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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.

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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