Toughness, trees, and walks

A graph is t-tough if the number of components of G\S is at most |S|/t for every cutset S ⊆ V (G). A k-walk in a graph is a spanning closed walk using each vertex at most k times. When k = 1, a 1-walk is a hamilton cycle, and a longstanding conjecture by Chvátal is that every sufficiently tough graph has a 1-walk. When k ≥ 3, Jackson and Wormald used a result of Win to show that every sufficiently tough graph has a k-walk. We fill in the gap between k = 1 and k ≥ 3 by showing that when k = 2 every sufficiently tough (specifically, 4-tough) graph has a 2-walk. To do this we first provide a new proof for and generalize a result by Win on the existence of a k-tree, a spanning tree with every vertex of degree at most k. We also provide new examples of tough graphs with no k-walk for k ≥ 2. * Supported by NSF Grant Number DMS-9622780