Small Maximal Independent Sets and Faster Exact Graph Coloring [chapter]

David Eppstein
2006 Graph Algorithms and Applications 4  
We show that, for any n-vertex graph G and integer parameter k, there are at most 3 4k−n 4 n−3k maximal independent sets I ⊂ G with |I| ≤ k, and that all such sets can be listed in time O(3 4k−n 4 n−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 + 3 4/3 /4) n ) ≈ 2.4150 n , improving a previous O((1 + 3 1/3 ) n ) ≈ 2.4422 n algorithm of Lawler (1976).
doi:10.1142/9789812773296_0006 fatcat:ctme3hxfwrcsnbj22loviyzw3u