What does this research mean for the field?

Ordinary circulant Ramsey(5,5;42) and Ramsey(5,5;43) colorings do not exist. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To determine whether valid two-color Ramsey(5,5) colorings exist within ordinary circulant graphs on 42 and 43 vertices.

September 5, 2026Open Access

Exhaustive Nonexistence of Circulant Ramsey(5,5;42/43) Colorings and Exploratory Symmetry-Constrained Search

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Authors

VIVladi Ivanov

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Implication

Computational analysis demonstrates the nonexistence of circulant Ramsey(5,5) colorings on 42 and 43 vertices, indicating that symmetric structures cannot improve existing lower bounds.

Key Points

To determine whether valid two-color Ramsey(5,5) colorings exist within ordinary circulant graphs on 42 and 43 vertices.
Conducted exhaustive computational exclusions of ordinary circulant graphs of orders 42 and 43.
Implemented exploratory symmetry-constrained and unconstrained tabu searches on complete graphs K42 and K43.
Applied neighborhood degree bounds establishing that 17 ≤ d_red(v), d_blue(v) ≤ 24 for n = 42 based on Ramsey(4,5) subgraphs.
Exhaustively eliminated ordinary circulant graphs as candidates for Ramsey(5,5;42) and Ramsey(5,5;43) colorings.
Identified best exploratory tabu search monochromatic clique values of 84 for K42 and 129 for K43.

Cite This Study

Vladi Ivanov (2026) studied this question.

synapsesocial.com/papers/6a9bd3726b95aff0620eaba8 https://doi.org/10.5281/zenodo.22272633

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Implication

Computational analysis demonstrates the nonexistence of circulant Ramsey(5,5) colorings on 42 and 43 vertices, indicating that symmetric structures cannot improve existing lower bounds.

Exhaustive Nonexistence of Circulant Ramsey(5,5;42/43) Colorings and Exploratory Symmetry-Constrained Search

Discussion

Member takes

Implication

Key Points

Cite This Study

Exhaustive Nonexistence of Circulant Ramsey(5,5;42/43) Colorings and Exploratory Symmetry-Constrained Search

Discussion

Member takes

Implication

Key Points

Cite This Study

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