Hamiltonian cycles in 2‐connected claw‐free‐graphs

Abstract

AbstractM. Matthews and D. Sumner have proved that of G is a 2‐connected claw‐free graph of order n such that δ ≧ (n − 2)/3, then G is hamiltonian. We prove that the bound for the minimum degree δ can be reduced to n/4 under the additional condition that G is not in F, where F is the set of all graphs defined as follows: any graph H in F can be decomposed into three vertex disjoint subgraphs H1, H2, H3 such that magnified image, where ui, vi ϵ V(Hi), uj vj ϵ V(Hj) 1 ϵ i ≦ j ≦ 3. Examples are given to show that the bound n/4 is sharp. © 1995 John Wiley & Sons, Inc.