bibkey: kok2025jaco authors: Johan Kok year: 2025 title: Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs doi: 10.48550/arXiv.2507.16500 claim: Conjecture 2.9 bounds a primary minimal dom-path by the graph diameter plus one, while Conjecture 2.12 says that A000149 indexes a gamma-set of J_infinity(x) and is p-graphical for the finite J_n(x). strata_touched:

• D5/S0/Certificates/JacoExponentialDominationRefutation
• D5/S3/ConceptDynamics/GraphColoring/JacoDomPathDiameterRefutation license: citation-only triage: anchor

Integer sequences and linear Jaco graph parameters

Version 1 of the preprint was submitted on 22 July 2025. Section 2.3, Conjecture 2.12, on printed page 9 states these two clauses verbatim:

The vertex subscripts of a γ-set X = ”{v1 }” ∪ {v2 , v7 , v20 , v54 , . . . } of the infinite linear Jaco graph J∞ (x) is given by the sequence A000149: a(t) = ⌊e^t⌋, t = 0, 1, 2, . . . where e ≈ 2.71828 is the Euler number (or Napier’s constant).

Furthermore, it implies that sequence A000149 is p-graphical where p(G) = {i : j the subscript of vj ∈ X with X some γ-set of G} and F = {G : G = Jn(x), n = 1, 2, 3, . . . }.

The first clause identifies A000149 with a gamma-set, hence with a minimum dominating set. The formal target retains only its necessary domination assertion. It does not encode gamma-set minimality or the second, p-graphical clause. Refuting the necessary assertion refutes the gamma-set assertion, not the conjecture’s separate p-graphical clause.

Dom-path diameter conjecture

Observation 2.7 on printed pages 6–7 states verbatim:

For any finite linear Jaco graph Jn (x), n ≥ 2 there exists a pair of vertices i.e. v1 , vn for which a minimal (v1 , vn )-path (not necessarily a diam-path) i.e. Pd (Jn (x)) exists such that a γ-set of Pd (Jn (x)) is a γ-set of Jn (x). We call the path Pd (Jn (x)) the primary minimal dom-path.

Conjecture 2.9 on printed page 7 states verbatim:

For any linear Jaco graph Jn (x), n ≥ 1 the length of a diam-path and a primary minimal dom-path Pd satisfy |Pd | − |diam(Jn (x))| ≤ 1.

The paper measures path length in edges: its six-vertex path for J_8(x) is said to have length 5. The formal assertion uses the weakest consequence of the conjecture, namely that at least one dom-path has length at most the graph diameter plus one. The finite graph at n = 33 refutes even this consequence: its diameter is 7, its domination number is 4, and every dom-path has at least 10 vertices. The path with vertices 1,2,3,4,7,11,12,20,32,33 and shared gamma-set {2,7,20,33} shows that the dom-path predicate is inhabited.

The arXiv API and searches for "Jaco" "Conjecture 2.9", "Jaco" "dom-path", "linear Jaco graphs" conjecture proof, and "linear Jaco graphs" 33 domination were checked on 2026-09-18. No proof, refutation, correction, later arXiv version, journal reference, or DOI beyond the arXiv DOI was found in those checked surfaces. Semantic Scholar returned HTTP 429, Google Scholar was not checked, and the full text of Kok’s cited domination research note was not available; this is therefore a bounded literature report rather than an exhaustive priority claim.

The arXiv version history, the current OEIS A000149 entry, and the following searches were checked on 2026-09-12: "2507.16500" proof, "Integer sequences with conjectured relation", "Jaco graphs" "Euler", "Jaco" "Conjecture 2.12", "Jaco" "2.12" "counterexample", and "Jaco graphs" "domination" "2026". A proof or refutation was not found in the checked surfaces. The paper’s conclusion leaves its conjectures for future work. Kok’s earlier Research note: Domination of exact deg-centric Jaco graphs provides background but does not settle this domination assertion. This is a bounded search report; no claim of exhaustive coverage or priority is made.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2507.16500
• Preprint: https://arxiv.org/abs/2507.16500
• Version and scope: https://arxiv.org/pdf/2507.16500v1, section 2.3, Observation 2.7 and Conjecture 2.9 on printed pages 6–7, and Conjecture 2.12 on printed page 9.
• Sequence: https://oeis.org/A000149