Quasi-transitive $K_\infty$-minor free graphs

TL;DR

This paper proves that certain infinite, locally finite, quasi-transitive graphs excluding an infinite complete minor are quasi-isometric to planar graphs, resolving a problem and extending previous results on graph quasi-isometries.

Contribution

It establishes that such graphs are quasi-isometric to planar graphs, advancing understanding of their geometric structure and solving an open problem.

Findings

Graphs are quasi-isometric to planar graphs

Extends previous results to infinite minors

Solves a problem posed by Esperet and Giocanti

Abstract

We prove that every locally finite quasi-transitive graph that does not contain $K_\infty$ as a minor is quasi-isometric to some planar quasi-transitive locally finite graph. This solves a problem of Esperet and Giocanti and improves their recent result that such graphs are quasi-isometric to some planar graph of bounded degree.