Planar Ramsey Numbers of Four Cycles Versus Wheels

TL;DR

This paper determines the exact planar Ramsey numbers for pairs of graphs involving a 4-cycle and wheels, by analyzing structural properties of $C_4$-free planar graphs.

Contribution

It characterizes structural properties of $C_4$-free planar graphs and computes all $PR(C_4, W_n)$ for $n \u2265 3$, advancing understanding of planar Ramsey numbers.

Findings

Determined all $PR(C_4, W_n)$ for $n \u2265 3$.

Characterized structural properties of $C_4$-free planar graphs.

Abstract

For two given graphs $G$ and $H$ the planar Ramsey number $PR(G,H)$ is the smallest integer $n$ such that every planar graph $F$ on $n$ vertices either contains a copy of $G$, or its complement contains a copy of $H$. In this paper, we first characterize some structural properties of $C_4$-free planar graphs, and then we determine all planar Ramsey numbers $PR(C_4, W_n)$, for $n\ge 3$.