Constrained Colouring and σ-Hypergraphs

Caro, Yair ; Lauri, Josef ; Zarb, Christina

Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 1, pp. 171-189

Résumé

A constrained colouring or, more specifically, an (α, β)-colouring of a hypergraph H, is an assignment of colours to its vertices such that no edge of H contains less than α or more than β vertices with different colours. This notion, introduced by Bujtás and Tuza, generalises both classical hypergraph colourings and 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 (α, β)-spectrum, that is, for k_1 lt; k_2 lt; k_3, H would be (α, β)-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 σ being a partition of r, a very versatile type of r-uniform hypergraph which they called σ-hypergraphs. They showed that, by simple manipulation of the parameters of a σ-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 σ 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 α = β = 2. In this paper we extend many of the results of the first two authors to more general (α, β)-colourings, and we study the phenomenon of the disappearance 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 σ-hypergraphs.

Détail

Keywords: σ-hypergraphs, constrained colourings, hypergraph colourings