Finite graphs whose every ball is an expander

Definitions and mathematical statement

For a finite connected graph $H$ with at least two vertices, define its edge-expansion constant by \[ h_E(H)=\min_{0<|A|\le|V(H)|/2} \frac{|E_H(A,V(H)\setminus A)|}{|A|}, \] where each edge crossing the indicated cut is counted once. Set $h_E(H)=+\infty$ for a singleton, so that trivial balls impose no restriction.

Does there exist an integer $d\ge2$, a constant $h>0$, and a sequence of finite connected simple $d$-regular graphs $G_n$ with $|V(G_n)|\to\infty$, such that \[ h_E\left(G_n[B_{G_n}(v,r)]\right)\ge h \] for every $n$, every vertex $v\in V(G_n)$, and every integer radius $r\ge0$? Here balls are defined using the graph metric of $G_n$, and $G_n[B]$ retains exactly those original edges with both endpoints in $B$.

The source conjectures that no such family exists. The degree $d$ and expansion constant $h$ must remain fixed throughout the family and across all ball radii and centers. The induced balls are generally not regular, even though the ambient graphs are.

Taking a radius at least the diameter shows that the ambient graphs would themselves form an expander family, but that alone is far weaker than the requirement. Boundary edges leaving a ball cannot help its internal expansion. In particular, tree-like neighborhoods of increasing radius provide a natural obstruction: their large branches can be separated by very small internal cuts. The question asks whether a bounded-degree family can avoid such obstructions at every location and every scale.

Short English statement

Can arbitrarily large regular graphs have every induced metric ball expand by one uniform positive edge-expansion constant?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 10000061 (AMR-099-0061), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Benjamini - Euclidean vs. Graph Metric, Question 5.4, PDF page 6. Original source indexed by AMR-099-0061. https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=6.
• S3 I. Benjamini, Euclidean vs. Graph Metric, primary numbered question and conventions. https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf.