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Establish structural and computational constraints on hypothetical 43-vertex Ramsey(5,5)-free graphs to restrict candidate search spaces.

August 14, 2026Open Access

Extension Coherence and Near-Extremal Phase Reduction for R(5, 5)

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CLChengjun Lyu CLChang Lyu

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Implication

Computational graph-theoretic analysis demonstrates structural constraints on hypothetical 43-vertex R(5,5)-free graphs, narrowing the balanced search space.

Key Points

Establish structural and computational constraints on hypothetical 43-vertex Ramsey(5,5)-free graphs to restrict candidate search spaces.
Formulated an extension-deletion coherence theory analyzing defect supports and card gluing at graph orders 41 to 43.
Reparameterized the McKay–Radziszowski excess identity with exact edge deficits and evaluated quartic potentials across 352,366 published R(4,5,24) classes using two motif-counting implementations.
Conducted an exhaustive search over 19,398,656 candidate extension rows across 74 order-18 cores and classified 62,382 order-18/order-24 pairs via endpoint-capacity tests.
Demonstrated that validity on five vertex-deleted cards strictly forces a genuine global 43-vertex extension.
Excluded the balanced 18/24 complement pair across all 352,366 R(4,5,24) catalogue classes, narrowing surviving balanced 43-vertex counterexamples strictly to regular degrees 20 and 22.
Found zero legal extension rows among 19,398,656 candidates at order-18 edge-85 cores, compressing the equality-five frontier to 67 rooted low-skeleton orbits.

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Lyu et al. (2026) studied this question.

synapsesocial.com/papers/6a7ec735b70b84ec8b913637 https://doi.org/10.5281/zenodo.21899446

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Implication

Computational graph-theoretic analysis demonstrates structural constraints on hypothetical 43-vertex R(5,5)-free graphs, narrowing the balanced search space.

Extension Coherence and Near-Extremal Phase Reduction for R(5, 5)

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Cite This Study

Extension Coherence and Near-Extremal Phase Reduction for R(5, 5)

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Implication

Key Points

Cite This Study

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