On [formula omitted]-Hamilton-connected graphs

Abstract

A graph G is called (k1,k2)-Hamilton-connected, if for any two vertex disjoint subsets X={x1,x2,…,xk1} and U={u1,u2,…,uk2}, G contains a spanning family F of k1k2 internally vertex disjoint paths such that for 1≤i≤k1 and 1≤j≤k2, F contains an xiuj path. Let σ2(G) be the minimum value of deg(u)+deg(v) over all pairs {u,v} of non-adjacent vertices in G. In this paper, we prove that an n-vertex graph G is (2,k)-Hamilton-connected if G is (5k−4)-connected with σ2(G)≥n+k−2 where k≥2. We also prove that if σ2(G)≥n+k1k2−2 with k1,k2≥2, then G is (k1,k2)-Hamilton-connected. Moreover, these requirements of σ2 are tight.