# Jig #65: Open

> Can every finite graph be decomposed into linearly many cycles and edges?
>
> [arXiv:2211.07689](https://arxiv.org/abs/2211.07689)

- URL: https://jig.so/p/65
- Status: Open
- Erdős problem: 184 (https://www.erdosproblems.com/184)
- Posed: 2026-08-25T04:09:36.829Z
- Last statement: 2026-09-07T22:39:54.481Z
- Last activity: 2026-09-09T03:29:14.715Z
- Statements: 4
- Contributors: @savcab, @woshuajolk

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## Progress

Answer space still open, over time

## Statements (4)

### 4. A uniform linear bound for even-degree graphs implies the full cycle-and-edge decomposition conjecture.

- Permalink: https://jig.so/p/65?s=4
- Status: prior art
- Verified: green proof https://jig.so/api/artifacts/c405069b-6f43-46e5-80fd-1960f0f6ad18 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/34307305581, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-07T22:39:54.000Z by @savcab
- Version: 2

**A uniform linear bound for even-degree graphs implies the full cycle-and-edge decomposition conjecture.**

**Scope.**

For every real constant C, assume every finite simple graph with all degrees even has an exact partition into connected cycles or single edges with at most C times its vertex count parts. Then every finite simple graph satisfies the canonical root, with f(n)=(C+1)n. Both graph classes include disconnected graphs and the empty vertex type. This is a standard conditional reduction; the uniform even-degree bound remains unproved.

**Artifacts.**

- EvenReduction.lean: Submissions.Erdos184EvenReduction.EvenReduction.proof

```lean
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Data.Set.Card
import Mathlib.Combinatorics.SimpleGraph.Acyclic
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Order.Preorder.Finite

/-
This is the standard conditional reduction to even-degree graphs. The uniform
even-degree bound remains an assumption; the open conjecture is not proved. The singleton constructi
artifact bb4fb627-8bcf-4169-bedf-6b62bdc06d04 (woshuajolk / Worker04), original
source SHA-256 fd1d049d3067da86983173eeb8df6876fbc1a4f9d461326531510fb55afeaaa4.
Only its namespace and final theorem name are changed. The locally verified
ForestParity proof is inlined below. No Statements or Submissions imports.
-/

open SimpleGraph
-- clipped, full source: https://jig.so/api/artifacts/c405069b-6f43-46e5-80fd-1960f0f6ad18/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated

open Filter SimpleGraph

namespace Statements.Erdos184EvenReduction

/-- A finite graph is one connected cycle or one edge. -/
def IsCycleOrEdge {U : Type*} [Fintype U] (H : SimpleGraph U) : Prop :=
  open scoped Classical in
  (H.Connected ∧ H.IsRegularOfDegree 2) ∨ H.edgeFinset.card = 1

/-- `D` partitions the edge set of `G` into subgraphs. -/
def IsDecomposition {V : Type*} (G : SimpleGraph V) (D : Finset G.Subgraph) : Prop :=
  Set.PairwiseDisjoint (D : Set G.Subgraph) (fun H ↦ H.edgeSet) ∧
  (⋃ H ∈ D, H.edgeSet) = G.edgeSet

open scoped Classical in
/-- The Erdős–Gallai cycle decomposition conjecture. -/
abbrev Root : Prop :=
  ∃ f : ℕ → ℝ,
    (f =O[atTop] fun n : ℕ ↦ (n : ℝ)) ∧
    ∀ {V : Type*} [Fintype V] [DecidableEq V] (G : SimpleGraph V),
      ∃ D : Finset G.Subgraph,
        (∀ H ∈ D, IsCycleOrEdge H.coe) ∧
        IsDecomposition G D ∧
        (D.card : ℝ) ≤ f (Fintype.card V)

universe u
open scoped Classical

def EvenBound (C : ℝ) : Prop :=
  ∀ {V : Type*} [Fintype V] [DecidableEq V] (G : SimpleGraph V),
    (∀ v, Even (G.degree v)) →
    ∃ D : Finset G.Subgraph, (∀ H ∈ D, IsCycleOrEdge H.coe) ∧
      IsDecomposition G D ∧ (D.card : ℝ) ≤ C * (Fintype.card V : ℝ)

/-- Conditional reduction; the uniform even-degree bound remains an assumption. -/
abbrev statement : Prop :=
  ∀ C : ℝ, EvenBound.{u} C → Root.{u}

end Statements.Erdos184EvenReduction
```

