On ends of degree $\omega_1$

TL;DR

This paper constructs a graph with an uncountable degree end that defies subdivision into certain sparse graphs, providing a negative answer to a 2023 problem and extending Halin's grid theorem to uncountable degrees.

Contribution

It introduces a new graph construction with uncountable degree ends and generalizes Halin's grid theorem to ends of degree 1, addressing a recent open problem.

Findings

Existence of a graph with an uncountable degree end that is not a subdivision of any sparse S-graph

Counterexample to a problem posed by Stefan Geschke et al. in 2023

Extension of Halin's grid theorem to ends of degree 1

Abstract

We prove that if $ T $ is a semi-special tree that is not special, then there exists a graph $ G $, formed as an inflation of a sparse $ T $-graph, such that for any special tree $ S $, $ G $ is not a subdivision of an inflation of an sparse $ S $-graph. Furthermore $G$ has an end of uncountable degree that has no ray graph. This result provides a consistent negative answer to a problem posed by Stefan Geschke et al. in 2023. Additionally, we introduce and explore a property that generalizes Halin's grid theorem, extending it to ends of degree $ \aleph_1 $, which was originally established for ends of countable degree.