What does this research mean for the field?

For m ≥ 2 and n ≥ 4tm, the Ramsey number R(C_n, tW_{2m}) for a cycle on n vertices and t disjoint copies of a wheel on 2m vertices is exactly 2n + t - 2. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The aim is to determine the Ramsey number for cycles versus multiple wheels and generalize prior results.

May 30, 2026Open Access

Ramsey numbers of cycles versus multiple wheels

Key Points

The aim is to determine the Ramsey number for cycles versus multiple wheels and generalize prior results.
Provided mathematical proof for R(C_n, tW_{2m}) when m ≥ 2 and n ≥ 4tm.
Analyzed edge-coloring in complete graphs to establish Ramsey number relationships.
Generalized findings from several previous studies in graph theory.
R(C_n, tW_{2m}) equals 2n + t - 2 for the specified conditions.
Confirms Sudarsana’s conjecture regarding wheels of odd order.
Result builds on and extends previous research in graph colorings.

Abstract

For given graphs G and H, the Ramsey number R (G, H) is defined as the smallest positive integer N such that every red-blue edge-coloring of the complete graph KN contains either a red copy of G or a blue copy of H. We denote by Cₙ the cycle on n vertices and by tW₂₌ the disjoint union of t copies of the wheel W₂₌. We prove that for m ≥ 2 and n ≥ 4tm, R (Cₙ, tW₂₌) = 2n + t-2. This result generalizes previous findings by Surahmat, Baskoro, and Tomescu (Discrete Math. , 2006), Chen, Cheng, Miao, and Ng (Appl. Math. Lett. , 2009), Zhang, Broersma, and Chen (Graph Combin. , 2015), as well as Sudarsana (Electron. J. Graph Theory Appl. , 2021). Furthermore, the result confirms Sudarsana’s conjecture for wheels of odd order.

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