Claw‐free circular‐perfect graphs

The circular chromatic number of a graph is a well‐studied refinement of the chromatic number. Circular‐perfect graphs form a superclass of perfect graphs defined by means of this more general coloring concept. This article studies claw‐free circular‐perfect graphs. First, we prove that if G is a connected claw‐free circular‐perfect graph with χ(G)>ω(G), then min{α(G), ω(G)}=2. We use this result to design a polynomial time algorithm that computes the circular chromatic number of claw‐free circular‐perfect graphs. A consequence of the strong perfect graph theorem is that minimal imperfect graphs G have min{α(G), ω(G)}=2. In contrast to this result, it is shown in Z. Pan and X. Zhu [European J Combin 29(4) (2008), 1055–1063] that minimal circular‐imperfect graphs G can have arbitrarily large independence number and arbitrarily large clique number. In this article, we prove that claw‐free minimal circular‐imperfect graphs G have min{α(G), ω(G)}≤3. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 163–172, 2010     

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 .     

Claw‐free 3‐connected P11‐free graphs are hamiltonian

We show that every 3‐connected claw‐free graph which contains no induced copy of P₁₁ is hamiltonian. Since there exist non‐hamiltonian 3‐connected claw‐free graphs without induced copies of P₁₂ this result is, in a way, best possible. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 111–121, 2004     

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.     

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.     

Designs with mutually orthogonal resolutions and decompositions of edge‐colored graphs

Two resolutions R and R^′ of a combinatorial design are called orthogonal if |R_i∩R|≤1 for all R_i∈R and R∈R^′. A set Q={R¹, R², …, R^d} of d resolutions of a combinatorial design is called a set of mutually orthogonal resolutions (MORs) if the resolutions of Q are pairwise orthogonal. In this paper, we establish necessary and sufficient conditions for the asymptotic existence of a (v, k, 1)‐BIBD with d mutually orthogonal resolutions for d≥2 and k≥3 and necessary and sufficient conditions for the asymptotic existence of a (v, k, k?1)‐BIBD with d mutually orthogonal near resolutions for d≥2 and k≥3. We use complementary designs and the most general form of an asymptotic existence theorem for decompositions of edge‐colored complete digraphs into prespecified edge‐colored subgraphs. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 425–447, 2009     

Tree‐Chromatic Number Is Not Equal to Path‐Chromatic Number*

For a graph G and a tree‐decomposition of G, the chromatic number of is the maximum of , taken over all bags . The tree‐chromatic number of G is the minimum chromatic number of all tree‐decompositions of G. The path‐chromatic number of G is defined analogously. In this article, we introduce an operation that always increases the path‐chromatic number of a graph. As an easy corollary of our construction, we obtain an infinite family of graphs whose path‐chromatic number and tree‐chromatic number are different. This settles a question of Seymour (J Combin Theory Ser B 116 (2016), 229–237). Our results also imply that the path‐chromatic numbers of the Mycielski graphs are unbounded.     

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.     

Edge‐decompositions of Kn,n into isomorphic copies of a given tree

We construct an incidence structure using certain points and lines in finite projective spaces. The structural properties of the associated bipartite incidence graphs are analyzed. These n × n bipartite graphs provide constructions of C₆‐free graphs with Ω(n^4/3 edges, C₁₀‐free graphs with Ω(n^6/5) edges, and Θ(7,7,7)‐free graphs with Ω(n^8/7) edges. Each of these bounds is sharp in order of magnitude. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 1–10, 2005     

Fractional clique decompositions of dense graphs

For each , we show that any graph G with minimum degree at least has a fractional K_r‐decomposition. This improves the best previous bounds on the minimum degree required to guarantee a fractional K_r‐decomposition given by Dukes (for small r) and Barber, Kühn, Lo, Montgomery, and Osthus (for large r), giving the first bound that is tight up to the constant multiple of r (seen, for example, by considering Turán graphs). In combination with work by Glock, Kühn, Lo, Montgomery, and Osthus, this shows that, for any graph F with chromatic number , and any , any sufficiently large graph G with minimum degree at least has, subject to some further simple necessary divisibility conditions, an (exact) F‐decomposition.     

Orienting Triangulations

We prove that any triangulation of a surface different from the sphere and the projective plane admits an orientation without sinks such that every vertex has outdegree divisible by three. This confirms a conjecture of Barát and Thomassen and is a step toward a generalization of Schnyder woods to higher genus surfaces.     

Primitive decompositions of Johnson graphs

A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the parts.     

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     

Commuting decompositions of complete graphs

We say that two graphs G and H with the same vertex set commute if their adjacency matrices commute. In this article, we show that for any natural number r, the complete multigraph K is decomposable into commuting perfect matchings if and only if n is a 2‐power. Also, it is shown that the complete graph K_n is decomposable into commuting Hamilton cycles if and only if n is a prime number. © 2006 Wiley Periodicals, Inc. J Combin Designs     

Critical graphs for subpancyclicity of 3‐connected claw‐free graphs

Let be the family of graphs G such that all sufficiently large k ‐connected claw‐free graphs which contain no induced copies of G are subpancyclic. We show that for every k≥3 the family is infinite and make the first step toward the complete characterization of the family . © 2009 Wiley Periodicals, Inc. J Graph Theory 62, 263–278, 2009     

Belief functions contextual discounting and canonical decompositions

In this article, the contextual discounting of a belief function, a classical discounting generalization, is extended and its particular link with the canonical disjunctive decomposition is highlighted. A general family of correction mechanisms allowing one to weaken the information provided by a source is then introduced, as well as the dual of this family allowing one to strengthen a belief function.     

Coloring quasi‐line graphs

A graph G is a quasi‐line graph if for every vertex v, the set of neighbors of v can be expressed as the union of two cliques. The class of quasi‐line graphs is a proper superset of the class of line graphs. A theorem of Shannon's implies that if G is a line graph, then it can be properly colored using no more than 3/2 ω(G) colors, where ω(G) is the size of the largest clique in G. In this article, we extend this result to all quasi‐line graphs. We also show that this bound is tight. © 2006 Wiley Periodicals, Inc. J Graph Theory     

Stanley decompositions and partitionable simplicial complexes

We study Stanley decompositions and show that Stanley’s conjecture on Stanley decompositions implies his conjecture on partitionable Cohen–Macaulay simplicial complexes. We also prove these conjectures for all Cohen–Macaulay monomial ideals of codimension 2 and all Gorenstein monomial ideals of codimension 3. Dedicated to Takayuki Hibi on the occasion of his fiftieth birthday.