Left-cut-percolation and induced-Sidorenko bigraphs

TL;DR

This paper introduces a new chain of implications involving left-sided concepts in bigraph theory, leading to induced-Sidorenko bigraphs, and improves bounds related to the Sidorenko property using these ideas.

Contribution

It generalizes the chain of implications for bigraphs by introducing left-sided versions, and enhances bounds on Sidorenko property criteria.

Findings

Left-reflection bigraph implies left-cut-percolating bigraph.

Left-cut-percolating bigraph implies induced-Sidorenko bigraph.

Left-sided weakly H"{o}lder property improves bounds on Sidorenko property criteria.

Abstract

A Sidorenko bigraph is one whose density in a bigraphon $W$ is minimized precisely when $W$ is constant. Several techniques of the literature to prove the Sidorenko property consist of decomposing (typically in a tree decomposition) the bigraph into smaller building blocks with stronger properties. One prominent such technique is that of $N$-decompositions of Conlon--Lee, which uses weakly H\"{o}lder (or weakly norming) bigraphs as building blocks. In turn, to obtain weakly H\"{o}lder bigraphs, it is typical to use the chain of implications reflection bigraph $\implies$ cut-percolating bigraph $\implies$ weakly H\"{o}lder bigraph. In an earlier result by the author with Razborov, we provided a generalization of $N$-decompositions, called reflective tree decompositions, that uses much weaker building blocks, called induced-Sidorenko bigraphs, to also obtain Sidorenko bigraphs. In this paper, we show that "left-sided" versions of the concepts of reflection bigraph and cut-percolating bigraph yield a similar chain of implications: left-reflection bigraph $\implies$ left-cut-percolating bigraph $\implies$ induced-Sidorenko bigraph. We also show that under mild hypotheses, the "left-sided" analogue of the weakly H\"{o}lder property (which is also obtained via a similar chain of implications) can be used to improve bounds on another result of Conlon--Lee that roughly says that bigraphs with enough vertices on the right side of each realized degree have the Sidorenko property.