A new sufficient condition for pancyclability of graphs

Let G be a graph of order n and S be a set of s vertices. We call G, S-pancyclable, if for every i with 3≤i≤s, 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 2 in S, denoted by dS(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: Let G be a 2-connected graph of order n and S be a subset of V(G) with |S|≥3. If max{d(u),d(v)}≥n/2 for all pairs of vertices u, v of S with dS(u,v)=2, then G is S-pancyclable or else |S|=4r and G[S] is a spanning subgraph of F4r, or else |S|=n is even and G is the complete bipartite graph Kn/2,n/2, or else |S|=n≥6 is even and G is Kn/2,n/2′, or else G[S]=K2,2 and the structure of G is well characterized. This generalizes a result of Benhocine and Wojda for the case when S=V(G). [A. Benhocine, A.P. Wojda, The Geng-Hua Fan conditions for pancyclic or Hamilton-connected graph, J. Combin. Theory B 42 (1987) 167-180].