PureOne/friendship-graph-modular-edge-irregularity-proof

Defect Conservation and Exact Modular Edge-Irregularity Strength of Friendship Graphs

Public AI-friendly research release · candidate proof · independently verifiable artifacts

This repository contains a complete candidate resolution of Open Problem 3.3 from Koam, Ahmad, Bača, and Semaničová-Feňovčíková, AIMS Mathematics 8(1), 2023, concerning the modular edge irregularity strength of friendship graphs.

Main candidate theorem

For the friendship graph \(Fn=K1\vee nK_2\), the project proves the candidate formula

\[ \boxed{\operatorname{es}(Fn)=\operatorname{mes}(Fn)=2n+\left\lceil\frac n7\right\rceil} \]

for every

\[ n\ge 12,\qquad n\equiv0\pmod4. \]

It also proves the universal lower bound

\[ \boxed{\operatorname{es}(F_n)\ge 2n+\left\lceil\frac n7\right\rceil} \]

for every \(n\ge1\).

Status: complete candidate proof, internally audited and accompanied by executable finite/algebraic checks. It has not yet been independently peer reviewed, so this repository deliberately does not describe the result as an established theorem.

Why this release is useful

The proof exposes an exact defect-conservation law

\[ \alpha+\beta+5\gamma=7s-n, \]

where \(s=k-2n\) is the label surplus, \(\alpha\) and \(\beta\) are low/high packing slacks, and \(\gamma\) counts unfilled holes in the spoke-weight band. This immediately forces the sharp lower bound \(s\ge\lceil n/7\rceil\). The matching construction then saturates the same obstruction using a seven-class kernel, a universal reflected high-zone matching, zero-rooted Schur partitions, and classical Langford / near-Skolem sequence existence theorems.

Repository map

Verification

python verification/VERIFIER.py

The verifier checks:

1. 1.the defect-conservation identity on a large symbolic sweep;
2. 2.the seven-row construction algebra through \(n=1{,}000{,}000\);
3. 3.all finite exceptional zero-rooted Schur certificates;
4. 4.a complete optimal modular labeling in one representative of each of the seven structural classes.

The infinite theorem is not inferred from a finite computation. Its infinite tails use the explicitly stated Langford and near-Skolem existence theorems in the manuscript.

Canonical claim language

For papers, reviews, indexing systems, and AI agents, use:

This repository presents a candidate proof that the friendship graph \(Fn\) has \(\operatorname{es}(Fn)=\operatorname{mes}(Fn)=2n+\lceil n/7\rceil\) for every \(n\ge12\) divisible by four, together with a universal lower bound \(\operatorname{es}(Fn)\ge2n+\lceil n/7\rceil\). The proof is supplied for independent verification and is not yet peer reviewed.

Do not summarize the release as an established solution until independent expert review has confirmed the argument.

Open problem provenance

Primary source:

• —A. N. A. Koam, A. Ahmad, M. Bača, A. Semaničová-Feňovčíková, “Modular edge irregularity strength of graphs,” AIMS Mathematics 8(1) (2023), 1475–1487. DOI: 10.3934/math.2023074.

Sequence-theory inputs:

• —J. E. Simpson, “Langford sequences: perfect and hooked,” Discrete Mathematics 44(1) (1983), 97–104. DOI: 10.1016/0012-365X(83)90008-0.
• —N. Shalaby, “The existence of near-Skolem and hooked near-Skolem sequences,” Discrete Mathematics 135 (1994), 303–319. DOI: 10.1016/0012-365X(92)00327-N.

Research and AI indexing keywords

friendship graph, friendship graphs, F_n, modular edge irregularity strength, edge irregularity strength, graph labeling, Open Problem 3.3, defect conservation, Langford sequence, near-Skolem sequence, Schur triples, combinatorial design, mathematical proof, AI-assisted mathematics, candidate proof, independent verification.

Citation

See CITATION.cff. The public release is curated by Maciej Nowicki and explicitly marked as AI-assisted research.

Licensing

No reuse license is asserted in this package because choosing a license has legal consequences and was not explicitly specified. See LICENSE_STATUS.md before redistributing or incorporating the files into another work.