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The Erdos-Gallai cycle decomposition conjecture (Erdos Problem #184)

Theorem 1.1: an absolute constant $C$ exists such that the edges of every finite simple graph on $n$ vertices partition into at most $Cn$ simple cycles and single edges. Corollary 1.2: every Eulerian graph partitions into at most $Cn$ cycles, and under a quasirandom cut condition into $\Delta/2+\delta n$ cycles. The constant is not computed or optimized; the conjectured sharp values (Erdos suggested $n-1$ ; $K_{3,n-3}$ -type lower bounds of about $3n/2$ ) are not addressed.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Erdos-Gallai cycle decomposition conjecture (Erdos Problem #184) · Erdos-Gallai cycle decomposition · Problem 184

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: The Erdos-Gallai cycle decomposition conjecture (Erdos Problem #184)

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture and against erdosproblems.com/184; it gives an absolute constant $C$ with every finite simple graph on $n$ vertices partitioned into at most $Cn$ cycles and single edges. Lean: lean/formalization.yaml lists comparator CycleDecomposition with declaration OAI.ErdosGallai.erdos_gallai. The statement ComparatorChallenges/CycleDecomposition.lean was read here: there is a real $C>0$ such that for every $n$ and every SimpleGraph on Fin n there is a $k\le Cn$ and $k$ pairwise disjoint edge sets, each the edge set of a cycle (Walk.IsCycle) or a single edge, whose union is the edge set. This is exactly the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.