3 research outputs found
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
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 --- the problem of finding the number of homomorphisms from a given graph to the graph . 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 thesis, we consider a modification of , the problem, a prime number: Given a graph , find the number of homomorphisms from to modulo . In a series of papers Faben and Jerrum, and G"{o}bel et al. determined the complexity of in the case (or, in fact, a certain graph derived from ) is square-free, that is, does not contain a 4-cycle. Also, G"{o}bel et al. found the complexity of for an arbitrary prime when is a tree. Here we extend the above result to show that the problem is #_pP-hard whenever the derived graph associated with is square-free and is not a star, which completely classifies the complexity of for square-free graphs