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$$\hom(H,G)\geq v(G)^{v(H)}\left(\frac{\hom(K_2,G)}{v(G)^2}\right)^{e(H)},$$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 $K_q$ on $q$ vertices, for a $K_2$ 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 colorings of a graph $H$. For instance, for a bipartite graph $H$ the number of $q$-colorings $\mathrm{ch}(H,q)$ satisfies$$\mathrm{ch}(H,q)\geq q^{v(H)}\left(\frac{q-1}{q}\right)^{e(H)}.$$In fact, we will prove that in the last two cases (independent sets and Widom-Rowlinson colorings) 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.
Highlights
Let v(G) and e(G) denote the number of vertices and edges of a graph G
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
Summary
Let v(G) and e(G) denote the number of vertices and edges of a graph G. In this paper we follow another route: we prove Sidorenko’s conjecture when G is fixed, namely when G is a complete graph (counting colorings) or a complete graph on 2 vertices with a loop added at one of the vertices (counting independent sets), or a path on 3 vertices with a loop added at each vertex (counting Widom-Rowlinson configurations). When G is a K2 with a loop added at one vertex hom(H, G) = i(H) counts the number of independent sets of the graph H In this case Sidorenko’s conjecture states that i(H ). This model corresponds to the case when G is a path on 3 vertices with a loop added at each vertex (see Fig. 1) We denote this graph by P3◦, hom(H, P3◦) = wr(H) counts the number of 3–colorings of H with colors white, red and blue such that a red and a blue vertex cannot be adjacent, but there is no other restriction. We conclude the paper with an interesting open problem
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