An Improved Exact Algorithm for the Domatic Number Problem

The 3-domatic number problem asks whether a given graph can be partitioned into three dominating sets. We prove that this problem can be solved by a deterministic algorithm in time 2.695ⁿ (up to polynomial factors). This result improves the previous bound of 2.8805 ⁿ, which is due to Fomin et al. (2005). To prove our result, we combine an algorithm by Fomin et al. (2005) with Yamamoto´s algorithm for the satisfiability problem. In addition, we show that the 3-domatic number problem can be solved for graphs G with bounded maximum degree Delta (G) by a randomized algorithm, whose running time is better than the previous bound due to Riege and Rothe (2005) whenever Delta(G) > 5. Our new randomized algorithm employs Schoning´s approach to constraint satisfaction problems by U. Schoning (1999)