Edge-dominating cycles, k-walks and Hamilton prisms in $2K_2$-free graphs

TL;DR

This paper demonstrates polynomial-time algorithms for finding edge-dominating cycles and k-walks in $2K_2$-free graphs, proves a long-standing conjecture, and shows that certain tough $2K_2$-free graphs are prism-Hamiltonian.

Contribution

It provides the first polynomial-time algorithms for edge-dominating cycles and k-walks in $2K_2$-free graphs and confirms a conjecture about their Hamiltonian properties.

Findings

Polynomial-time algorithm for edge-dominating cycles in $2K_2$-free graphs.

Every 1/(k-1)-tough $2K_2$-free graph admits a k-walk.

$(1+\epsilon)$-tough $2K_2$-free graphs are prism-Hamiltonian.

Abstract

We show that an edge-dominating cycle in a $2K_2$-free graph can be found in polynomial time; this implies that every 1/(k-1)-tough $2K_2$-free graph admits a k-walk, and it can be found in polynomial time. For this class of graphs, this proves a long-standing conjecture due to Jackson and Wormald (1990). Furthermore, we prove that for any \epsilon>0 every (1+\epsilon)-tough $2K_2$-free graph is prism-Hamiltonian and give an effective construction of a Hamiltonian cycle in the corresponding prism, along with few other similar results.