Abstract
We show that, for any n- vertex graph G and integer parameter k, there are at most 34k-n4n-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).
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Eppstein, D. (2001). Small Maximal Independent Sets and Faster Exact Graph Coloring. In: Dehne, F., Sack, JR., Tamassia, R. (eds) Algorithms and Data Structures. WADS 2001. Lecture Notes in Computer Science, vol 2125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44634-6_42
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DOI: https://doi.org/10.1007/3-540-44634-6_42
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