On almost-regular edge colourings of hypergraphs

Darryn Bryant¹

¹University of Queensland

The electronic journal of combinatorics, Tome 23 (2016) no. 4

Résumé

We prove that if ${\cal{H}}=(V({\cal{H}}),{\cal{E}}({\cal{H}}))$ is a hypergraph, $\gamma$ is an edge colouring of ${\cal{H}}$, and $S\subseteq V({\cal{H}})$ such that any permutation of $S$ is an automorphism of ${\cal{H}}$, then there exists a permutation $\pi$ of ${\cal{E}}({\cal{H}})$ such that $|\pi(E)|=|E|$ and $\pi(E)\setminus S=E\setminus S$ for each $E\in{\cal{H}}({\cal{H}})$, and such that the edge colouring $\gamma'$ of ${\cal{H}}$ given by $\gamma'(E)=\gamma(\pi^{-1}(E))$ for each $E\in{\cal{E}}({\cal{H}})$ 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.

Détail

DOI : 10.37236/5826

Classification : 05C15, 05C65
Mots-clés : hypergraphs, edge colouring