Result 181, Combinatorics

The Erdős–Gallai cycle-decomposition conjecture

Proves that the edges of every finite simple undirected graph on n vertices can be partitioned into at most CnCn simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai cycle-decomposition conjecture, bounding the number of pieces linearly even for dense graphs.

Lean formalization Proof

The bigger picture

Why it matters

Can a network's connections be grouped into surprisingly few loops and individual links? The manuscript reports that every finite simple undirected graph admits such a grouping, with the number of groups bounded by a constant times its number of vertices.

What changes?

A finite simple undirected graph consists of vertices joined by edges, with no self-links, duplicate edges or directions. The manuscript claims that, for every such graph on n vertices, its edges can be partitioned into at most C times n pieces. Each piece is either a simple cycle, a closed path with no repeated vertices except its start and end, or one edge. The constant C is independent of the graph.

What does that help mathematicians do?

The claimed result resolves the Erdős–Gallai cycle-decomposition conjecture by controlling the number of pieces through vertices rather than edges. This matters especially for dense graphs, where the number of edges can grow quadratically with the number of vertices. Every edge belongs to exactly one piece, so this is an exact decomposition, not an approximation. Allowing single edges also accommodates graphs containing no cycles.

Are there practical applications?

Its immediate value is foundational: it supplies a uniformly small set of pieces for studying arbitrary graphs. In arguments that handle cycles and individual edges separately, researchers could track only linearly many pieces. The supplied abstract does not give an algorithm or running-time guarantee, so it does not establish an efficient practical procedure for finding the partition.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A linear cycle-and-edge decomposition of every graph

September 24, 2026 32 pages Main result formalized in Lean

We prove that every finite simple undirected graph on n vertices has an edge partition into at most CnCn simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai cycle decomposition conjecture positively.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026,
  author = {{OpenAI}},
  title = {{A linear cycle-and-edge decomposition of every graph}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf}{OAI:A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/181.md.

The Erdős–Gallai cycle-decomposition conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The Erdős–Gallai cycle-decomposition conjecture asks for a linear bound on the number of cycles and single edges needed to partition a graph's edges. The formalized result gives one absolute constant C>0C>0 such that every finite simple graph on nn vertices has an edge-disjoint decomposition into at most CnCn cycles or singleton edges. Edgeless and small graphs are included. The optimal value of CC is not determined.

Comparator links

Result Comparator statement
Linear cycle-and-edge decomposition CycleDecomposition.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.