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

The degree d(H) of a subgraph H of a graph G is |∪u∈V(H)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.