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arXiv · 1909.00457

Equitable colorings of hypergraphs with few edges

Margarita Akhmejanova·Dmitry Shabanov
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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 $r<\sqrt[5]{\ln n}$.

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Margarita Akhmejanova, Dmitry Shabanov. 2019-09-01. Equitable colorings of hypergraphs with few edges. https://doi.org/10.1016/j.dam.2019.03.024

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