Constrained colouring and $\sigma$-hypergraphs

A constrained colouring or, more specifically, an $(\alpha,\beta)$-colouring of a hypergraph $H$, is an assignment of colours to its vertices such that no edge of $H$ contains less than $\alpha$ or more than $\beta$ vertices with different colours. This notion, introduced by B{\'u}jtas and Tuza, generalises both classical hypergraph colourings and the more general Voloshin colourings of hypergraphs. In fact, for $r$-uniform hypergraphs, classical colourings correspond to $(2,r)$-colourings while an important instance of Voloshin colourings of $r$-uniform hypergraphs gives $(2, r-1)$-colourings. One intriguing aspect of all these colourings, not present in classical colourings, is that $H$ can have gaps in its $(\alpha,\beta)$-spectrum, that is, for $k_1 < k_2 < k_3$, $H$ would be $(\alpha,\beta)$-colourable using $k_1$ and using $k_3$ colours, but not using $k_2$ colours. In an earlier paper, the first two authors introduced, for $\sigma$ a partition of $r$, a very versatile type of $r$-uniform hypergraph which they called $\sigma$-hypergraphs. They showed that, by simple manipulation of the parameters of a $\sigma$-hypergraph $H$, one can obtain families of hypergraphs which have $(2,r-1)$-colourings exhibiting various interesting chromatic properties. They also showed that, if the smallest part of $\sigma$ is at least 2, then $H$ will never have a gap in its $(2,r-1)$-spectrum but, quite surprisingly, they found examples where gaps re-appear when $\alpha=\beta=2$. In this paper we extend many of the results of the first two authors to more general $(\alpha,\beta)$-colourings, and we study the phenomenon of the disappearanace and re-appearance of gaps and show that it is not just the behaviour of a particular example but we place it within the context of a more general study of constrained colourings of $\sigma$-hypergraphs.