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Theorems 1.1 and 6.1, Proposition 2.1, Corollary 5.1, Appendices A–B and computational provenance.
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The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.
A second, nonisomorphic 112-vertex graph has automorphism group D₃ of order six and separately checked coloring certificates.
Passage ID: petersen-coloring:entry:alternative:v1
Source item: alternative
Source record: petersen-coloring:source:paper:v1
The finite counterexample together with the cited universal-target theorem implies infinitely many connected simple bridgeless cubic counterexamples.
Passage ID: petersen-coloring:entry:infinite:v1
Source item: infinite
Source record: petersen-coloring:source:paper:v1
The edge-star formulation is equivalent to the existence of a normal five-edge-coloring. The original conjecture concerns every bridgeless cubic graph.
Passage ID: petersen-coloring:entry:question:v1
Source item: question
Source record: petersen-coloring:source:paper:v1
The main graph has 112 vertices, 168 edges, girth five and edge- and vertex-connectivity three. Section 3 specifies its recursive construction.
Passage ID: petersen-coloring:entry:witness:v1
Source item: witness
Source record: petersen-coloring:source:paper:v1
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Problem publication v7
Source assessment
Proved
Second nonisomorphic counterexample reported
Exact D₃-symmetric graph in Theorem 6.1.
Source date Aug 8, 2026 · Observed Oct 14, 2026
Source assessment
Proved
Infinite family consequence reported
Uses the separate Ma–Mattiolo–Steffen–Wolf theorem.
Source date Aug 8, 2026 · Observed Oct 14, 2026
Source assessment
Proved
Jaeger equivalence recalled and proved
Finite loopless cubic multigraphs.
Source date Aug 8, 2026 · Observed Oct 14, 2026
Connection
Jaeger’s theorem gives equivalent existence conditions for each loopless cubic graph.
Statement
Source assessment
Disproved
Counterexample reported with checked SAT certificates
Universal Petersen coloring assertion; minimum counterexample order is not determined.
Source date Aug 8, 2026 · Observed Oct 14, 2026
Statement
Source assessment
Disproved
Counterexample reported with checked SAT certificates
Universal Petersen coloring assertion; minimum counterexample order is not determined.
Source date Aug 8, 2026 · Observed Oct 14, 2026
Source assessment
Proved
Explicit main graph and noncolorability reported
Exact main labeled graph in Theorem 1.1.
Source date Aug 8, 2026 · Observed Oct 14, 2026
Proof explanation
Two independent coloring encodings
History
Two explicit counterexamples and certificates
Question
Petersen coloring conjecture
Definition
Incident edge-star
Definition
Bridgeless cubic graph
History
Jaeger’s coloring question
History
The normal-five formulation
Further reading
Read the coloring definition and Jaeger equivalence.
Further reading
Read the main construction and exact labeled graph.
Recorded authorship
Bryce Putman
Putman reports extensive OpenAI-model assistance in discovery, search, verification and preparation; he reviewed the final claims and artifacts and accepts responsibility.
OpenMath source records preserve exact citations. Observation dates order these records; they do not establish the latest upstream paper or proof.
A 112-Vertex Counterexample to the Petersen Coloring Conjecture
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