Equitable colorings of hypergraphs with few edges

TL;DR

This paper establishes conditions under which sparse uniform hypergraphs can be equitably colored with a given number of colors, extending understanding of hypergraph coloring in extremal cases.

Contribution

It proves that hypergraphs with a limited number of edges are equitably r-colorable when r is below a certain threshold, advancing extremal hypergraph coloring theory.

Findings

Hypergraphs with few edges are equitably r-colorable.

The threshold for r depends on the number of vertices and edges.

Provides bounds for equitable colorability in sparse hypergraphs.

Abstract

The paper deals with an extremal problem concerning equitable colorings of uniform hyper\-graph. Recall that a vertex coloring of a hypergraph $H$ is called proper if there are no monochro-matic edges under this coloring. A hypergraph is said to be equitably $r$-colorable if there is a proper coloring with $r$ colors such that the sizes of any two color classes differ by at most one. In the present paper we prove that if the number of edges $|E(H)|\leq 0.01\left(\frac{n}{\ln n}\right)^{\frac {r-1}{r}}r^{n-1}$ then the hypergraph $H$ is equitably $r$-colorable provided $r<\sqrt[5]{\ln n}$.