On [formula omitted]-Hamilton-connected graphs.

A graph G is called (k 1 , k 2) -Hamilton-connected, if for any two vertex disjoint subsets X = { x 1 , x 2 , ... , x k 1 } and U = { u 1 , u 2 , ... , u k 2 } , G contains a spanning family F of k 1 k 2 internally vertex disjoint paths such that for 1 ≤ i ≤ k 1 and 1 ≤ j ≤ k 2 , F contains an x i u j path. Let σ 2 (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 (5 k − 4) -connected with σ 2 (G) ≥ n + k − 2 where k ≥ 2. We also prove that if σ 2 (G) ≥ n + k 1 k 2 − 2 with k 1 , k 2 ≥ 2 , then G is (k 1 , k 2) -Hamilton-connected. Moreover, these requirements of σ 2 are tight. • This paper is motivated by k -fan connected graphs, we define (k 1 , k 2) -Hamilton-connected graphs. • This paper gives a sufficient condition for a graph to be (2 , k) -Hamilton-connected. • The results provide tight, sufficient, σ 2 (G) -conditions. • We extend the properties of Hamilton-connectedness and being k -linked. [ABSTRACT FROM AUTHOR]