Biregularity in Sidorenko's Conjecture

TL;DR

This paper provides an elementary proof that Sidorenko's Conjecture can be reduced to biregular bigraphons, and introduces new classes of graphs satisfying the conjecture using reflective tree decompositions.

Contribution

It offers a simpler proof of the biregularity reduction and extends the class of graphs known to satisfy Sidorenko's Conjecture via reflective tree decompositions.

Findings

Elementary proof of biregularity reduction

New class of graphs satisfying Sidorenko's Conjecture

Unification of tree decomposition concepts

Abstract

Sidorenko's Conjecture says that the minimum density of a bigraph $G$ in a bigraphon $W$ of a given edge density is attained when $W$ is a constant function. A consequence of a result by B. Szegedy is that it is enough to show Sidorenko's Conjecture under the further assumption that $W$ is biregular. In this paper, we retrieve this result with a more elementary proof. With this biregularity result and some ideas of its proof, we also obtain simple proofs of several other results related to Sidorenko's Conjecture. Furthermore, we also show that bigraphs that have a special type of tree decomposition, called reflective tree decomposition, satisfy Sidorenko's conjecture. This both unifies and generalizes the notions of strong tree decompositions and $N$-decompositions from the literature.