Toughness and spanning trees in K4mf graphs

For an integer k, a k-tree is a tree with maximum degree at most k. More generally, if f is an integer-valued function on vertices, an f -tree is a tree in which each vertex v has degree at most f(v). Let c(G) denote the number of components of a graph G. We show that if G is a connected K4-minor-free graph and c(G− S) ≤ ∑ v∈S (f(v)− 1) for all S ⊆ V (G) with S 6= ∅ then G has a spanning f -tree. Consequently, if G is a 1 k−1 -tough K4-minor-free graph, then G has a spanning k-tree. These results are stronger than results for general graphs due to Win (for k-trees) and Ellingham, Nam and Voss (for f -trees). The K4-minorfree graphs form a subclass of planar graphs, and are identical to graphs of treewidth at most 2, and also to graphs whose blocks are series-parallel. We provide examples to show that the inequality above cannot be relaxed by adding 1 to the right-hand side, and also to show that our result does not hold for general planar graphs. Our proof uses a technique where we incorporate toughness-related information into weights associated with vertices and cutsets.