On almost-regular edge colourings of hypergraphs
Summary: We prove that if \(\mathcal{H}=(V(\mathcal{H}),\mathcal{E}(\mathcal{H}))\) is a hypergraph, \(\gamma\) is an edge colouring of \(\mathcal{H}\), and \(S\subseteq V(\mathcal{H})\) such that any permutation of \(S\) is an automorphism of \(\mathcal{H}\), then there exists a permutation \(\pi\) of \(\mathcal{E}(\mathcal{H})\) such that \(|\pi(E)|=|E|\) and \(\pi(E)\backslash S=E\backslash S\) for each \(E\in\mathcal{H}(\mathcal{H})\), and such that the edge colouring \(\gamma'\) of \(\mathcal{H}\) given by \(\gamma'(E)=\gamma(\pi^{-1}(E))\) for each \(E\in\mathcal{E}(\mathcal{H})\) is almost regular on \(S\). The proof is short and elementary. We show that a number of known results, such as Baranyai's theorem on almost-regular edge colourings of complete \(k\)-uniform hypergraphs, are easy corollaries of this theorem.
scientific article; zbMATH DE number 6661359
scientific article