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. © 2000 John Wiley & Sons, Inc. J Graph Theory 33:125–137, 2000