Delegate and conquer: an LP-based approximation algorithm for minimum degree MSTs
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
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A graph is t-tough if the number of components ofG\S is at most |S|-t for every cutsetS ⊆ V (G). A k-walk in a graph isa spanning closed walk using each vertex at most k times.When k = 1, a 1-walk is a Hamilton cycle, and a longstandingconjecture by Chvátal is that every sufficiently tough graphhas a 1-walk. When k ≥ 3, Jackson and Wormald used aresult of Win to show that every sufficiently tough graph has ak-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 firstprovide a new proof for and generalize a result by Win on theexistence of a k-tree, a spanning tree with every vertex ofdegree at most k. We also provide new examples of toughgraphs with no k-walk for k ≥ 2. © 2000 JohnWiley & Sons, Inc. J Graph Theory 33:125137, 2000