bibkey: vemuri2019domination authors: Harish Vemuri year: 2019 title: Domination in Direct Products of Complete Graphs doi: 10.48550/arXiv.1908.02445 claim: “Domination numbers of direct products of complete graphs have known lower bounds; Theorem 1.1 cites Mekis’s bound gamma >= t+1 for t >= 4 factors.” strata_touched:

• D5/S3/ConceptDynamics/GraphColoring/AnnorCoverRefutation license: citation-only triage: anchor

Known product-graph ingredient

Checked https://arxiv.org/html/1908.02445v1 on 2026-09-07 Asia/Singapore. The introduction, Theorem 1.1, and Theorem 1.2 were read. Theorem 1.1 attributes to Mekis (2010), Theorem 2.1, the lower bound gamma >= t+1 for a direct product of t >= 4 complete graphs with factor sizes at least two, with equality when the smallest factor is at least t+1. Theorem 1.2 cites stronger results of Defant and Iyer (2018).

The repository only needs gamma >= t when each factor has at least t vertices. Its diagonal argument is elementary and is not claimed as new mathematics. The bound, connectedness and degree counting are formalized for the existing Mathlib SimpleGraph type. No corresponding Lean theorem was found in the ordered D5, pinned Mathlib and ecosystem searches.

The repository parameterization is t=r+1 factors of size 2t+1. The initial revised preregistration used r factors of size 2r; the shift removes zero coordinate and subtraction bookkeeping and preserves the same density and unbounded-domination argument. This implementation adjustment is disclosed after the probes, not represented as an earlier preregistration.

Verified locator

DOI: https://doi.org/10.48550/arXiv.1908.02445. Independently resolved and checked on 2026-09-07 Asia/Singapore against https://arxiv.org/html/1908.02445v1 (2019-08-07), Introduction, Theorems 1.1 and 1.2. Theorem 1.1 attributes the bound gamma >= t+1 for t >= 4 complete factors of size at least two to Mekis (2010), Theorem 2.1; Theorem 1.2 cites Defant and Iyer. This locates established product-graph ingredients, not the repository’s cover refutation or a priority claim.