bibkey: bastienkhormali2025digraphs authors: Alexander Bastien and Omid Khormali year: 2025 title: On Link-irregular Digraphs doi: null url: https://arxiv.org/abs/2512.20494v1 claim: Conjecture 6 states that a link-irregular tournament exists on n vertices if and only if n is at least 6; the paper proves the assertion through order 8 and reports computational verification through order 100. strata_touched:

• D5/S3/ConceptDynamics/GraphIrregularity/LinkIrregularTournamentExistence license: citation-only triage: anchor

Link-irregular digraphs

The paper defines the directed link of a vertex v in a digraph D as the subdigraph induced on the union of the out-neighbors and in-neighbors of v. A digraph is link-irregular when the directed links at every two distinct vertices are non-isomorphic. Conjecture 6 states that a link-irregular tournament exists on n vertices if and only if n >= 6.

The source proves the conjecture for n <= 8. Its order-six tournament has the following one-based arcs:

(1,6), (1,3), (1,4), (2,1), (3,2), (3,4), (3,6), (4,5), (4,6), (4,2), (5,1), (5,2), (5,3), (5,6), (6,2).

The paper also reports computational verification through order 100. It does not contain a proof for all orders. The repository theorem uses the displayed order-six tournament and a symbolic construction for every order at least seven; it does not rely on the reported search.

Verified locator

• arXiv abstract and version record: https://arxiv.org/abs/2512.20494v1
• Source HTML: https://arxiv.org/html/2512.20494v1