Spanning trails with maximum degree at most 4 in $2K_2$-free graphs

TL;DR

This paper confirms a conjecture that certain tough $2K_2$-free graphs always have a spanning trail with maximum degree at most 4, and provides counterexamples for graphs with lower toughness.

Contribution

It proves Mou and Pasechnik's conjecture for $rac{3}{2}$-tough $2K_2$-free graphs and constructs examples showing the conjecture does not hold for toughness less than $rac{5}{4}$.

Findings

Confirmed the conjecture for $rac{3}{2}$-tough graphs.

Provided counterexamples for $t$-tough graphs with $t<rac{5}{4}$.

Established bounds on toughness for the existence of such spanning trails.

Abstract

A graph is called $2K_2$-free if it does not contain two independent edges as an induced subgraph. Mou and Pasechnik conjectured that every $\frac{3}{2}$-tough $2K_2$-free graph with at least three vertices has a spanning trail with maximum degree at most $4$. In this paper, we confirm this conjecture. We also provide examples for all $t < \frac{5}{4}$ of $t$-tough graphs that do not have a spanning trail with maximum degree at most $4$.