Let G be a 2-connected claw-free graph on n vertices. For a vertex \(v\in V(G)\) and an integer \(r\ge 1\) , \(M^r(v)\) 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 \(d(v)\ge \frac{n-2}{3}\) for every vertex \(v\in V(G)\) . 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 \(d(v)\ge \frac{|M^r(v)|-2}{3}\) for every vertex \(v\in V(G)\) . 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 \(d(v)\ge \frac{|M^4(v)|-2}{3}\) 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 \(d(v)\ge \frac{n-2}{3}\) for every vertex v that is an end-vertex of an induced copy of a net.
