Selective Hypergraph Colourings

TL;DR

This paper introduces a new framework for hypergraph colourings using $Q$-colourings and $\\Sigma$-hypergraphs, exploring their properties, spectrum gaps, and connections to known problems like Ramsey theory.

Contribution

It defines and analyzes $Q$-colourings and $\\Sigma$-hypergraphs, establishing results on tight colourings, spectrum gaps, and their relation to existing hypergraph colouring problems.

Findings

Many known hypergraph colouring problems can be expressed as $Q$-colourings.

Existence of $\\Sigma$-hypergraphs with large $Q$-chromatic and chromatic numbers but bounded clique number.

Characterization of $Q$ sets that lead to spectrum gaps in hypergraphs.

Abstract

We look at colourings of $r$-uniform hypergraphs, focusing our attention on unique colourability and gaps in the chromatic spectrum. The pattern of an edge $E$ in an $r$-uniform hypergraph $H$ whose vertices are coloured is the partition of $r$ induced by the colour classes of the vertices in $E$. Let $Q$ be a set of partitions of $r$. A $Q$-colouring of $H$ is a colouring of its vertices such that only patterns appearing in $Q$ are allowed. We first show that many known hypergraph colouring problems, including Ramsey theory, can be stated in the language of $Q$-colourings. Then, using as our main tools the notions of $Q$-colourings and $\Sigma$-hypergraphs, we define and prove a result on tight colourings, which is a strengthening of the notion of unique colourability. $\Sigma$-hypergraphs are a natural generalisation of $\sigma$-hypergraphs introduced by the first two authors in an earlier paper. We also show that there exist $\Sigma$-hypergraphs with arbitrarily large $Q$-chromatic number and chromatic number but with bounded clique number. Dvorak et al. have characterised those $Q$ which can lead to a hypergraph with a gap in its $Q$-spectrum. We give a short direct proof of the necessity of their condition on $Q$. We also prove a partial converse for the special case of $\Sigma$-hypergraphs. Finally, we show that, for at least one family $Q$ which is known to yield hypergraphs with gaps, there exist no $\Sigma$-hypergraphs with gaps in their $Q$-spectrum.