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 k1 < k2 < k3, H would be (α, β)-colourable using k1 and using k3 colours, but not using k2 colours. In an earlier paper, the first two authors introduced, for being a partition of r, a very versatile type of r-uniform hypergraph which they called -hypergraphs. They showed that, by simple manipulation of the param- eters of a σ -hypergraph H, one can obtain families of hypergraphs which have (2, r − 1)-colourings exhibiting various interesting chromatic proper- ties. They also showed that, if the smallest part of 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.
@article{bwmeta1.element.doi-10_7151_dmgt_1789,
author = {Yair Caro and Josef Lauri and Christina Zarb},
title = {Constrained Colouring and $\sigma$-Hypergraphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {35},
year = {2015},
pages = {171-189},
zbl = {1307.05065},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1789}
}
Yair Caro; Josef Lauri; Christina Zarb. Constrained Colouring and σ-Hypergraphs. Discussiones Mathematicae Graph Theory, Tome 35 (2015) pp. 171-189. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1789/
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