A Note on Random Greedy Coloring of Uniform Hypergraphs

The smallest number of edges forming an n-uniform hypergraph which is not r-colorable is denoted by m(n, r). Erdos and Lovasz conjectured that m(n, 2) = circle minus (n2(n)). The best known lower bound m(n, 2) = Omega(root n/ln(n)2(n)) was obtained by Radhakrishnan and Srinivasan in 2000. We present a simple proof of their result. The proof is based on the analysis of a random greedy coloring algorithm investigated by Pluhar in 2009. The proof method extends to the case of r-coloring, and we show that for any fixed r we have m(n, r) = Omega((n/ln(n))((r-1)/r) r(n)) improving the bound of Kostochka from 2004. We also derive analogous bounds on minimum edge degree of an n-uniform hypergraph that is not r-colorable. (C) 2014Wiley Periodicals, Inc.