A Fan-type condition for cycles in 1-tough and k-connected (P2 ∪ kP1)-free graphs

Abstract

For a graph G, define μk(G):=min⁡{maxx∈S⁡dG(x):S∈Sk}, where Sk is the set consisting of all independent sets {u1,…,uk} of G such that some vertex, say ui (1≤i≤k), is at distance two from every other vertex in it. A graph G is called 1-tough if for each cut set S⊆V(G), G−S has no more than |S| components. Recently, Shi and Shan [19] conjectured that for each integer k≥4, being 2k-connected is sufficient for 1-tough (P2∪kP1)-free graphs to be hamiltonian, which was confirmed by Xu et al. [20] and Ota and Sanka [16], respectively. In this article, we generalize the above results through the following Fan-type theorem: If G is a 1-tough and k-connected (P2∪kP1)-free graph and satisfies μk+1(G)≥7k−65, where k≥2 is an integer, then G is hamiltonian or the Petersen graph.