The Geng-Hua Fan conditions for pancyclic or Hamilton-connected graphs
Journal of Combinatorial Theory Series B
A cycle structure theorem for Hamiltonian graphs
Journal of Combinatorial Theory Series A
Cycles through prescribed vertices with large degree sum
Discrete Mathematics
Sequences, claws and cyclability of graphs
Journal of Graph Theory
Journal of Combinatorial Theory Series B
Graph Theory With Applications
Graph Theory With Applications
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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 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: 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 d"S(u,v)=2, then G is S-pancyclable or else |S|=4r and G[S] is a spanning subgraph of F"4"r, or else |S|=n is even and G is the complete bipartite graph K"n"/"2","n"/"2, or else |S|=n=6 is even and G is K"n"/"2","n"/"2^', or else G[S]=K"2","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].