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

Vertex-disjoint triangles in K1,t-free graphs with minimum degree at least t

AbstractA graph is said to be K1,t-free if it does not contain an induced subgraph isomorphic to K1,t. Let h(t,k) be the smallest integer m such that every K1,t-free graph of order greater than m and with minimum degree at least t contains k vertex-disjoint triangles. In this paper, we obtain a lower bound of h(t,k) by a constructive method. According to the lower bound, we totally disprove the conjecture raised by Hong Wang [H. Wang, Vertex-disjoint triangles in claw-free graphs with minimum degree at least three, Combinatorica 18 (1998) 441–447]. We also obtain an upper bound of h(t,k) which is related to Ramsey numbers R(3,t). In particular, we prove that h(4,k)=9(k−1) and h(5,k)=14(k−1)