We show that an edge-dominating cycle in a 2K2-free graph can be found in polynomial time; this implies that every 1 k−1 -tough 2K2free 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 ǫ > 0 every (1 + ǫ)-tough 2K2-free graph is prism-Hamiltonian and give an effective construction of a Hamiltonian cycle in the corresponding prism, along with few other similar results.

We prove that every 2-tough 2K 2 -free graph admits a 2-walk.