Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Potts model and chromatic functions of graphs   The $U$-polynomial of Noble and Welsh is known to have intimate connections
with the Potts model as well as with several important graph polynomials. For
each graph $G$, $U(G)$ is equivalent to Stanley's symmetric bad colouring
polynomial $XB(G)$. Moreover Sarmiento established the equivalence between $U$
and the polychromate of Brylawski. Loebl defined the $q$-dichromate
$B_q(G,x,y)$ as a function of a graph $G$ and three independent variables
$q,x,y$, proved that it is equal to the partition function of the Potts model
with variable number of states and with a certain external field contribution,
and conjectured that the $q$-dichromate is equivalent to the $U$-polynomial. He
also proposed a stronger conjecture on integer partitions. The aim of this
paper is two-fold. We present a construction disproving Loebl's integer
partitions conjecture, and we introduce a new function $B_{r,q}(G;x,k)$ which
is also equal to the partition function of the Potts model with variable number
of states and with a (different) external field contribution, and we show that
$B_{r,q}(G;x,k)$ is equivalent to the $U$-polynomial and to Stanley's symmetric
bad colouring polynomial.

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The categories of this paper are: bayes-an
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The categories of this paper are: gr-qc
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The categories of this paper are: eess
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The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an
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The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
11
The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: mtrl-th
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The categories of this paper are: econ
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The categories of this paper are: astro-ph
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The categories of this paper are: adap-org
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: comp-gas