Compactness for localization number in locally finite graphs with finitely or countably many ends

Ascertain whether a compactness property holds for locally finite graphs with finitely or countably many ends: given an integer M such that every finite subgraph of a locally finite graph G has localization number at most M, determine whether ζ(G) can be bounded by an integer-valued function of M, or construct a locally finite graph with finitely or countably many ends that fails this compactness property.

Background

The authors contrast results for finite graphs with those for locally finite graphs and show that subdivisions can reduce the localization number to 1. They then raise a broader question of compactness: if every finite subgraph of G has ζ at most M, is ζ(G) bounded by a function of M?

They note that compactness fails for graphs with uncountably many ends, via a constructed tree Tω whose finite subgraphs have ζ ≤ 2 but ζ(Tω) = ℵ0. However, they explicitly state that they do not know of any such counterexamples when restricting to locally finite graphs with finitely or countably many ends.

References

One such question is compactness: if every finite subgraph of a graph G has localization number at most M, can we bound ζ(G) by some integer-valued function of M? Theorem 3 implies that this is not true for locally finite graphs with uncountably many ends as finite subgraphs of T ω have localization number at most 2 and yet ζ(T ω ) = ℵ 0. However, we know of no such examples of locally finite graphs failing compactness with finitely (or even countably) many ends.

— Locally finite graphs and their localization numbers (2404.02409 - Bonato et al., 3 Apr 2024) in Section 3, final paragraph