On the Chromatic Index of Random Uniform Hypergraphs

Let H(n,N), where k ≥ 2, be a random hypergraph on the vertex set [n] = {1, 2, . . . , n} with N edges drawn independently with replacement from all subsets of [n] of size k. For d̄ = kN/n and any ε > 0 we show that if k = o(ln(d̄/ lnn)) and k = o(ln(n/ ln d̄)), then with probability 1−o(1) a random greedy algorithm produces a proper edge-colouring of H(n,N) with at most d̄(1+ε) colours. This yields the asymptotic chromatic number of the corresponding uniform random intersection graph. keywords: random uniform hypergraph, chromatic index, random intersection graph, greedy algorithm.