On the Ramsey number of 4-cycle versus wheel

Abstract

For any fixed graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest positive integer $n$ such that for every graph $F$ on $n$ vertices must contain $G$ or the complement of $F$ contains $H$. The girth of graph $G$ is a length of the shortest cycle. A $k$-regular graph with the girth $g$ is called a $(k,g)$-graph. If the number of of vertices in $(k,g)$-graph is minimized then we call this graph a $(k,g)$-cage. In this paper, we derive the bounds of Ramsey number $R(C_4,W_n)$ for some values of $n$. By modifying $(k, 5)$-graphs, for $k = 7$ or $9$, we construct these corresponding $(C_4,W_n)$-good graphs.