Long cycles through specified vertices

Let sigma(k)(G) = min {Sigma(k)(i=1) d(G)(v(i)) : {v(1), v(2),..., v(k)}is an independent set of G} for a graph G and an integer k >= 1. In 1991, Egawa et al. (1991) generalized a famous result on the existence of a hamiltonian cycle given by Ore (1960). In this paper, we will enhance the result of Egawa et al.. by providing the following result: Let G be a k-connected graph (k >= 2) and let S be an any vertex set with |S| <= k. Then G has a cycle of length at least min {|V(G)|, 2/k sigma(k)(G)} passing through S. (C) 2020 Elsevier B.V. All rights reserved.