Cycle Extendability of Hamiltonian Interval Graphs

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.