bibkey: kok2014jaco
authors: Johan Kok, Paul Fisher, Bettina Wilkens, Mokhwetha Mabula, Vivian Mukungunugwa
year: 2014
title: Characteristics of Jaco Graphs, J_infinity(a), a in N
doi: null
url: https://arxiv.org/abs/1404.1714v1
claim: Proposition 4.4 conjectures the equivalence of both adjacent infinite-outdegree inequalities and both adjacent shortest-directed-path-count inequalities for every k at least seven in every Fibonacci bracket.
strata_touched:
  - D5/S3/ConceptDynamics/GraphColoring/JacoDirectedPathCount
license: citation-only
triage: anchor

The original directed Jaco path-count question

Source: https://arxiv.org/abs/1404.1714v1, Section 4, Proposition 4.4 and the final open-problem list. All five authors are listed above in source order. The versioned original source is distinct from the later Jaco integer-sequence paper.

Proposition 4.4 states:

Let $k\ge 7$ and $f_i<k\le f_{i+1}$, and $f_i,f_{i+1}\in \mathbb{F}$. If and only if $d^{+}(v_k)$ is non-repetitive (meaning $d^{+}(v_{k-1})\ne d^{+}(v_k)\ne d^{+}(v_{k+1})$) then $\psi (v_k)$ is non-repetitive (meaning $\psi (v_{k-1})\ne \psi (v_k)\ne \psi (v_{k+1})$).

Its proof placeholder is “(Conjectured)”, and the source explicitly asks “[Open problem: Prove Proposition 4.4]”. Each chained inequality denotes the conjunction of its two adjacent comparisons, with no comparison required between the first and third terms. The full question universally quantifies every $k\ge 7$ and every bracket index $i$ satisfying $f_i<k\le f_{i+1}$.

The infinite directed graph has positive labels and an arc $(v_h,v_j)$ exactly when $h<j$ and $j\le (a+1)h-d^{-}(v_h)$. At $a=1$ its right endpoint is $2h-d^{-}(v_h)$. The finite graph $J_n(1)$ retains vertices $1,\dots ,n$ and these forward arcs. The degree $d^{+}(v_h)$ is the full infinite outdegree. The quantity $\psi (v_n)$ is the number of distinct shortest directed paths from $v_1$ to $v_n$ in $J_n(1)$; every endpoint uses its own finite truncation and path length counts edges. An undirected graph or a count defined by a recurrence would change the question. The formal result does not use the converse of Proposition 4.3.

Verified locator

https://arxiv.org/abs/1404.1714v1, Section 4, Proposition 4.4, states the full all-index equivalence quoted above. Its proof is marked “(Conjectured)”. The authors list “[Open problem: Prove Proposition 4.4]” in the final open-problem list. This locator identifies the original author-posed question; it does not establish worldwide solution status or priority.

Bounded literature scope

The inspected material comprises arXiv:1404.0484v1, arXiv:1506.06538v1, arXiv:2507.16500v1, an eighteen-record arXiv Jaco-title catalog, a five-result OpenAlex screen, displayed domination material, critical-review metadata and bibliography, linear-Jaco landing material and a preface. No whole-equivalent answer was authenticated in this bounded material. Matching MathDB answer bodies were not acquired: the public entry and supplied access clue returned HTTP 403 with a Cloudflare challenge. Semantic Scholar and a public search index returned HTTP 429; another public index returned HTTP 500. Search-engine challenges and unread journal or follow-up full texts remain gaps. These observations establish neither worldwide absence of an earlier solution nor novelty or priority. The original paper itself explicitly poses the question as open.