Small Maximal Independent Sets and Faster Exact Graph Coloring

  title={Small Maximal Independent Sets and Faster Exact Graph Coloring},
  author={David Eppstein},
  journal={J. Graph Algorithms Appl.},
We show that, for anyn-vertex graphG and integer parameter k, there are at most 3 4k−n4n−3k maximal independent sets I ⊂ G with |I| ≤ k, and that all such sets can be listed in time O(34k−n4n−3k). These bounds are tight when n/4 ≤ k ≤ n/3. As a consequence, we show how to compute the exact chromatic number of a graph in time O((4/3 + 34/3/4)n) ≈ 2.4150n, improving a previous O((1 + 31/3)n) ≈ 2.4422n algorithm of Lawler (1976). 
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