Sidorenko's Conjecture, Colorings and Independent Sets

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) ( hom(K2, G) v(G)2 )e(H) , 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 ( q − 1 q )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, the electronic journal of combinatorics 24(1) (2017), #P1.2 1 we first prove a certain correlation inequality which implies Sidorenko’s conjecture in a stronger form.