Further results on Hendry's Conjecture

TL;DR

This paper investigates Hendry's Conjecture in graph theory, demonstrating its failure under various conditions and exploring related models, thereby deepening understanding of cycle extendibility in complex graphs.

Contribution

The paper extends the disproof of Hendry's Conjecture to strongly chordal graphs, highly connected graphs, and under relaxed definitions, also analyzing subtree intersection models.

Findings

Hendry's Conjecture fails for strongly chordal graphs.

The conjecture does not hold in highly connected graphs.

A subtree intersection model result is nearly optimal.

Abstract

Recently, a conjecture due to Hendry was disproved which stated that every Hamiltonian chordal graph is cycle extendible. Here we further explore the conjecture, showing that it fails to hold even when a number of extra conditions are imposed. In particular, we show that Hendry's Conjecture fails for strongly chordal graphs, graphs with high connectivity, and if we relax the definition of "cycle extendible" considerably. We also consider the original conjecture from a subtree intersection model point of view, showing that a result of Abuieda et al is nearly best possible.