Corrigendum: The complexity of counting graph homomorphisms

We close a gap in the proof of Theorem 4.1 in our paper “The complexity of counting graph homomorphisms” [Random Structures and Algorithms 17 (2000), 260– 289]. Our paper [2] analysed the complexity of counting graph homomorphisms from a given graph G into a fixed graph H. This problem was called #H. A crucial step in our argument, Theorem 4.1, was to prove that the counting problem #H is #P -complete if H has a connected component H` such that the counting problem #H` associated with H` is #P -complete. The first step of our proof claimed the existence of a positive integer r such that every entry of A` r was positive, where A` is the adjacency matrix of H`. However, as Leslie Goldberg has recently pointed out to us, no such r exists when H` is bipartite. This claim was not used elsewhere in the paper, but its invalidity leaves a gap in the proof of Theorem 4.1. We give here an amended proof which shows how to deal with bipartite connected components of H. Before presenting the proof we note that Bulatov and Grohe [1] have extended our result to the problem of computing a weighted sum of homomorphisms to a weighted graph H. This problem is equivalent to the problem of computing the partition function of a spin system from statistical physics, and to the problem of counting the solutions to a constraint satisfaction problem whose constraint language consists of two equivalence relations. In the rest of the paper, the equation numbers correspond to those in [2]. The following interpolation result is well-known (a proof was given in [2]) and we state it again here for ease of reference. Lemma 3.2 Let w1, . . . , wr be known distinct nonzero constants. Suppose that we know values f1, . . . , fr such that fs = r ∑