Bigraph percolation problems

TL;DR

This paper investigates the properties of bigraphs in relation to weakly norming conditions, identifying key obstacles to cut-percolation and establishing equivalences with fold-stability and colorings.

Contribution

It introduces the concept of fold-stability as a key obstacle and links cut-percolation of bigraphs to the non-existence of fold-stable colorings.

Findings

Existence of cut-percolation is equivalent to non-existence of fold-stable colorings.

Identifies fold-stability as a fundamental obstacle to cut-percolation.

Provides a new perspective on bigraph properties related to weakly norming conditions.

Abstract

A bigraph $G$ is weakly norming if the $e(G)$th root of the density of $G$ in $\lvert W\rvert$ is a norm in the space of bounded measurable functions $W\colon\Omega\times\Lambda\to\mathbb{R}$. The only known technique, due to Conlon--Lee, to show that a bigraph $G$ is weakly norming is to present a cut-percolation sequence of $G$. In this paper, we identify a key obstacle for cut-percolation, which we call fold-stability and we show that existence of a cut-percolating of a bigraph $G$ is equivalent to non-existence of non-monochromatic fold-stable colorings of the edges of $G$.