Toughness threshold for the existence of 2-walks in K4K4-minor-free graphs

We show that every K4K4-minor-free graph with toughness greater than 4/74/7 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 4/74/7-tough K4K4-minor-free graph with no 2-walk.