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Abstract. The b-chromatic number b(G) of a graph G = (V, E) is the largest integer k such that G admits a vertex partition into k independent sets Xi (i = 1, . . . , k) such that each Xi contains a vertex xi adjacent to at least one vertex of each Xj, j = i. We discuss on the tightness of some bounds on b(G) and on the complexity of determining b(G). We also determine the asymptotic behavior of b(Gn,p) for the random graph, within the accuracy of a multiplicative factor 2 + o(1) as n → ∞.

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A graph G is (a, b)-choosable if for any assignment of a list of a colors to each of its vertices there is a subset of b colors of each list so that subsets corresponding to adjacent vertices are disjoint. It is shown that for every graph G, the minimum ratio a/b where a, b range over all pairs of integers for which G is (a, b)-choosable is equal to the fractional chromatic number of G.

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For a graph G on vertex set V = {1, . . . , n} let k = (k 1 , . . . , k n ) be an integral vector suchFor k 1 = · · · = k n = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found by N. Alon and J. H. Spencer.If k i = d i for i = 1, . . . , n, then the notion of k-dominating set corresponds to the complement of an independent set. A function f k (p) is defined, and it will be proved that γ k (G) = min f k (p), where the minimum is taken over the n-dimensional cube C n = {p = (p 1 , . . . , p n ) | p i ∈ R, 0 6 p i 6 1, i = 1, . . . , n} . An O(∆ 2 2 ∆ n)-algorithm is presented, where ∆ is the maximum degree of G, with INPUT: p ∈ C n and OUTPUT: a k-dominating set D k of G with |D k | 6 f k (p).

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