bibkey: bastienkhormali2026link authors: Alexander Bastien and Omid Khormali year: 2026 title: On the regularity, planarity and edge bounds of link-irregular graphs doi: 10.7151/dmgt.2619 license: citation-only triage: anchor claim: No regular link-irregular graph exists on at most nine vertices; a seven-regular link-irregular graph on twelve vertices exists, refuting the Ali-Chartrand-Zhang conjecture that no regular link-irregular graph exists. strata_touched:

• D5/S3/ConceptDynamics/GraphIrregularity/RegularLinkIrregularEleven

Link-irregular graphs: regularity, planarity, edge bounds

Discussiones Mathematicae Graph Theory 46(2) (2026) 555-568. Preprint arXiv:2503.21916, version 2 of 12 June 2025.

Definitions used by the repository module, verbatim from the paper: “A graph G is a link-irregular graph if every two distinct vertices of G have non-isomorphic links. The link of a vertex v in G is the subgraph induced by the neighbors of v in G.”

Section 3 of the preprint closes with a printed conjecture: “Conjecture 17. There exists a regular link-irregular graph on n vertices if and only if n >= 12.” A first-hand inspection of the published version, in which every occurrence of the string “onjecture” was examined, found only Conjecture 1, the Ali-Chartrand-Zhang conjecture that the paper refutes; on that reading Conjecture 17 was removed in revision. An independent web-enabled review seat was unable to load the publisher page or the published PDF and therefore could not corroborate that reading.

Verified locator

10.7151/dmgt.2619 - Discussiones Mathematicae Graph Theory 46(2) (2026) 555-568. The conjecture quoted in this note is Conjecture 17 of Section 3 of the preprint arXiv:2503.21916 version 2; the first-hand inspection described above did not find it in the published version, and that reading remains independently uncorroborated. Retrieved 2026-09-11.