All Ramsey Numbers for Brooms in Graphs

  • Pei Yu
  • Yusheng Li
Keywords: Ramsey number, Tree, Broom


For $k,\ell\ge 1$, a broom $B_{k,\ell}$ is a tree on $n=k+\ell$ vertices obtained by connecting the central vertex of a star $K_{1,k}$ with an end-vertex of a path on $\ell-1$ vertices. As $B_{n-2,2}$ is a star and $B_{1,n-1}$ is a path, their  Ramsey number have been determined among rarely known $R(T_n)$ of trees $T_n$ of order $n$. Erdős, Faudree, Rousseau and Schelp determined the value of  $R(B_{k,\ell})$ for $\ell\ge 2k\geq2$. We shall determine all other $R(B_{k,\ell})$ in this paper, which says that, for fixed $n$, $R(B_{n-\ell,\ell})$ decreases first on $1\le\ell \le 2n/3$ from $2n-2$ or $2n-3$ to $\lceil\frac{4n}{3}\rceil-1$, and then it increases  on $2n/3 < \ell\leq n$ from $\lceil\frac{4n}{3}\rceil-1$ to $\lfloor\frac{3n}{2}\rfloor -1$. Hence $R(B_{n-\ell,\ell})$ may attain the maximum  and minimum values of $R(T_n)$ as $\ell$ varies.
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