On proper colorings of hypergraphs

Abstract

Let ℋ be a hypergraph with maximal vertex degree Δ such that each its hyperedge has at least δ vertices. Let k = [2Δ/δ]. We prove that ℋ admits a proper vertex coloring with k + 1 colors (i.e., such that any hyperedge contains at least two vertices of different colors). For k ≥ 3 and δ ≥ 3 we prove that ℋ admits a proper vertex coloring with k colors.For a graph G set k = [2Δ (G)/δ (G)]. As a corollary, we prove that there exists a dynamic coloring of the graph G with k + 1 colors in general and with k colors if δ(G) ≥ 3 and k ≥ 3. Bibliography: 16 titles. © 2012 Springer Science+Business Media, Inc.