Approximating the domatic number
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Let G be an undirected graph with a set of vertices V and a set of edges E. Given an integer r, we assign at most r labels (representing "resources") to each vertex. We say that such an assignment is an r-configuration if, for each label c, the vertices labeled by c form a dominating set for G. In this paper, we are interested in the maximum number, Dr(G), of labels that can be assigned to the vertices of graph G by an r-configuration. The decision problem ``$D_1(G)\geq K$?" (known as the domatic number problem) is NP-complete.We first investigate Dr(G) for general graphs and establish bounds on Dr(G) in terms of D1(G) and the minimum vertex degree of G. We then discuss Dr(G) for d-regular graphs. We clearly have $D_r (G) \leq r(d+1)$. We show that the problem of testing if Dr(G) = r(d+1) is solvable in polynomial time for d-regular graphs with |V| = 2(d+1), but is NP-complete for those with |V| = a(d+1) for some integer $a \geq 3$. Finally, we discuss cubic (i.e., 3-regular) graphs. It is easy to show $2r \leq D_r (G) \leq 4r$ for cubic graphs. We show that the decision problem for D1 (G) = K is co-NP-complete for K = 2 and is NP-complete for K = 4. Although there are many cubic graphs G with D1 (G) = 2, surprisingly, every cubic graph has a 2-configuration with five labels, i.e., $D_2 (G) \geq 5$ and such a 2-configuration can be constructed in polynomial time. We use this fact to show $D_r (G) \geq \lfloor 5r/2 \rfloor$ in general.