Hamiltonian chordal graphs are not cycle extendible

TL;DR

This paper disproves Hendry's 1990 conjecture that all Hamiltonian chordal graphs are cycle extendible by providing counterexamples for all sufficiently large graphs and exploring conditions involving forbidden subgraphs.

Contribution

It constructs the first known counterexamples to the conjecture for any n ≥ 15 and analyzes cycle extendibility under specific forbidden subgraph conditions.

Findings

Counterexamples exist for all n ≥ 15

Non-extendible cycles can be arbitrarily small proportion of the graph

Cycle extendibility varies with forbidden subgraphs

Abstract

In 1990, Hendry conjectured that every Hamiltonian chordal graph is cycle extendible; 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 $n \geq 15$. Furthermore, we show that there exist counterexamples where the ratio of the length of a non-extendible cycle to the total number of vertices can be made arbitrarily small. We then consider cycle extendibility in Hamiltonian chordal graphs where certain induced subgraphs are forbidden, notably $P_n$ and the bull.