Star-Critical Ramsey Numbers of Cycles Versus Wheels

Abstract

For two graphs $G_1$ and $G_2$, the star-critical Ramsey number $r_*(G_1,G_2)$ is the minimum integer k such that any red/blue edge-coloring of $K_{r-1}\sqcup K_{1,k}$ contains a red copy of $G_1$ or a blue copy of $G_2$, where r is the classical Ramsey number $R(G_1,G_2)$ and $K_{r-1}\sqcup K_{1,k}$ is the graph obtained from a $K_{r-1}$ and an additional vertex v by joining v to k vertices of $K_{r-1}$. Let $C_n$ denote a cycle of order n and $W_m$ a wheel of order $m+1$. Hook (2010) proved that $r_*(C_n,W_3)=2n$ for $n\ge 5$. In this paper, it is shown that $r_*(C_n,W_m)=2n$ for m odd, $n\ge m\ge 5$ and $n\ge 60$.