Left-Cut-Percolation and Induced-Sidorenko Bigraphs

SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1586-1629, June 2024.
Abstract. A Sidorenko bigraph is one whose density in a bigraphon [math] is minimized precisely when [math] is constant. Several techniques in 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 [math]-decompositions of Conlon and Lee, which uses weakly Hölder (or weakly norming) bigraphs as building blocks. In turn, to obtain weakly Hölder bigraphs, it is typical to use the chain of implications reflection bigraph [math] cut-percolating bigraph [math] weakly Hölder bigraph. In an earlier result by the author with Razborov, we provided a generalization of [math]-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 [math] left-cut-percolating bigraph [math] induced-Sidorenko bigraph. We also show that under mild hypotheses the “left-sided” analogue of the weakly Hölder property (which is also obtained via a similar chain of implications) can be used to improve bounds on another result of Conlon and Lee that roughly says that bigraphs with enough vertices on the right side of each realized degree have the Sidorenko property.