On harmonious coloring of hypergraphs

TL;DR

This paper investigates harmonious colorings of hypergraphs, providing a new proof for an upper bound on the harmonious number using the local cut lemma, which advances understanding of hypergraph colorings.

Contribution

It offers a novel proof of an upper bound on the harmonious number of hypergraphs employing the local cut lemma, enhancing theoretical bounds in hypergraph coloring.

Findings

Established an upper bound h(H)=O(√[k]{k!m}) for the harmonious number.

Applied the local cut lemma of A. Bernshteyn to hypergraph coloring.

Provided a new proof technique for bounds on harmonious colorings.

Abstract

A harmonious coloring of a $k$-uniform hypergraph $H$ is a vertex coloring such that no two vertices in the same edge have the same color, and each $k$-element subset of colors appears on at most one edge. The harmonious number $h(H)$ is the least number of colors needed for such a coloring. The paper contains a new proof of the upper bound $h(H)=O(\sqrt[k]{k!m})$ on the harmonious number of hypergraphs of maximum degree $\Delta$ with $m$ edges. We use the local cut lemma of A. Bernshteyn.