bibkey: kok2014finitejaco
authors: Johan Kok; Paul Fisher; Bettina Wilkens; Mokhwetha Mabula; Vivian Mukungunugwa
year: 2014
title: Characteristics of Finite Jaco Graphs, J_n(1), n in N
doi: 10.48550/arXiv.1404.0484
claim: Theorem 2.10 (Conjectured) gives a two-branch maximum-total-degree addition law for all distinct orders at least three.
strata_touched:
  - D5/S3/ConceptDynamics/GraphColoring/JacoAdditionLawRefutation
license: CC-BY-3.0
triage: anchor

Characteristics of finite Jaco graphs

Verified locator

• DOI: https://doi.org/10.48550/arXiv.1404.0484

Johan Kok, Paul Fisher, Bettina Wilkens, Mokhwetha Mabula and Vivian Mukungunugwa, arXiv:1404.0484v1, Characteristics of Finite Jaco Graphs, J_n(1), n in N. The primary PDF locates the conjectured addition assertion at Theorem 2.10 on printed page 7 and the explicit open problem on printed page 11. Definitions 1.1–1.4 and 1.7 on printed page 2 specify the recursive graph, its Jaconian vertices and ordinary total degree.

Source interpretation

The original theorem states, for distinct n,m at least three, that the maximum degree of J_(n+m)(1) equals the sum of the two maximum degrees if J_n(1) or J_m(1) has a unique Jaconian vertex, and equals that sum plus one otherwise. The source’s open prompt reads: “Prove Theorem (conjectured) 2.10.”

Vertices have positive labels. A forward arc from i to j exists exactly when i<j and j<=2i-d^-(v_i); finite graphs truncate vertices and arcs beyond their order. A Jaconian vertex attains the maximum ordinary total degree. Uniqueness concerns a vertex within one graph. Forgetting orientation counts each forward arc once as a simple edge and preserves ordinary total degree.

The source metadata links CC BY 3.0. The statement and notation are adapted with attribution to the five authors. The source includes a Creative Commons attribution notice. No executable code is copied from the article.

Kok’s later arXiv:2507.16500v1, Table 1, is prior published numerical data containing the degree values used in the counterexample. The proof uses existing exact recursive-graph suppliers, not the table as an assumption. That paper’s Claim 2.10 concerns domination paths and is a different assertion.

Bounded source screening found no specific complete resolution in its successfully inspected sources. MathDB live contents, complete citation enumeration, the special issue Reflections on Jaco Graphs, the named domination research note and unindexed sources remain coverage gaps. No worldwide openness, first computation, or publication-priority claim follows.