Ramsey numbers for trees II

TL;DR

This paper determines specific Ramsey numbers involving trees and stars, providing explicit formulas and new results for various tree structures and their combinations.

Contribution

It introduces new formulas and exact values for Ramsey numbers involving certain trees and stars, expanding the understanding of these combinatorial parameters.

Findings

Explicit formulas for r(K_{1,m-1},S(n_1,n_2))

Exact values for r(T_m',T_n), r(T_n,T_n), r(T_n',T_n), r(P_n,T_n)

Determination of r for specific tree structures under certain conditions

Abstract

Let $r(G_1, G_2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. For $n_1\ge n_2\ge 1$ let $S(n_1,n_2)$ be the double star given by $V(S(n_1,n_2))=\{v_0,v_1,\ldots,v_{n_1},w_0,w_1,\ldots,w_{n_2}\}$ and $E(S(n_1,n_2))=\{v_0v_1,\ldots,v_0v_{n_1},v_0w_0,w_0w_1,\ldots,w_0w_{n_2}\}$. In this paper we determine $r(K_{1,m-1},$ $S(n_1,n_2))$ under certain conditions. For $n\ge 6$ let $T_n^3=S(n-5,3)$, $T_n^{''}=(V,E_2)$ and $T_n^{'''} =(V,E_3)$, where $V=\{v_0,v_1,\ldots,v_{n-1}\}$, $E_2=\{v_0v_1,\ldots,v_0v_{n-4},v_1v_{n-3},v_1v_{n-2},$ $v_2v_{n-1}\}$ and $E_3=\{v_0v_1,\ldots,v_0v_{n-4},v_1v_{n-3},v_2v_{n-2},v_3v_{n-1}\}$. We also obtain explicit formulas for $r$ $(K_{1,m-1},T_n)$, $r(T_m',T_n)$ $(n\ge m+3)$, $r(T_n,T_n)$, $r(T_n',T_n)$ and $r(P_n,T_n)$, where $T_n\in\{T_n'',T_n''',T_n^3\}$, $P_n$ is the path on $n$ vertices and $T_n'$ is the unique tree with $n$ vertices and maximal degree $n-2$.