On the Hamiltonicity, traceability and toughness of complements of line graphs

TL;DR

This paper investigates the Hamiltonicity and related properties of complements of line graphs, providing new proofs and characterizations, especially focusing on tough coline graphs and their Hamiltonian cycles.

Contribution

It offers an alternative proof for existing characterizations of Hamiltonian coline graphs and introduces new results on tough coline graphs and those with Hamiltonian paths.

Findings

Tough coline graphs are Hamiltonian except for four specific cases.

Characterizations of coline graphs with Hamiltonian paths are provided.

The Petersen graph is identified as a non-Hamiltonian tough coline graph.

Abstract

A coline graph $\text{co}(G)$ of a graph $G$ is the graph with vertex set $E(G)$ for which two vertices $e$ and $e'$ of $\text{co}(G)$ are adjacent if and only if they are not adjacent as edges in $G$. A graph $G$ is tough if the number of connected components of $G-S$ is at most $|S|$ for all cut sets $S$. Wu and Meng, and Liu independently gave similar characterisations of coline graphs that are Hamiltonian. In this paper we give an alternate proof of Wu and Meng's and Liu's results using the longest cycle method. We in fact prove the following reformation of their results. A tough coline graph $\text{co}(G)$ is Hamiltonian unless $G$ is one of four examples, one of which is $K_5$, since $\text{co}(K_5)$ is the Petersen graph. Characterisations of tough coline graphs and coline graphs which contain a Hamiltonian path are also given.