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

David Eppstein
2000 arXiv   pre-print
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 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).
arXiv:cs/0011009v1 fatcat:tcbdf2rbvnan5k7i7x2nx5ihpm