Chvátal–Erdös Type Conditions for Hamiltonicity of Claw-Free Graphs

Abstract

For a graph H , let α(H ) and α (H ) denote the independence number and the matching number, respectively. Let k ≥ 2 and r > 0 be given integers. We prove that if H is a k-connected claw-free graph with α(H ) ≤ r , then either H is Hamiltonian or the Ryjácek's closure cl(H ) = L(G) where G can be contracted to a k-edge-connectedand G 0 does not have a dominating closed trail containing all the vertices that are obtained by contracting nontrivial subgraphs. As corollaries, we prove the following:(a) A 2-connected claw-free graph H with α(H ) ≤ 3 is either Hamiltonian or cl(H ) = L(G) where G is obtained from K 2,3 by adding at least one pendant edge on each degree 2 vertex; (b) A 3-connected claw-free graph H with α(H ) ≤ 7 is either Hamiltonian or cl(H ) = L(G) where G is a graph with α (G) = 7 that is obtained from the Petersen graph P by adding some pendant edges or subdividing some edges of P.Case (a) was first proved by Xu et al. [19]. Case (b) is an improvement of a result proved by Flandrin and Li [12]. For a given integer r > 0, the number of graphs of order at most max{4r − 5, 2r + 1} is fixed. The main result implies that improvements to case (a) or (b) by increasing the value of r and by enlarging the collection of exceptional graphs can be obtained with the help of a computer. Similar results involved degree or neighborhood conditions are also discussed.

The Independence Number Conditions for 2-Factors of a Claw-Free Graph

“…For results related to Hamiltonicity of claw-free graphs, the reader may refer to the literature; see [5].…”