Sidorenko's conjecture, colorings and independent sets

TL;DR

This paper proves Sidorenko's conjecture for specific graphs related to colorings and independent sets, using correlation inequalities to establish stronger results for certain classes of graphs.

Contribution

The paper establishes Sidorenko's conjecture for special graphs like complete graphs, loops, and paths with loops, extending results to non-bipartite graphs in some cases.

Findings

Proved Sidorenko's conjecture for complete graphs and certain looped graphs.

Derived lower bounds for colorings and independent sets.

Established correlation inequalities that imply stronger forms of Sidorenko's conjecture.

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 $\textrm{ch}(H,q)$ satisfies $$\textrm{ch}(H,q)\geq q^{v(H)}\left(\frac{q-1}{q}\right)^{e(H)}.$$ 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.