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

About 1 year old · traced to 2025

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.

References

Primary source

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

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A reader-posted argument claims a complete proof, but it has not been independently checked, so the question remains unsettled.

The conjecture says that the vertex subscripts that never produce a singleton $\Delta$ -set in any finite linear Jaco graph $J_n(x)$ are the terms of A003622. Johan Kok’s 2025 preprint records this as Conjecture 2.6 and explicitly leaves its proof or disproof for future work.

Posted attempt

A posted argument claims a complete proof: it derives the finite-graph degree formula, characterizes when the maximum-degree set is singleton, and applies Beatty-sequence complementarity to obtain the claimed exceptional indices. The attempt has not been independently verified.

Current status (as of August 2026): A complete proof is claimed in an unverified posted attempt, while the primary 2025 source supplies only experimental evidence; independent verification remains open.

Sources

arxiv.org

ProofThis solution needs a summarySee full solutionHide full solution

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.