On Ramsey Numbers of Short Paths versus Large Wheels

Abstract

For two given graphs $G_1$ and $G_2$, the $Ramseynumber$ $R(G_1,G_2)$ is the smallest integer $n$ such that for any graph $G$ of order $n$, either $G$ contains $G_1$ or the complement of $G$ contains $G_2$. Let $P_n$ denote a path of order $n$ and $W_m$ a wheel of order $m+1$. Chen et al. determined all values of $R(P_n,W_m)$ for $n\ge m-1$. In this paper, we establish the best possible upper bound and determine some exact values for $R(P_n,W_m)$ with $n\le m-2$.