Sidorenko's conjecture, colorings and independent sets

Abstract

Let hom(H, G) denote the number of homomorphisms from a graph H to a graph G. Sidorenko’s conjecture asserts that for any bipartite graph H, and a graph G we have hom(H, G) > v(G)[superscript v(H)](hom(K[subscript 2], G)[superscript e(H)]/v(G)[superscript 2], where v(H), v(G) and e(H), e(G) denote the number of vertices and edges of the graph H and G, respectively. In this paper we prove Sidorenko’s conjecture for certain special graphs G: for the complete graph Kq on q vertices, for a K2 with a loop added at one of the end vertices, and for a path on 3 vertices with a loop added at each vertex. These cases correspond to counting colorings, independent sets and Widom-Rowlinson configurations of a graph H. For instance, for a bipartite graph H the number of q-colorings ch(H, q) satisfies ch(H, q) ≥ q[superscript v(H)](q − 1/q)[superscript e(H)]. In fact, we will prove that in the last two cases (independent sets and WidomRowlinson configurations) the graph H does not need to be bipartite. In all cases, we first prove a certain correlation inequality which implies Sidorenko’s conjecture in a stronger form.