Proved
Jaeger equivalence recalled and proved ↗
Finite loopless cubic multigraphs.
Source dated Aug 8, 2026 · Observed Oct 14, 2026
Equivalence · version 1
Petersen and normal-five colorings are equivalent
Statement
For each finite loopless cubic multigraph, the existence of a Petersen coloring is equivalent to the existence of a normal five-edge-coloring.
Assumptions
- The paper gives an explicit bijection using P = KG(5,2). Bridgelessness is not needed for this equivalence.
Evidence
Source reports
Connections
In this problem · publication v4 · Selected context
Petersen coloring conjecture →
Supporting result · Proved
Equivalent statements
equivalent to ↕Normal five-edge-colorings for bridgeless cubic graphs
Jaeger’s theorem gives equivalent existence conditions for each loopless cubic graph.
The positive universal assertions are both refuted by the reported witness; equivalence remains a valid theorem.
Details & sources
Every bridgeless cubic graph admits a Petersen coloring →
Disproved · version 1
Every finite loopless bridgeless cubic graph admits an edge map to the Petersen graph that maps each incident edge-star bijectively onto a target vertex’s edge-star.
- Parallel edges are allowed; the simple connected witness already lies in this class.
Scope: Parallel edges are allowed; the simple connected witness already lies in this class.
Normal five-edge-colorings for bridgeless cubic graphs →
Disproved · version 1
Every finite loopless bridgeless cubic graph has a proper edge-coloring with at most five colors in which the union of the palettes at the endpoints of every edge has either three or five colors.
- A poor edge has three colors across its endpoint stars; a rich edge has five. Four is forbidden.
Scope: A poor edge has three colors across its endpoint stars; a rich edge has five. Four is forbidden.
Petersen and normal-five colorings are equivalent →
Relationship statement · Proved · version 1
For each finite loopless cubic multigraph, the existence of a Petersen coloring is equivalent to the existence of a normal five-edge-coloring.
- The paper gives an explicit bijection using P = KG(5,2). Bridgelessness is not needed for this equivalence.
Scope: The paper gives an explicit bijection using P = KG(5,2). Bridgelessness is not needed for this equivalence.
Relationship not independently reviewed.
Connection in Petersen coloring conjecture →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v7
Petersen coloring conjecture →
Supporting result · Proved
Equivalent statements
equivalent to ↕Normal five-edge-colorings for bridgeless cubic graphs
Jaeger’s theorem gives equivalent existence conditions for each loopless cubic graph.
The positive universal assertions are both refuted by the reported witness; equivalence remains a valid theorem.
Details & sources
Every bridgeless cubic graph admits a Petersen coloring →
Disproved · version 1
Every finite loopless bridgeless cubic graph admits an edge map to the Petersen graph that maps each incident edge-star bijectively onto a target vertex’s edge-star.
- Parallel edges are allowed; the simple connected witness already lies in this class.
Scope: Parallel edges are allowed; the simple connected witness already lies in this class.
Normal five-edge-colorings for bridgeless cubic graphs →
Disproved · version 1
Every finite loopless bridgeless cubic graph has a proper edge-coloring with at most five colors in which the union of the palettes at the endpoints of every edge has either three or five colors.
- A poor edge has three colors across its endpoint stars; a rich edge has five. Four is forbidden.
Scope: A poor edge has three colors across its endpoint stars; a rich edge has five. Four is forbidden.
Petersen and normal-five colorings are equivalent →
Relationship statement · Proved · version 1
For each finite loopless cubic multigraph, the existence of a Petersen coloring is equivalent to the existence of a normal five-edge-coloring.
- The paper gives an explicit bijection using P = KG(5,2). Bridgelessness is not needed for this equivalence.
Scope: The paper gives an explicit bijection using P = KG(5,2). Bridgelessness is not needed for this equivalence.
Relationship not independently reviewed.
Connection in Petersen coloring conjecture →Relationship not independently reviewed. This concerns its use in this problem.
Sources
A 112-Vertex Counterexample to the Petersen Coloring Conjecture →
- Introduction; Proposition 2.1; references to Jaeger (1985, 1988) → · Original passage ↗
Recorded wording
The edge-star formulation is equivalent to the existence of a normal five-edge-coloring. The original conjecture concerns every bridgeless cubic graph.
paraphrase · Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.
- Theorem 1.1; Section 3; Appendix A → · Original passage ↗
Recorded wording
The main graph has 112 vertices, 168 edges, girth five and edge- and vertex-connectivity three. Section 3 specifies its recursive construction.
paraphrase · Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.
Bryce Putman · Preprint · Publication date Aug 8, 2026
Source-reported results and external checks are separate from independent OpenMath assessment.
Revision & attribution
Source revision: v1. Observed Oct 14, 2026.
Putman reports extensive OpenAI-model assistance in discovery, search, verification and preparation; he reviewed the final claims and artifacts and accepts responsibility.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Jaeger equivalence recalled and proved ↗
Finite loopless cubic multigraphs.
Source dated Aug 8, 2026 · Observed Oct 14, 2026 · Source passage →
Record details
- Claim ID
- claim-petersen-normal-five-equivalence
- Statement version
- 1
- SHA-256
- a141512647d61f533f312e2763b62f0572ebe1c7af3cc97fe747077eb1b773f6