Sidorenko's conjecture for subdivisions and theta substitutions

TL;DR

This paper proves Sidorenko's conjecture for a class of graphs formed by replacing edges with generalized theta graphs, extending previous results and providing new cases where the conjecture holds.

Contribution

It introduces a new class of graphs satisfying Sidorenko's conjecture, specifically those formed by substituting edges with generalized theta graphs under divisibility conditions.

Findings

Graphs with edges replaced by generalized theta graphs satisfy Sidorenko's conjecture.

Extension of previous results to complete graphs with theta substitutions.

Unconditional proof for bipartite graphs derived from complete graphs with theta replacements.

Abstract

The famous Sidorenko's conjecture asserts that for every bipartite graph $H$, the number of homomorphisms from $H$ to a graph $G$ with given edge density is minimized when $G$ is pseudorandom. We prove that for any graph $H$, a graph obtained from replacing edges of $H$ by generalized theta graphs consisting of even paths satisfies Sidorenko's conjecture, provided a certain divisibility condition on the number of paths. To achieve this, we prove unconditionally that bipartite graphs obtained from replacing each edge of a complete graph with a generalized theta graph satisfy Sidorenko's conjecture, which extends a result of Conlon, Kim, Lee and Lee [J. Lond. Math. Soc., 2018].