Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: H-colouring bipartite graphs   For graphs $G$ and $H$, an {\em $H$-colouring} of $G$ (or {\em homomorphism}
from $G$ to $H$) is a function from the vertices of $G$ to the vertices of $H$
that preserves adjacency. $H$-colourings generalize such graph theory notions
as proper colourings and independent sets.
  For a given $H$, $k \in V(H)$ and $G$ we consider the proportion of vertices
of $G$ that get mapped to $k$ in a uniformly chosen $H$-colouring of $G$. Our
main result concerns this quantity when $G$ is regular and bipartite. We find
numbers $0 \leq a^-(k) \leq a^+(k) \leq 1$ with the property that for all such
$G$, with high probability the proportion is between $a^-(k)$ and $a^+(k)$, and
we give examples where these extremes are achieved. For many $H$ we have
$a^-(k) = a^+(k)$ for all $k$ and so in these cases we obtain a quite precise
description of the almost sure appearance of a randomly chosen $H$-colouring.
  As a corollary, we show that in a uniform proper $q$-colouring of a regular
bipartite graph, if $q$ is even then with high probability every colour appears
on a proportion close to $1/q$ of the vertices, while if $q$ is odd then with
high probability every colour appears on at least a proportion close to
$1/(q+1)$ of the vertices and at most a proportion close to $1/(q-1)$ of the
vertices.
  Our results generalize to natural models of weighted $H$-colourings, and also
to bipartite graphs which are sufficiently close to regular. As an application
of this latter extension we describe the typical structure of $H$-colourings of
graphs which are obtained from $n$-regular bipartite graphs by percolation, and
we show that $p=1/n$ is a threshold function across which the typical structure
changes.
  The approach is through entropy, and extends work of J. Kahn, who considered
the size of a randomly chosen independent set of a regular bipartite graph.

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The categories of this paper are: chao-dyn
37
The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs
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The categories of this paper are: acc-phys
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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: econ
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The categories of this paper are: hep-th
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The categories of this paper are: ao-sci
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org