Equitable colorings of hypergraphs with limitation on vertex degree

The article deals with a problem of equitable colorings of uniform hypergraphs with bounded maximal vertex degrees. Let H = (V, E) be a hypergraph. A vertex r-coloring of the vertex set V is said to be equitable if each of the edges of E is not monochromatic under this coloring and moreover the cardinalities of every two color classes differ at most by one. We improve the bound for maximal vertex degree of a k-uniform hypergraph, which guarantees the existence of an equitable coloring with r > 2 colors provided that the number of vertices is large enough.