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 ⌈n/r⌉ or ⌊n/r⌋. We prove that for all a ≥ 1, if the maximum degree of H satisfies Δ (H) ≤ ka, 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 nonsymmetric version of the Lovász local lemma.
ASJC Scopus subject areas
- Mathematics (all)