known that every n‐vertex graph has at most  maximal induced matchings, and this bound is the best possible. We prove that every n‐vertex triangle‐free graph has at most  maximal induced matchings; this bound is attained by every disjoint union of copies of the complete bipartite graph K_{3, 3}. Our result implies that all maximal induced matchings in an n‐vertex triangle‐free graph can be listed in time , yielding the fastest known algorithm for finding a maximum induced matching in a triangle‐free graph.     

关于图的3-条件色数

For integers k0,r0,a(k,r)-coloring of a graph G is a proper k-coloring of the vertices such that every vertex of degree d is adjacent to vertices with at least min{d,r}diferent colors.The r-hued chromatic number,denoted byχr(G),is the smallest integer k for which a graph G has a(k,r)-coloring.Define a graph G is r-normal,ifχr(G)=χ(G).In this paper,we present two sufcient conditions for a graph to be 3-normal,and the best upper bound of 3-hued chromatic number of a certain families of graphs.     

On factors of 4‐connected claw‐free graphs*

We consider the existence of several different kinds of factors in 4‐connected claw‐free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4‐connected line graph is hamiltonian, i.e., has a connected 2‐factor. Conjecture 2 (Matthews and Sumner): Every 4‐connected claw‐free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass‐free graphs, i.e., graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjectures 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 125–136, 2001     

Local Clique Covering of Claw‐Free Graphs

A clique covering of a simple graph G is a collection of cliques of G covering all the edges of G such that each vertex is contained in at most k cliques. The smallest k for which G admits a clique covering is called the local clique cover number of G and is denoted by lcc(G). Local clique cover number can be viewed as the local counterpart of the clique cover number that is equal to the minimum total number of cliques covering all edges. In this article, several aspects of the local clique covering problem are studied and its relationships to other well‐known problems are discussed. In particular, it is proved that the local clique cover number of every claw‐free graph is at most , where Δ is the maximum degree of the graph and c is a constant. It is also shown that the bound is tight, up to a constant factor. Moreover, regarding a conjecture by Chen et al. (Clique covering the edges of a locally cobipartite graph, Discrete Math 219(1–3)(2000), 17–26), we prove that the clique cover number of every connected claw‐free graph on n vertices with the minimum degree δ, is at most , where c is a constant.     

关于有限群的强幂图的谱

令$G$是一个阶为$n$的有限群, $G$上的强幂图定义为: 以$G$为顶点集, 对于两个不同的元素$x$和$y$, 如果存在两个不超过$n$的正整数$n_1, n_2$使得$x^{n_1}=y^{n_2}$, 则$x$和$y$ 之间连一条边. 本文给出了$G$上强幂图的距离矩阵和邻接矩阵的特征多项式, 并且计算了其距离谱和邻接谱.     

Line graphs of multigraphs and Hamilton‐connectedness of claw‐free graphs

We introduce a closure concept that turns a claw‐free graph into the line graph of a multigraph while preserving its (non‐)Hamilton‐connectedness. As an application, we show that every 7‐connected claw‐free graph is Hamilton‐connected, and we show that the well‐known conjecture by Matthews and Sumner (every 4‐connected claw‐free graph is hamiltonian) is equivalent with the statement that every 4‐connected claw‐free graph is Hamilton‐connected. Finally, we show a natural way to avoid the non‐uniqueness of a preimage of a line graph of a multigraph, and we prove that the closure operation is, in a sense, best possible. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:152‐173, 2011     

Maximal K3's and Hamiltonicity of 4‐connected claw‐free graphs

Let cl(G) denote Ryjá?ek's closure of a claw‐free graph G. In this article, we prove the following result. Let G be a 4‐connected claw‐free graph. Assume that G[N_G(T)] is cyclically 3‐connected if T is a maximal K₃ in G which is also maximal in cl(G). Then G is hamiltonian. This result is a common generalization of Kaiser et al.'s theorem [J Graph Theory 48(4) (2005), 267–276] and Pfender's theorem [J Graph Theory 49(4) (2005), 262–272]. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Hourglasses and Hamilton cycles in 4‐connected claw‐free graphs

We show that if G is a 4‐connected claw‐free graph in which every induced hourglass subgraph S contains two non‐adjacent vertices with a common neighbor outside S, then G is hamiltonian. This extends the fact that 4‐connected claw‐free, hourglass‐free graphs are hamiltonian, thus proving a broader special case of a conjecture by Matthews and Sumner. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 267–276, 2005