The Ramsey numbers for trees of order $n$ with maximum degree at least $n-5$ versus the wheel graph of order nine

Abstract

The Ramsey numbers $R(T_n,W_8)$ are determined for each tree graph $T_n$ of order $n\geq 7$ and maximum degree $\Delta(T_n)$ equal to either $n-4$ or $n-5$. These numbers indicate strong support for the conjecture, due to Chen, Zhang and Zhang and to Hafidh and Baskoro, that $R(T_n,W_m) = 2n-1$ for each tree graph $T_n$ of order $n\geq m-1$ with $\Delta(T_n)\leq n-m+2$ when $m\geq 4$ is even.