# The Ramsey numbers of large star-like trees versus large odd wheels

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## Abstract

For two given graphs $G$ and $H$, the \textit{Ramsey number} $R(G,H)$ is the smallest positive integer $N$ such that for every graph $F$ of order $N$ the following holds: either $F$ contains $G$ as a subgraph or the complement of $F$ contains $H$ as a subgraph. In this paper, we shall study the Ramsey number $R(T_n,W_{m})$ for a star-like tree $T_n$ with $n$ vertices and a wheel $W_m$ with $m+1$ vertices and $m$ odd. We show that the Ramsey number $R(S_{n},W_{m})=3n-2$ for $n\geq 2m-4, m\geq 5$ and $m$ odd, where $S_n$ denotes the star on $n$ vertices. We conjecture that the Ramsey number is the same for general trees on $n$ vertices, and support this conjecture by proving it for a number of star-like trees.
Original language English Enschede University of Twente, Department of Applied Mathematics 9 Published - 2002

### Publication series

Name Memorandum Department of Applied Mathematics, University of Twente 1621 0169-2690

• MSC-05C55
• IR-65808
• EWI-3441
• MSC-05D10

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