SIAM J. DISCRETE MATH. c  2006 Society for Industrial and Applied Mathematics Vol. 20, No. 3, pp. 682–689 CYCLE EXTENDABILITY OF HAMILTONIAN INTERVAL GRAPHS ∗ GUANTAO CHEN † , RALPH J. FAUDREE ‡ , RONALD J. GOULD § , AND MICHAEL S. JACOBSON ¶ Abstract. A graph G of order n is pancyclic if it contains cycles of all lengths from 3 to n. A graph is called cycle extendable if for every cycle C of less than n vertices there is another cycle C ∗ containing all vertices of C plus a single new vertex. Clearly, every cycle extendable graph is pancyclic if it contains a triangle. Cycle extendability has been intensively studied for dense graphs while little is known for sparse graphs, even very special graphs. We show that all Hamiltonian interval graphs are cycle extendable. This supports a conjecture of Hendry that all Hamiltonian chordal graphs are cycle extendable. Key words. interval graph, Hamiltonian, cycle extendable AMS subject classification. 05C38 DOI. 10.1137/S0895480104441450 1. Introduction. All graphs considered in this paper are finite and simple. We will generally follow the notation and definitions of West [14]. Let G be a graph. We use V (G) and E(G) to denote its vertex set and edge set, respectively. For any vertex v of G, N (v) (or N G (v)) denotes the neighborhood of v (neighborhood of v in G) and d(v) (or d G (v)) denotes the degree of v (degree of v in G). For any X ⊆ V (G), let G[X] denote the subgraph induced by X. If H is a subgraph of G, we define G[H] := G[V (H)]. A graph is chordal if every cycle of length at least 4 contains a chord. An interval graph is a graph whose vertices correspond to a family of intervals so that vertices are adjacent if and only if the corresponding intervals intersect. It is well known that all interval graphs are chordal graphs. In a graph G,a Hamiltonian cycle is a cycle containing all vertices of G. A graph is Hamiltonian if it has a Hamiltonian cycle. Determining when graphs are Hamiltonian is one of the fundamental problems in graph theory. Although it is NP- hard to decide whether a graph is Hamiltonian, finding conditions sufficient to imply a graph is Hamiltonian has been intensively studied in the last thirty years. While studying Hamiltonicity, many related properties have also been heavily explored. For example, a graph G of order n is pancyclic if it contains cycles of all lengths from 3 to n. Clearly, every pancyclic graph is Hamiltonian, but the converse is not true. Being pancyclic provides a lot more cycle structure to graphs. Although there are many ∗ Received by the editors February 21, 2004; accepted for publication (in revised form) October 17, 2005; published electronically September 15, 2006. http://www.siam.org/journals/sidma/20-3/44145.html † Department of Computer Science and Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, and Faculty of Mathematics and Statistics, Huazhong Normal Uni- versity, Wuhan, China (gchen@gsu.edu). The research of this author was partially supported by NSA grant H98230-04-1-0300 and NSF grant DMS-0500951. ‡ University of Memphis, Memphis, TN 38152 (rfaudree@memphis.edu). § Department of Math and Computer Science, Emory University, Atlanta, GA 30322 (rg@mathcs. emory.edu). ¶ Department of Mathematics, University of Colorado at Denver, Denver, CO 80217-3364 (msj@ math.cudenver.edu). 682