RAMSEY NUMBERS OF CYCLES VERSUS MULTIPLE ODD WHEELS

Abstract Given two graphs G and H , the Ramsey number $R(G,H)$ is the smallest positive integer N such that every graph of order N contains G or its complement contains H as a subgraph. Let $C_n$ denote the cycle on n vertices and let $tW_{2m+1}$ denote the disjoint union of t copies of the $(2m+2)$ -vertex wheel $W_{2m+1}$ . We show that for integers $m\ge 1$ , $t\ge 2$ and $n\ge (6m+3)t-6m+999$ , $$ \begin{align*} R(C_n, tW_{2m+1})=3n+t-3. \end{align*} $$ This result extends several previous results and settles a conjecture posed by Sudarsana [‘A note on the Ramsey number for cycle with respect to multiple copies of wheels’, Electron. J. Graph Theory Appl. 9 (2) (2021), 561–566].