The Lovász Local Lemma is a tool that enables one to show that certain events hold with pos-itive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for converting some of these existence proofs into efficient algorithmic procedures, at the cost of loosing a little in the estimates. His method does not seem to be parallelizable. Here we modify his technique and achieve an algorithmic version that can be parallelized, thus obtaining deterministic NC 1 algorithms for several interesting algorithmic problems.

Numerous problems in Graph Theory and Combinatorics can be formulated in terms of the existence of certain colorings of graphs or hypergraphs. Many of these problems can be solved or partially solved by applying probabilistic arguments. In this paper we discuss several examples that illustrate the methods used. This is mainly a survey paper, but it contains some new results as well.

A digraph is m-labelled if every arcs is labelled by an integer in {1,..., m}. Motivated by wavelength assignment for multicasts in optical star networks, we study n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v, for each colour in λ, in(v, λ) + out(v, λ) ≤ n with in(v, λ) the number of arcs coloured λ entering v and out(v, λ) the number of labels l such that there exists an arc leaving v coloured λ. One likes to find the minimum number of colours λn(D) such that an m-labbelled digraph D has an n-fiber colouring. In the particular case, when D is 1-labelled then λn(D) is the directed star arboricty of D, denoted dst(D). We first show that dst(D) ≤ 2 ∆ − (D) + 1 and conjecture that if ∆ − (D) ≥ 2 then dst(D) ≤ 2 ∆ − (D). We also prove that if D is subcubic then dst(D) ≤ 3 and that if ∆ + (D), ∆ − (D) ≤ 2 then dst(D) ≤ 4. Finally, we study λn(m, k) = max{λn(D) | D is m-labelled and ∆ − ‰ ‰ ı (D) ≤ k}. We show that if m ≥ n m k then + n n k ı ‰ ‰ ı m k ≤ λn(m, k) ≤ +

In a directed graph, a star is an arborescence with at least one arc, in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. In this paper, we consider the Spanning Galaxy Problem of deciding whether a digraph D has a spanning galaxy or not. We show that although this problem is NP-complete (even when restricted to acyclic digraphs), it becomes polynomial-time solvable when restricted to strongly connected digraphs. We prove indeed that in the strongly connected case, the problem is equivalent to find a strong subgraph with an even number of vertices. As a consequence of this work, we improve some results concerning the notion of directed star arboricity of a digraph D, which is the minimum number of galaxies needed to cover all the arcs of D. We show in particular that dst(D) ≤ ∆(D) + 1 for every digraph D and that dst(D) ≤ ∆(D) for every acyclic digraph D.