On (2, k)-Hamilton-connected graphs

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 sigma 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 (5k - 4)-connected with sigma 2(G) >= n + k - 2 where k >= 2. We also prove that if sigma 2(G) >= n + k1k2 - 2 with k1, k2 >= 2, then G is (k1, k2)-Hamilton-connected. Moreover, these requirements of sigma 2 are tight. (c) 2023 Elsevier B.V. All rights reserved.