Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counting Homomorphisms Modulo a Prime Number   Counting problems in general and counting graph homomorphisms in particular
have numerous applications in combinatorics, computer science, statistical
physics, and elsewhere. One of the most well studied problems in this area is
#GraphHom(H) --- the problem of finding the number of homomorphisms from a
given graph G to the graph H. Not only the complexity of this basic problem is
known, but also of its many variants for digraphs, more general relational
structures, graphs with weights, and others.
  In this paper we consider a modification of #GraphHom(H), the #_p GraphHom(H)
problem, p a prime number: Given a graph G, find the number of homomorphisms
from G to H modulo p. In a series of papers Faben and Jerrum, and Goebel et al.
determined the complexity of #_2 GraphHom(H) in the case H (or, in fact, a
certain graph derived from H) is square-free, that is, does not contain a
4-cycle. Also, Goebel et al. found the complexity of #_p GraphHom(H) for an
arbitrary prime p when H is a tree. Here we extend the above result to show
that the #_p GraphHom(H) problem is #_p P-hard whenever the derived graph
associated with H is square-free and is not a star, which completely classifies
the complexity of #_p GraphHom(H) for square-free graphs H.

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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: q-bio
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The categories of this paper are: nlin
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-th
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: comp-gas