A cycle C in a graph G is extendable if there exists a cycle C' in G such that V(C) E V(C') and jV(C')l = IV(C)
1 + 1. A graph G is cycle extendable if G has at least one cycle and every nonhamiltonian cycle is extendable. A graph G of order p 2 3 has a pancyclic ordering if its vertices can be labelled ur, uz, . . , II,, so that the subgraph of G induced by ur, u2, . , uk contains a cycle of length k, for each k E {3,4, . . , p}. The theme of this paper is to investigate to what extent known sufficient conditions for a graph to be hamiltonian imply the extendability of cycles. A number of theorems and conjectures are stated. For example, it is shown that if C is a nonextendable cycle in a graph satisfying Ore's sufficient condition for a hamiltonian cycle then the subgraph induced by the vertices of C is either a complete graph or a regular complete bipartite graph. Results are also given relating to extremal problems, stability, graphs with forbidden induced subgraphs, squares of graphs and chordal graphs.
Definition. A cycle C in a graph G is extendable (in G) if there exists a cycle C' in G such that V(C) E V(C') and IV(C')l = IV(C)1 + 1. If such a cycle C' exists we will say that C can be extended to C' or that C' is an extension of C.