A sufficient condition for pancyclability of graphs

Let GG be a graph of order nn and SS be a vertex set of qq vertices. We call G,SG,S-pancyclable, if for every integer ii with 3≤i≤q3≤i≤q there exists a cycle CC in GG such that |V(C)∩S|=i|V(C)∩S|=i. For any two nonadjacent vertices u,vu,v of SS, we say that u,vu,v are of distance two in SS, denoted by dS(u,v)=2dS(u,v)=2, if there is a path PP in GG connecting uu and vv such that |V(P)∩S|≤3|V(P)∩S|≤3. In this paper, we will prove that if GG is 2-connected and for all pairs of vertices u,vu,v of SS with dS(u,v)=2dS(u,v)=2, max{d(u),d(v)}≥n2, then there is a cycle in GG containing all the vertices of SS. Furthermore, if for all pairs of vertices u,vu,v of SS with dS(u,v)=2dS(u,v)=2, max{d(u),d(v)}≥n+12, then GG is SS-pancyclable unless the subgraph induced by SS 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)S=V(G).