Interface threshold for localisation of partial automorphisms

Determine whether the interface bound $5-k$ in the non-localisation theorem for partial automorphisms of IPR fullerenes can be increased, ideally toward the cyclic edge-connectivity threshold.

Background

The main non-localisation theorem excludes a nontrivial partial automorphism of deficiency k∈{1,2,3}k\in\{1,2,3\} whose support lies behind an interface of at most $5-k$ edges to a sufficiently large pointwise-fixed remainder. The paper notes that stronger bounds might make the conclusion genuinely global, but leaves open whether the threshold can be raised.

References

Whether the interface bound $5-k$ can be raised---ideally towards the cyclic edge-connectivity threshold, which would make the conclusion global in a literal sense---we do not know.

— Extremal Asymmetric Depth of Planar Graphs and Hidden Near-Mirror Symmetries of IPR Fullerenes  (2609.02585 - Pastorek, 2 Sep 2026) in Remark 2, Section 4, “The scale of clause (ii)”

Whether \Cref{thm:unified-localisation} extends to non-IPR fullerenes we do not know. As recorded in the preliminaries to the proof, the facts that every $5$-cycle and every $6$-cycle bounds a face hold in every fullerene, so the IPR hypothesis enters in exactly one place: Cases~1 and~2 of clause~\ref{it:unified-interface} use that every cyclic $5$- or $6$-edge cut of $F$ is trivial, which by fails precisely when two pentagons are adjacent. Removing the hypothesis therefore reduces to analysing the nontrivial cyclic $5$- and $6$-edge cuts described by the Kardo\v{s}--\v{S}krekovski characterisation.

— Extremal Asymmetric Depth of Planar Graphs and Hidden Near-Mirror Symmetries of IPR Fullerenes  (2609.02585 - Pastorek, 2 Sep 2026) in Question 1, Section 4, “Sharpness: what the hypotheses are really doing”