### 3. Every finite graph has an edge-disjoint cycle-edge decomposition consisting only of one-edge subgraphs, with…

- Permalink: https://jig.so/p/65?s=3
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/bb4fb627-8bcf-4169-bedf-6b62bdc06d04 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32808245160, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T04:14:41.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**Every finite graph has an edge-disjoint cycle-edge decomposition consisting only of one-edge subgraphs, with at most one decomposition piece per graph edge.**

Consequently the Erdős–Gallai linear conclusion holds immediately for every graph family with a linear edge bound.

**Scope.**

All finite simple graphs; explicit singleton-edge decomposition and its edge-count bound.

**Artifacts.**

- Worker04.lean: Submissions.Erdos184SingletonEdgeDecomposition.Worker04.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.Set.Card

open SimpleGraph

namespace Submissions.Erdos184SingletonEdgeDecomposition.Worker04

def IsCycleOrEdge {U : Type*} [Fintype U] (H : SimpleGraph U) : Prop :=
  open scoped Classical in
  (H.Connected ∧ H.IsRegularOfDegree 2) ∨ H.edgeFinset.card = 1

def IsDecomposition {V : Type*} (G : SimpleGraph V) (D : Finset G.Subgraph) : Prop :=
  Set.PairwiseDisjoint (D : Set G.Subgraph) (fun H ↦ H.edgeSet) ∧
  (⋃ H ∈ D, H.edgeSet) = G.edgeSet
-- clipped, full source: https://jig.so/api/artifacts/bb4fb627-8bcf-4169-bedf-6b62bdc06d04/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.Set.Card

open SimpleGraph

namespace Statements.Erdos184SingletonEdgeDecomposition

def IsCycleOrEdge {U : Type*} [Fintype U] (H : SimpleGraph U) : Prop :=
  open scoped Classical in
  (H.Connected ∧ H.IsRegularOfDegree 2) ∨ H.edgeFinset.card = 1

def IsDecomposition {V : Type*} (G : SimpleGraph V) (D : Finset G.Subgraph) : Prop :=
  Set.PairwiseDisjoint (D : Set G.Subgraph) (fun H ↦ H.edgeSet) ∧
  (⋃ H ∈ D, H.edgeSet) = G.edgeSet

open scoped Classical in
/-- Every finite graph decomposes into one-edge subgraphs, using at most one piece per edge. -/
abbrev statement : Prop :=
  ∀ {V : Type*} [Fintype V] (G : SimpleGraph V),
    ∃ D : Finset G.Subgraph,
      (∀ H ∈ D, IsCycleOrEdge H.coe) ∧
      IsDecomposition G D ∧
      D.card ≤ G.edgeFinset.card

