OpenAI · 181-01 · 2026-09-24 · Combinatorics
A linear cycle-and-edge decomposition of every graph
preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026
Family 181. The Erdős–Gallai cycle-decomposition conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves that the edges of every finite simple undirected graph on vertices can be partitioned into at most simple cycles and single edges, for an absolute constant . This resolves the Erdős–Gallai cycle-decomposition conjecture, bounding the number of pieces linearly even for dense graphs.
We prove that every finite simple undirected graph on n vertices has an edge partition into at most simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai cycle decomposition conjecture positively.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 181: The Erdős–Gallai cycle-decomposition conjecture. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.