On proper colorings of hypergraphs.

Let ℋ be a hypergraph with maximal vertex degree Δ such that each its hyperedge has at least δ vertices. Let $ k = \left\lceil {\frac{{2\Delta }}{\delta }} \right\rceil $. 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 = \left\lceil {\frac{{2\Delta (G)}}{{\delta (G)}}} \right\rceil $. 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. [ABSTRACT FROM AUTHOR]