A Twelve Vertex Theorem for 3-Connected Claw-Free Graphs

Abstract

The cyclability of a graph H, denoted by C(H), is the largest integer r such that H has a cycle through any r vertices. For a claw-free graph H, by Ryjaă?ek (J Comb Theory Ser B 70:217---224, 1997) closure concept, there is a $$K_3$$K3-free graph G such that the closure $$cl(H)=L(G)$$cl(H)=L(G). In this note, we prove that for a 3-connected claw-free graph H with its closure $$cl(H)=L(G)$$cl(H)=L(G), $$C(H)\ge 12$$C(H)?12 if and only if G can not be contracted to the Petersen graph in such a way that each vertex in P is obtained by contracting a nontrivial connected $$K_3$$K3-free subgraph. This is an improvement of the main result in Gyori and Plummer (Stud Sci Math Hung 38:233---244, 2001).