On ( 2 , k ) -Hamilton-connected graphs
Résumé
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 is （2,k)-Hamilton-connected if 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.