Let J_n(x) be a finite linear Jaco graph and let X be its Delta-set. Non-singleton-Delta-vertex conjecture. The vertex subscripts i for which v_i does not yield a singleton Delta-set X in any finite linear Jaco graph are given by sequence A003622, the Wythoff compound sequence AA, with the stated formula i=lfloor fraci(1+√5)^24rfloor-1,qquad i=2,3,4,…. Equivalently, excluding the first term, sequence A003622 is p-graphical for the parameter identifying the vertex subscript i when v_i is not a Delta-set, over F=G:G=J_n(x), n=1,2,3,…. The claim is motivated by the listed exceptional subscripts; its status is unresolved in the supplied source.

Vertices that never form singleton Delta-sets in finite linear Jaco graphs

From papers

Let $J_n(x)$ be a finite linear Jaco graph and let $X$ be its $\Delta$ -set. Non-singleton-Delta-vertex conjecture. The vertex subscripts $i$ for which $v_i$ does not yield a singleton $\Delta$ -set $X$ in any finite linear Jaco graph are given by sequence A003622, the Wythoff compound sequence AA, with the stated formula

i=\left\lfloor\frac{i(1+\sqrt{5})^2}{4}\right\rfloor-1,\qquad i=2,3,4,\dots.

Equivalently, excluding the first term, sequence A003622 is $p$ -graphical for the parameter identifying the vertex subscript $i$ when $\{v_i\}$ is not a $\Delta$ -set, over $\mathcal{F}=\{G:G=J_n(x),\ n=1,2,3,\dots\}$ . The claim is motivated by the listed exceptional subscripts; its status is unresolved in the supplied source.

Progress summary

Open

A July 2025 paper records the proposed list but supplies no proof or counterexample, so the conjecture remains open.

The conjecture asserts that the vertex subscripts which never produce a singleton $\Delta$ -set in any finite linear Jaco graph $J_n(x)$ are the terms of A003622, the Wythoff compound sequence, via the stated floor formula. A July 2025 preprint labels this Conjecture 2.6 and leaves its proof or disproof for future work.

July 2025 conjecture record

The preprint reports the observed exceptional subscripts, including $4,6,9,12,14,17,19$ , but presents an experimental study rather than a proof. No retrieved source gives a counterexample, verification, referee report, withdrawal, or subsequent claimed solution.

Current status (as of August 2026): The A003622 characterization remains an unresolved conjecture; no proof or counterexample is publicly recorded in the supplied sources.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

Johan Kok, “Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs”, arXiv:2507.16500 (2025).

1 week ago

Put $\varphi=(1+\sqrt5)/2$ and

b_i=\left\lfloor\frac{i+1}{\varphi}\right\rfloor.

The known indegree formula for the Jaco graph gives the exact finite-graph degree

\deg_{J_n}(v_i)=\min\{i,n-b_i\}.

Write $d=\Delta(J_n)$ and $h=n-d$ . Since $b_i$ is nondecreasing, the complete maximum-degree set is

X_n=\{v_i:i\ge d,\ b_i\le h\}.

The indices satisfying $b_i=h$ form the consecutive plateau

\left\lfloor h\varphi\right\rfloor,\ldots, \left\lfloor(h+1)\varphi\right\rfloor-1.

Consequently $X_n=\{v_d\}$ if and only if

d=\left\lfloor(h+1)\varphi\right\rfloor-1, \qquad n=d+h.

Thus, apart from the initial singleton $\{v_1\}$ in $J_1$ , the vertex indices that occur as singleton maximum-degree sets are exactly

d=\left\lfloor t\varphi\right\rfloor-1,\qquad t\ge2.

Beatty's theorem partitions the positive integers into the two disjoint complementary sequences

\{\lfloor t\varphi\rfloor:t\ge1\} \quad\text{and}\quad \{\lfloor t\varphi^2\rfloor:t\ge1\}.

Therefore the indices $i\ge2$ that never occur as singleton maximum-degree sets are precisely

\boxed{\ i=\left\lfloor t\varphi^2\right\rfloor-1,\qquad t\ge2.\ }

This is exactly the conjectured sequence, with its artificial first term removed.

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Shivam Patel · 23 hours ago