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 $Δ(T_n)$ 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 $Δ(T_n)\leq n-m+2$ when $m\geq 4$ is even.

Figures (9)

• Figure 1: Examples of $S_n(\ell,m)$, $S_n(\ell)$ and $S_n[\ell]$
• Figure 2: Tree graphs of order $7$ with $\Delta(T_n)=n-4$.
• Figure 3: Three tree graphs with $\Delta(T_n)=n-4$.
• Figure 4: Tree graphs $T_n$ with $\Delta(T_n)=n-5$.
• Figure 5: $S_7(2,1)$ and $U$ in $G$.

Theorems & Definitions (88)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• Lemma 4.1
• Lemma 4.2
• Lemma 4.4