What does this research mean for the field?

The three known counterexamples to the Petersen coloring conjecture still satisfy the Berge-Fulkerson and 5-cycle double cover conjectures, and all exhibit a Petersen defect of exactly 2, consistent with a newly proven parity theorem. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To audit and verify the structural graph-theoretic properties and surviving conjectures on three recently discovered counterexamples to Jaeger's Petersen coloring conjecture.

September 5, 2026Open Access

Berge-Fulkerson covers, cycle double covers, flows and exact normality defects of the first counterexamples to the Petersen coloring conjecture

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Authors

FSFelipe Santibañez-LealOpen University of Cyprus

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Implication

Computational analysis reveals major structural invariants in three Petersen coloring counterexamples, indicating survival of related classical graph conjectures.

Key Points

To audit and verify the structural graph-theoretic properties and surviving conjectures on three recently discovered counterexamples to Jaeger's Petersen coloring conjecture.
Generated independent propositional satisfiability (SAT) encodings to verify the absence of Petersen colorings and normal 5-edge-colorings across three counterexamples (two 112-vertex graphs and one 52-vertex graph), validated with drat-trim proof checking.
Constructed and certified explicit witnesses for alternative structural conjectures and computed exact topological and coloring graph invariants.
Performed a counterexample-guided search over compositions of the Petersen 4-pole below 52 vertices.
All three counterexamples admit a Berge-Fulkerson cover, a 5-cycle double cover, a nowhere-zero 5-flow, a Fan-Raspaud triple, and have a perfect matching index of exactly 4 and normal chromatic index of 6.
The 52-vertex graph exhibits an oddness of 2 and resistance of 2, whereas both 112-vertex graphs exhibit an oddness of 4 and resistance of 3.
The Petersen defect is universally exactly 2 across all three graphs, with every vertex pair serving as a critical repair witness (1,326 pairs in the 52-vertex graph and 6,216 pairs in each 112-vertex graph).

Cite This Study

Felipe Santibañez-Leal (2026) studied this question.

synapsesocial.com/papers/6a9bd4126b95aff0620eb782 https://doi.org/10.5281/zenodo.22285165

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Implication

Computational analysis reveals major structural invariants in three Petersen coloring counterexamples, indicating survival of related classical graph conjectures.

Berge-Fulkerson covers, cycle double covers, flows and exact normality defects of the first counterexamples to the Petersen coloring conjecture

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Implication

Key Points

Cite This Study

Berge-Fulkerson covers, cycle double covers, flows and exact normality defects of the first counterexamples to the Petersen coloring conjecture

Discussion

Member takes

Implication

Key Points

Cite This Study

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