Characterization of finiteness for positively LLY-curved graphs

Determine which graph properties guarantee that only finitely many connected graphs with positive Lin–Lu–Yau curvature exist.

Background

The paper studies global finiteness phenomena for connected graphs with positive Lin–Lu–Yau curvature. It proves finiteness for several classes, including graphs of minimum degree at least s that exclude a fixed complete bipartite graph K_{s,t} as a minor, thereby generalizing earlier results for outerplanar and planar graphs.

These theorems provide sufficient conditions but do not characterize all graph properties that force finiteness. The broader question remains unresolved beyond the specific minor-exclusion and minimum-degree settings treated in the paper.

References

This raises the following question: What graph property forces that there is only finitely many connected graphs with positive LLY curvature?

— Positive Lin-Lu-Yau curvature, planar graphs, and forbidden minors  (2610.10307 - Esperet et al., 7 Oct 2026) in Section 1, Introduction