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 Conjecture 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. \u

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

Regular clique covers of graphs

A family of cliques in a graph G is said to be p-regular if any two cliquesin the family intersect in exactly p vertices. A graph G is said to have ap-regular k-clique cover if there is a p-regular family H of k-cliques of Gsuch that each edge of G belongs to a clique in H. Such a p-regular k-clique cover is separable if the complete subgraphs of order p that arise asintersections of pairs of distinct cliques of H are mutually vertex-disjoint.For any given integers p,k,l; p < k, we present bounds on the smallestorder of a graph that has a p-regular k-clique cover with exactly l cliques,and we describe all graphs that have p-regular separable k-clique coverswith l cliques

Hamiltonian index is NP-complete

In this paper we show that the problem to decide whether the hamiltonian index of a given graph is less than or equal to a given constant is NP-complete (although this was conjectured to be polynomial). Consequently, the corresponding problem to determine the hamiltonian index of a given graph is NP-hard. Finally, we show that some known upper and lower bounds on the hamiltonian index can be computed in polynomial time

Dirac's minimum degree condition restricted to claws

Let G be a graph on n 3 vertices. Dirac's minimum degree condition is the condition that all vertices of G have degree at least . This is a well-known sufficient condition for the existence of a Hamilton cycle in G. We give related sufficiency conditions for the existence of a Hamilton cycle or a perfect matching involving a restriction of Dirac's minimum degree condition to certain subsets of the vertices. For this purpose we define G to be 1-heavy (2-heavy) if at least one (two) of the end vertices of each induced subgraph of G isomorphic to K1,3 (a claw) has (have) degree at least . Thus, every claw-free graph is 2-heavy, and every 2-heavy graph is 1-heavy. We show that a 1-heavy or a 2-heavy graph G has a Hamilton cycle or a perfect matching if we impose certain additional conditions on G involving numbers of common neighbours, (local) connectivity, and forbidden induced subgraphs. These results generalize or extend previous work of Broersma & Veldman, Dirac, Fan, Faudree et al., Goodman & Hedetniemi, Las Vergnas, Oberly & Sumner, Ore, Shi, and Sumner

Unifying Results On Hamiltonian Claw-Free Graphs

This work was motivated by many (recent) papers on hamiltonicity of claw-free graphs, i.e. graphs that do not contain K 1;3 as an induced subgraph. By combining ideas from these papers with some new observations, we unify several of the existing sufficiency results, using a new sufficient condition consisting of seven subconditions. If each pair of vertices at distance two of a 2-connected claw-free graph G satisfies at least one of these subconditions, then G is hamiltonian. We also present infinite classes of examples of graphs showing that these subconditions are, in some sense, independent. AMS Subject Classifications (1991): 05 C 45 Key words: claw-free graph, hamiltonian graph, degree condition, local connectivity, forbidden subgraph Part of the work was done while subsets of the authors met at Aachen and Plzen. The work of the second and third author was supported by EC-grant No. 927. 1 1. TERMINOLOGY AND NOTATION We use [1] for terminology and notation not defined here and co..