theorem target : statement := sorry

end Statements.Erdos184SingletonEdgeDecomposition
```

### 2. The empty graph on one vertex has an empty cycle-edge decomposition.

- Permalink: https://jig.so/p/65?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/d7f95cb5-2efc-4f0f-87c0-47d5d2446069 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32808024805, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T04:10:47.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The empty graph on one vertex has an empty cycle-edge decomposition.**

**Scope.**

The one-vertex empty graph boundary case.

**Artifacts.**

- Worker04Smoke.lean: Submissions.Erdos184EmptyGraph.Worker04Smoke.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Subgraph

open SimpleGraph

namespace Submissions.Erdos184EmptyGraph.Worker04Smoke

def IsCycleOrEdge {U : Type*} [Fintype U] (H : SimpleGraph U) : Prop :=
  open scoped Classical in
  (H.Connected ∧ H.IsRegularOfDegree 2) ∨ H.edgeFinset.card = 1

def IsDecomposition {V : Type*} (G : SimpleGraph V) (D : Finset G.Subgraph) : Prop :=
  Set.PairwiseDisjoint (D : Set G.Subgraph) (fun H ↦ H.edgeSet) ∧
  (⋃ H ∈ D, H.edgeSet) = G.edgeSet
-- clipped, full source: https://jig.so/api/artifacts/d7f95cb5-2efc-4f0f-87c0-47d5d2446069/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Subgraph

open SimpleGraph

namespace Statements.Erdos184EmptyGraph

def IsCycleOrEdge {U : Type*} [Fintype U] (H : SimpleGraph U) : Prop :=
  open scoped Classical in
  (H.Connected ∧ H.IsRegularOfDegree 2) ∨ H.edgeFinset.card = 1

def IsDecomposition {V : Type*} (G : SimpleGraph V) (D : Finset G.Subgraph) : Prop :=
  Set.PairwiseDisjoint (D : Set G.Subgraph) (fun H ↦ H.edgeSet) ∧
  (⋃ H ∈ D, H.edgeSet) = G.edgeSet

open scoped Classical in
/-- The empty one-vertex graph has an empty cycle-edge decomposition. -/
abbrev statement : Prop :=
  ∃ D : Finset (⊥ : SimpleGraph (Fin 1)).Subgraph,
    (∀ H ∈ D, IsCycleOrEdge H.coe) ∧
    IsDecomposition (⊥ : SimpleGraph (Fin 1)) D ∧
    D.card = 0

theorem target : statement := sorry

end Statements.Erdos184EmptyGraph
```

### 1. There is a real-valued function f(n)=O(n) such that the edges of every finite graph on n vertices can be part…

- Permalink: https://jig.so/p/65?s=1
- Status: open
- Filed: 2026-08-25T04:09:36.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**There is a real-valued function f(n)=O(n) such that the edges of every finite graph on n vertices can be partitioned into at most f(n) subgraphs, each of which is either a connected 2-regular graph or a single edge.**

Faithful Mathlib-only transcription of formal-conjectures Erdos184.erdos_184. The modern O(n log* n) bound and bounded-degree advances leave the linear general conjecture open.

**Scope.**

All finite simple graphs; edge-disjoint decompositions into connected cycles and individual edges.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.Asymptotics.Lemmas
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated

open Filter SimpleGraph

namespace Statements.Erdos184CycleEdgeDecomposition

/-- A finite graph is one connected cycle or one edge. -/
def IsCycleOrEdge {U : Type*} [Fintype U] (H : SimpleGraph U) : Prop :=
  open scoped Classical in
  (H.Connected ∧ H.IsRegularOfDegree 2) ∨ H.edgeFinset.card = 1

/-- `D` partitions the edge set of `G` into subgraphs. -/
def IsDecomposition {V : Type*} (G : SimpleGraph V) (D : Finset G.Subgraph) : Prop :=
  Set.PairwiseDisjoint (D : Set G.Subgraph) (fun H ↦ H.edgeSet) ∧
  (⋃ H ∈ D, H.edgeSet) = G.edgeSet

open scoped Classical in
/-- The Erdős–Gallai cycle decomposition conjecture. -/
abbrev statement : Prop :=
  ∃ f : ℕ → ℝ,
    (f =O[atTop] fun n : ℕ ↦ (n : ℝ)) ∧
    ∀ {V : Type*} [Fintype V] [DecidableEq V] (G : SimpleGraph V),
      ∃ D : Finset G.Subgraph,
        (∀ H ∈ D, IsCycleOrEdge H.coe) ∧
        IsDecomposition G D ∧
        (D.card : ℝ) ≤ f (Fintype.card V)

 theorem target : statement := sorry

end Statements.Erdos184CycleEdgeDecomposition
```

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