Toughness threshold for the existence of 2-walks in -minor-free graphs

We show that every K 4 -minor-free graph with toughness greater than 4 / 7 has a 2-walk, i.e., a closed walk visiting each vertex at most twice. We show that the bound cannot be improved by constructing a 4 / 7 -tough K 4 -minor-free graph with no 2-walk.