QUADRILATERAL-TREE PLANAR RAMSEY NUMBERS

Abstract

For two given graphs $G_{1}$ and $G_{2}$, the planar Ramsey number $PR(G_{1},G_{2})$ is the smallest integer $N$ such that every planar graph $G$ on $N$ vertices either contains $G_{1}$, or its complement contains $G_{2}$. Let $C_{4}$ be a quadrilateral, $T_{n}$ a tree of order $n\geq 3$ with maximum degree $k$, and $K_{1,k}$ a star of order $k+1$. We show that $PR(C_{4},T_{n})=\max \{n+1,PR(C_{4},K_{1,k})\}$. Combining this with a result of Chen et al. [‘All quadrilateral-wheel planar Ramsey numbers’, Graphs Combin.33 (2017), 335–346] yields exact values of all the quadrilateral-tree planar Ramsey numbers.