Irregularity Strength of Uniform Hypergraphs

Abstract

A hypergraph is irregular if no two of its vertices have the same degree. It is shown that for all \(r \geq 3\) and \(n \geq r + 3\), there exist irregular \(r\)-uniform hypergraphs of order \(n\). For \(r \geq 6\) it is proved that almost all \(r\)-uniform hypergraphs are irregular. A linear upper bound is given for the irregularity strength of hypergraphs of order \(n\) and fixed rank. Furthermore, the irregularity strength of complete and complete equipartite hypergraphs is determined.