Toughness and spanning trees in K4mf graphs

A k-tree is a tree with maximum degree at most k, and a k-walk is a closed walk with each vertex repeated at most k times. A k-walk can be obtained from a k-tree by visiting each edge of the k-tree twice. Jackson and Wormald conjectured in 1990 that for k ≥ 2, every 1 k−1 -tough connected graph contains a spanning k-walk. This conjecture is open even for planar graphs. We confirm this conjecture for K4-minor-free graphs, an important subclass of planar graphs including series-parallel graphs. We first prove a general result for K4-minor-free graphs on the existence of spanning trees with a specified maximum degree for each vertex, given a condition on the number of components obtained when we delete a set of vertices. We provide examples for which this condition is best possible. It then follows that for k ≥ 2, every 1 k−1 -tough K4-minor-free graph has a spanning k-tree, and hence a spanning k-walk. Our main proof uses a technique where we incorporate toughness-related information into weights associated with vertices and cutsets.