Equitable Coloring of k-Uniform Hypergraphs

Let H be a k-uniform hypergraph with n vertices. A strong r-coloring is a partition of the vertices into r parts, such that each edge of H intersects each part. A strong r-coloring is called equitable if the size of each part is dn/re or bn/rc. We prove that for all a ≥ 1, if the maximum degree of H satisfies ∆(H) ≤ k then H has an equitable coloring with k a ln k (1 − ok(1)) parts. In particular, every k-uniform hypergraph with maximum degree O(k) has an equitable coloring with k ln k (1−ok(1)) parts. The result is asymptotically tight. The proof uses a double application of the non-symmetric version of the Lovász Local Lemma. Ams classification code: 05C15