Forcing finite minors in sparse infinite graphs by large-degree assumptions

TL;DR

This paper explores how large-degree conditions in sparse infinite graphs can guarantee the presence of specific finite minors or subgraphs, advancing the extremal theory of such graphs.

Contribution

It extends Stein's concept of relative end degrees to identify degree conditions that force finite minors or subgraphs in infinite graphs.

Findings

Degree assumptions can force finite minors in sparse infinite graphs

Extension of Stein's relative end degrees to new extremal conditions

Progress towards an extremal theory of sparse infinite graphs

Abstract

Developing further Stein's recent notion of relative end degrees in infinite graphs, we investigate which degree assumptions can force a locally finite graph to contain a given finite minor, or a finite subgraph of given minimum degree. This is part of a wider project which seeks to develop an extremal theory of sparse infinite graphs.