Local Degree Conditions for Hamiltonicity of Claw-Free Graphs

Let G be a 2-connected claw-free graph on n vertices. For a vertex v∈V(G) and an integer r≥1, Mr(v) denotes the set of vertices of G whose distances from v do not exceed r. Matthews and Sumner in 1985 proved that G is hamiltonian if d(v)≥n-23 for every vertex v∈V(G). In this paper we pay attention to localize the above Matthews-Sumner’s degree condition by determining the minimum integer r such that G is hamiltonian if d(v)≥|Mr(v)|-23 for every vertex v∈V(G). While we conjecture that r=3 is best possible, we settle the case r=4. In fact, we obtain a strong result that G is hamiltonian if d(v)≥|M4(v)|-23 for every vertex v that is an end-vertex of an induced copy of a net, which is a graph obtained from a triangle by adding three disjoint pendant edges. This generalizes a result of Chen which states that G is hamiltonian if d(v)≥n-23 for every vertex v that is an end-vertex of an induced copy of a net.