Extending perfect matchings to Hamiltonian cycles in line graphs

TL;DR

This paper investigates conditions under which the line graph of a given graph has the property that every perfect matching can be extended to a Hamiltonian cycle, providing new sufficient conditions for this property.

Contribution

It establishes new sufficient conditions for the line graph of a graph to have the PMH-property, including cases for Hamiltonian graphs with degree at most 3, complete graphs, and traceable graphs.

Findings

Line graphs of Hamiltonian graphs with max degree 3 have the PMH-property.

Complete graphs' line graphs have the PMH-property.

Traceable graphs' line graphs have the PMH-property.

Abstract

A graph admitting a perfect matching has the Perfect-Matching-Hamiltonian property (for short the PMH-property) if each of its perfect matchings can be extended to a Hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH-property. In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most $3$, (ii) a complete graph, or (iii) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.