Extending cycles in graphs

Abstract

A cycle C in a graph G is extendable if there exists a cycle C' in G such that V(C)⊆V(C') and |V(C')| = |V(C)| + 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⩾3 has a pancyclic ordering if its vertices can be labelled v1, v2,…, vp so that the subgraph of G induced by v1, v2,…, vk contains a cycle of length k, for each k∈ {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.