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• W2069323956 abstract "We resolve the problem posed as the main open question in [4]: letting δ(G), ∆(G) and D(G) respectively denote the minimum degree, maximum degree, and domatic number (defined below) of an undirected graph G = (V,E), we show that D(G) ≥ (1−o(1))δ(G)/ ln(∆(G)), where the “o(1)” term goes to zero as ∆(G) → ∞. A dominating set of G is any set S ⊆ V such that for all v ∈ V , either v ∈ S or some neighbor of v is in S. A domatic partition of V is a partition of V into dominating sets, and the number of these dominating sets is called the size of such a partition. The domatic number D(G) of G is the maximum size of a domatic partition; it is NP-hard to find a maximumsized domatic partition. This is a very well-studied problem especially for various special classes of (perfect) graphs: see, e.g., [2, 6, 7] and the references in [4]. Recent interesting work of [4] has given the first non-trivial approximation algorithm for the domatic partition problem, whose approximation guarantee is also shown to be essentially best-possible in [4]. Let n = |V |, δ = δ(G), and ∆ = ∆(G). It is easy to check that D(G) ≤ δ + 1. An efficient algorithm to find a domatic partition of size (1 − o(1))δ/ lnn is shown in [4], where the o(1) term goes to zero as n increases; thus, this is a (1 + o(1)) lnn approximation. It is also shown in [4] that for any fixed > 0, an (1 − ) lnnapproximation algorithm for D(G) would imply that NP ⊆ DTIME[n log ]; hence such an algorithm appears unlikely. An interesting point is that this seems to be the first natural maximization problem proven to have a Θ(logn) approximation threshold. Can we say something better for sparse graphs? It is shown in [4] that D(G) ≥ (1 − o(1))δ/(3 ln ∆), where the o(1) term is a function of ∆ that goes to zero as ∆ increases. (Among the very few such lower bounds known before was that D(G) ≥ dn/(n− δ)e [8]. This is relevant primarily for very dense graphs. For instance, when 1 ≤ δ ≤ n/2, this bound says that D(G) ≥ 2; however, D(G) ≥ 2 is readily seen to hold if (and only" @default.
• W2069323956 created "2016-06-24" @default.
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• W2069323956 date "2001-01-09" @default.
• W2069323956 modified "2023-10-01" @default.
• W2069323956 title "Domatic partitions and the Lovász local lemma" @default.
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• W2069323956 doi "https://doi.org/10.5555/365411.365810" @default.
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