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 degreeδ(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.

The degree d（H） of a subgraph H of a graph G is ■, 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.