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-n4^n-3k maximal independent sets I ⊂ G with |I| ≤ k, and that all such sets can be listed in time $\mathcal{O}(3^{4k - n} 4^{n - 3k} )$. 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 $\mathcal{O}((4/3 + 3^{4/3} /4)^n ) \approx 2.4150^n$, improving a previous $\mathcal{O}((1/3^{1/3} )^n ) \approx 2.4422^n$ lgorithm of Lawler (1976).