Bottlenecking in graphs and a coarse Menger-type theorem

TL;DR

This paper introduces a new perspective on graph connectivity through coarse bottlenecking, formulates a Coarse Menger-type theorem, and discusses related conjectures, advancing understanding of graph coarsening and connectivity measures.

Contribution

It extends the concept of bottlenecking to coarse graphs, formulates and proves a Coarse Menger-type theorem, and proposes a new coarse Erdős–Menger conjecture.

Findings

Coarse bottlenecking offers a new approach to graph coarsening.

A Coarse Menger-type theorem is formulated and proved.

A new coarse Erdős–Menger-type conjecture is proposed.

Abstract

We expand upon the notion of bottlenecking introduced in our earlier work, characterizing a spectrum of graphs and showing that this naturally extends to a concept of coarse bottlenecking. We show how the notion of bottlenecking provides a different approach to coarsening measures of connectedness than the Coarse Menger Conjecture proposed independently by Georgakopoulos and Papasoglu as well as Albrechtsen, Huynh, Jacobs, Knappe, and Wollan - which was recently disproved by a counterexample. We formulate and prove a Coarse Menger-type theorem, and also propose a coarse Erd\H{o}s-Menger-type Conjecture, in the spirit of the Erd\H{o}s-Menger conjecture which was proven after decades by Aharoni and Berger.