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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:10.7155/jgaa.00064 fatcat:qutjj4g3rbba3hjsji7dnazejy