Cycle Extendability of Hamiltonian Strongly Chordal Graphs

TL;DR

This paper investigates the cycle extendability of Hamiltonian strongly chordal graphs, providing negative answers to existing open questions and identifying new classes of graphs that are cycle extendable.

Contribution

It resolves key open questions about cycle extendability in Hamiltonian strongly chordal graphs and introduces two new classes of cycle extendable graphs.

Findings

Hamiltonian strongly chordal graphs are not cycle extendable.

Counterexamples exist for the conjecture in strongly chordal graphs.

Two new classes of cycle extendable graphs are identified.

Abstract

In 1990, Hendry conjectured that all Hamiltonian chordal graphs are cycle extendable. After a series of papers confirming the conjecture for a number of graph classes, the conjecture is yet refuted by Lafond and Seamone in 2015. Given that their counterexamples are not strongly chordal graphs and they are all only $2$-connected, Lafond and Seamone asked the following two questions: (1) Are Hamiltonian strongly chordal graphs cycle extendable? (2) Is there an integer $k$ such that all $k$-connected Hamiltonian chordal graphs are cycle extendable? Later, a conjecture stronger than Hendry's is proposed. In this paper, we resolve all these questions in the negative. On the positive side, we add to the list of cycle extendable graphs two more graph classes, namely, Hamiltonian $4$-\textsc{fan}-free chordal graphs where every induced $K_5 - e$ has true twins, and Hamiltonian $\{4\textsc{-fan}, \overline{A} \}$-free chordal graphs.