In 1990, Hendry Conjectured that every Hamiltonian chordal graph is cycle extendable; that is, the vertices of any non-Hamiltonian cycle are contained in a cycle of length one greater. We disprove this conjecture by constructing counterexamples on n vertices for any n ≥ 15. Furthermore, we show that there exist counterexamples where the ratio of the length of a nonextendable cycle to the total number of vertices can be made arbitrarily small. We then consider cycle extendability in Hamiltonian chordal graphs where certain induced subgraphs are forbidden, notably Pn and the bull.
Introduction.All graphs considered here are simple, finite, and undirected. A graph is Hamiltonian if it has a cycle containing all vertices; such a cycle is a Hamiltonian cycle. A graph G on n vertices is pancyclic if G contains a cycle of length m for every integer 3 ≤ m ≤ n. Let C and C be cycles in G of length m and m + 1, respectively, such that V (C ) \ V (C) = {v}. We say that C is an extension of C and that C is extendable (or, C extends through v to C ). If every non-Hamiltonian cycle of G is extendable, then G is cycle extendable. If, in addition, every vertex of G is contained in a triangle, then G is fully cycle extendable. The study of pancyclic graphs was initiated by Bondy [3], who recognized that most of the sufficient conditions for Hamiltonicity known at the time in fact implied a more complex cycle structure. Hendry [12] introduced the concept of cycle extendability and proved that many known sufficient conditions for a graph to be pancyclic in fact were sufficient for a graph to be (fully) cycle extendable.Given a graph G and a set of vertices U ⊆ V (G), we denote by G[U ] the subgraph obtained by deleting from G all vertices except those in U ; G[U ] is the subgraph induced by U , and a subgraph of G is an induced subgraph if it is induced by some U ⊆ V (G). A graph is chordal if it contains no induced cycles of length 4 or greater. It is not hard to show that every Hamiltonian chordal graph is pancyclic (see Proposition 3.4); however, the question of whether not every Hamiltonian chordal graph is cycle extendable has remained open since 1990.Conjecture 1.1 (Hendry's Conjecture [12]). If G is a Hamiltonian chordal graph, then G is fully cycle extendable.In this paper, we settle Hendry's Conjecture in the negative. In section 2, we show that (a) for any n ≥ 15 there exists a counterexample to Hendry's Conjecture on n vertices and (b) for every real number α > 0 there exists a counterexample G with a nonextendable cycle C such that |V (C)| < α|V (G)|. The question then remains: *
