Published online by Cambridge University Press: 30 January 2018
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.
The first author is partially supported by NSFC under grant number 11601176 and NSF of Hubei Province under grant number 2016CFB146; the second author is partially supported by NSFC under grant numbers 11671198 and 11571168; the third author is partially supported by NSFC under grant number 11601527.