Small Maximal Independent Sets and Faster Exact Graph Coloring

Abstract

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) ≈ 2.4150ⁿ, improving a previous O((1+3^1/3)ⁿ) ≈ 2.4422ⁿ algorithm of Lawler (1976).