Asymptotic half-grid and full-grid minors

TL;DR

This paper proves that certain infinite graphs with specific structural properties contain large grid minors as asymptotic or diverging minors, advancing understanding of graph minors in infinite graph theory.

Contribution

It establishes the existence of full-grid and half-grid minors in classes of infinite graphs with thick ends and bounded cycle lengths, partially solving open problems.

Findings

Locally finite, quasi-transitive graphs with thick ends contain full-grid minors.

Graphs with finite maximum degree and thick ends contain half-grid minors.

Results apply to Cayley graphs of finitely presented groups that are not virtually free.

Abstract

We prove that every locally finite, quasi-transitive graph with a thick end whose cycle space is generated by cycles of bounded length contains the full-grid as an asymptotic minor and as a diverging minor. This in particular includes all locally finite Cayley graphs of finitely presented groups that are not virtually free, and partially solves problems of Georgakopoulos and Papasoglu and of Georgakopoulos and Hamann. Additionally, we show that every (not necessarily quasi-transitive) graph of finite maximum degree which has a thick end and whose cycle space is generated by cycles of bounded length contains the half-grid as an asymptotic minor and as a diverging minor.