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) boolean AND 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) boolean AND 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). (C) 2008 Elsevier B.V. All rights reserved.