The Hamilton-Connectivity with the Degree Sum of Non-adjacent Subgraphs of Claw-free Graphs

Abstract

The degree d(H) of a subgraph H of a graph G is $$|\underset{u \in V(H)}{\cup} N(u)-V(H)|$$ , where N(u) denotes the neighbor set of the vertex u of G. In this paper, we prove the following result on the condition of the degrees of subgraphs. Let G be a 2-connected claw-free graph of order n with minimum degree δ(G) ≥ 3. If for any three non-adjacent subgraphs H1, H2, H3 that are isomorphic to K1, K1, K2, respectively, there is d(H1) + d(H2) + d(H3) ≥ n + 3, then for each pair of vertices u,v ∈ G that is not a cut set, there exists a Hamilton path between u and v.