A sufficient condition for pancyclability of graphs

Let G be a graph of order n and S be a vertex set of q vertices. We call G , S -pancyclable, if for every integer i with 3 ≤ i ≤ q there exists a cycle C in G such that | V ( C ) ∩ S | = i . For any two nonadjacent vertices u , v of S , we say that u , v are of distance two in S , denoted by d S ( u , v ) = 2 , if there is a path P in G connecting u and v such that | V ( P ) ∩ S | ≤ 3 . In this paper, we will prove that if G is 2-connected and for all pairs of vertices u , v of S with d S ( u , v ) = 2 , max { d ( u ) , d ( v ) } ≥ n 2 , then there is a cycle in G containing all the vertices of S . Furthermore, if for all pairs of vertices u , v of S with d S ( u , v ) = 2 , max { d ( u ) , d ( v ) } ≥ n + 1 2 , then G is S -pancyclable unless the subgraph induced by S is in a class of special graphs. This generalizes a result of Fan [G. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory B 37 (1984) 221–227] for the case when S = V ( G ) .