Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: cs
21
The categories of this paper are: adap-org
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: plasm-ph