# Graph cycle decomposition research toolkit

Family 181: The Erdős–Gallai cycle-decomposition conjecture. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Graph-analysis teams could compare edge decompositions into cycles and singleton edges with the claimed linear number of pieces.

## Applicability and commercial boundary

The universal constant, construction cost and allowed singleton pieces matter. This does not solve arbitrary constrained flow or routing objectives.

## Initial business decision

Research or conditional engineering only until an effective implementation and a recurring buyer problem are identified. No commercial demand or profitability is established. Where a direct product bridge is weak, the legitimate use is a research reference or evidence adapter, rather than a new standalone company.

## Next verification action

Extract the decomposition algorithm and a finite baseline before any integration.

## Evidence scope

The catalog statement was individually reviewed. The manuscript proof and selected formal statement have not yet been compared in depth for this family. No independent proof verification was run. Pinned source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[A linear cycle-and-edge decomposition of every graph](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf).