Counting Homomorphisms to Trees Modulo a Prime

Many important graph-theoretic notions can be encoded as counting graph homomorphism problems, such partition functions in statistical physics, particular independent sets and colourings. In this article, we study the complexity of # p H OMS T O , problem homomorphisms from an input to a modulo prime number . Dyer Greenhill proved dichotomy stating that tractability non-modular depends on structure target graph. intractable cases become tractable modular due common phenomenon cancellation. subsequent studies 2, however, influence was shown persist, which yields similar dichotomies. Our main result states for every tree is either polynomial time computable or P-complete. This relates conjecture Faben Jerrum holds when 2. contrast previous results counting, are essentially same all values tree. To prove result, structural properties homomorphism. As interim our weighted bipartite some These first suggesting dichotomies hold not only 2 case but also primes