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

TL;DR

This work analyzes Hamiltonicity, traceability, and toughness of coline graphs $co(G)$, the edge-complements of line graphs. Using a longest-cycle approach, it provides an alternate proof that every tough coline graph is Hamiltonian unless the root graph $G$ is one of $K_5$, $H_1$, $H_2$, or $H_3$, and it characterizes when coline graphs fail to be tough or traceable via the pseudo-tough framework and the $L^*$ construction. The paper also connects these results to cyclic matching sequenceability and presents detailed classifications of exceptional graphs, including explicit graphs and configurations that prevent Hamiltonicity or traceability. These findings deepen the understanding of how structural graph properties of $G$ translate into Hamiltonian and traceable properties of $co(G)$, with implications for graph reliability and the study of line/coline graphs.

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.

Figures (3)

• Figure 1: The graphs $H_1$, $H_2$, $H_3$
• Figure 3: The 18 exceptions of Corollary \ref{['c:exceptions']} with the largest exceptional subgraph in blue and a cutset $S$ of their coline graph $L$ such that $c(L-S)>|S|$ in green.
• Figure 4: The 9 exceptions of Corollary \ref{['c:Trace']}

Theorems & Definitions (23)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof