All Quadrilateral-Wheel Planar Ramsey Numbers

Abstract

For two given graphs $$G_1$$G1 and $$G_2$$G2, the planar Ramsey number $${\textit{PR}}(G_1,G_2)$$PR(G1,G2) is the smallest integer n such that every planar graph G on n vertices either contains a copy of $$G_1$$G1, or its complement contains a copy of $$G_2$$G2. Let $$C_l$$Cl denote a cycle of length l and $$W_m$$Wm a wheel of order $$m+1$$m+1. A quadrilateral is a $$C_4$$C4. A graph is called $$C_l$$Cl-free if it has no $$C_l$$Cl and $$\delta (n,C_4)$$?(n,C4) denotes the maximum values of the minimum degrees in all $$C_4$$C4-free planar graphs of order n. In this paper, we first show that $$\delta (n,C_4)=2$$?(n,C4)=2 if $$5\le n\le 9$$5≤n≤9, $$\delta (n,C_4)=3$$?(n,C4)=3 if $$10\le n\le 43$$10≤n≤43 and $$n\notin \{30,36,39,42\}$$n?{30,36,39,42}, and $$\delta (n,C_4)=4$$?(n,C4)=4 otherwise. Based on this result, it is shown that $${\textit{PR}}(C_4, W_n)=n+\mu $$PR(C4,Wn)=n+μ, where $$\mu =3$$μ=3 if $$n=6$$n=6, $$\mu =4$$μ=4 if $$7\le n\le 39$$7≤n≤39 and $$n\notin \{26,32,35,38\}$$n?{26,32,35,38}, and $$\mu =5$$μ=5 otherwise.