Ramsey numbers of cycles versus multiple wheels

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 $K_N$ contains either a red copy of $G$ or a blue copy of $H$. We denote by $C_n$ the cycle on $n$ vertices and by $tW_{2m}$ the disjoint union of $t$ copies of the wheel $W_{2m}$. We prove that for $m\ge 2$ and $n\ge 4tm$,
\[R(C_n, tW_{2m})=2n+t-2.\]
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.