Regular homogeneously traceable nonhamiltonian graphs

TL;DR

This paper investigates the existence of regular homogeneously traceable nonhamiltonian graphs, providing constructions for cubic and 4-regular cases, and highlighting open problems in the area.

Contribution

It proves the existence of cubic and 4-regular homogeneously traceable nonhamiltonian graphs for certain orders, advancing understanding of their structure.

Findings

Existence of cubic homogeneously traceable nonhamiltonian graphs for even n ≥ 10.

Existence of 4-regular homogeneously traceable graphs with circumference p-4 for p ≥ 18.

Poses open problems related to regular homogeneously traceable nonhamiltonian graphs.

Abstract

A graph is called homogeneously traceable if every vertex is an endpoint of a Hamilton path. In 1979 Chartrand, Gould and Kapoor proved that for every integer $n\ge 9,$ there exists a homogeneously traceable nonhamiltonian graph of order $n.$ The graphs they constructed are irregular. Thus it is natural to consider the existence problem of regular homogeneously traceable nonhamiltonian graphs. We prove two results: (1) For every even integer $n\ge 10,$ there exists a cubic homogeneously traceable nonhamiltonian graph of order $n;$ (2) for every integer $p\ge 18,$ there exists a $4$-regular homogeneously traceable graph of order $p$ and circumference $p-4.$ Unsolved problems are posed.