Some advances on Sidorenko's conjecture

TL;DR

This paper identifies three new families of bipartite graphs that possess Sidorenko's property, expanding understanding of graph homomorphism probabilities through novel methods involving tree decompositions, subdivisions, and Cartesian products.

Contribution

It introduces three distinct methods to prove Sidorenko's property for new classes of bipartite graphs, including those with specific tree decompositions, subdivisions, and Cartesian products.

Findings

Bipartite graphs with certain tree decompositions have Sidorenko's property.

Subdivisions of graphs like cliques also have Sidorenko's property.

Cartesian product of a Sidorenko graph with an even cycle retains the property.

Abstract

A bipartite graph $H$ is said to have Sidorenko's property if the probability that the uniform random mapping from $V(H)$ to the vertex set of any graph $G$ is a homomorphism is at least the product over all edges in $H$ of the probability that the edge is mapped to an edge of $G$. In this paper, we provide three distinct families of bipartite graphs that have Sidorenko's property. First, using branching random walks, we develop an embedding algorithm which allows us to prove that bipartite graphs admitting a certain type of tree decomposition have Sidorenko's property. Second, we use the concept of locally dense graphs to prove that subdivisions of certain graphs, including cliques, have Sidorenko's property. Third, we prove that if $H$ has Sidorenko's property, then the Cartesian product of $H$ with an even cycle also has Sidorenko's property.