All Quadrilateral-Wheel Planar Ramsey Numbers

Abstract

For two given graphs $G_1$ and $G_2$, the planar Ramsey number ${\textit{PR}}(G_1,G_2)$ is the smallest integer n such that every planar graph G on n vertices either contains a copy of $G_1$, or its complement contains a copy of $G_2$. Let $C_l$ denote a cycle of length l and $W_m$ a wheel of order $m+1$. A quadrilateral is a $C_4$. A graph is called $C_l$-free if it has no $C_l$ and $\delta (n,C_4)$ denotes the maximum values of the minimum degrees in all $C_4$-free planar graphs of order n. In this paper, we first show that $\delta (n,C_4)=2$ if $5\le n\le 9$, $\delta (n,C_4)=3$ if $10\le n\le 43$ and $n\notin \{30,36,39,42\}$, and $\delta (n,C_4)=4$ otherwise. Based on this result, it is shown that ${\textit{PR}}(C_4, W_n)=n+\mu$, where $\mu =3$ if $n=6$, $\mu =4$ if $7\le n\le 39$ and $n\notin \{26,32,35,38\}$, and $\mu =5$ otherwise.