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 [email protected][email protected]?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)}>=n2, 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+12, 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).