Equitable colorings of hypergraphs with few edges
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 $r<\sqrt[5]{\ln n}$.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2019
- DOI:
- 10.48550/arXiv.1909.00457
- arXiv:
- arXiv:1909.00457
- Bibcode:
- 2019arXiv190900457A
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- \emph{Discrete Applied Mathematics}, (2019), https://doi.org/10.1016/j.dam.2019.03.024