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

Research output: Book/ReportReportOther research output

### 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

### Fingerprint

Ramsey number
Wheel
Odd
Subgraph
Graph in graph theory
Star
Complement
Denote
Integer

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

### Cite this

Surahmat, Baskoro, E. T., & Broersma, H. J. (2002). The Ramsey numbers of large star-like trees versus large odd wheels. (Memorandum; No. 1621). Enschede: University of Twente, Department of Applied Mathematics.
Surahmat ; Baskoro, E.T. ; Broersma, H.J. / The Ramsey numbers of large star-like trees versus large odd wheels. Enschede : University of Twente, Department of Applied Mathematics, 2002. 9 p. (Memorandum; 1621).
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title = "The Ramsey numbers of large star-like trees versus large odd wheels",
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.",
keywords = "MSC-05C55, IR-65808, EWI-3441, MSC-05D10",
author = "Surahmat and E.T. Baskoro and H.J. Broersma",
note = "Imported from MEMORANDA",
year = "2002",
language = "English",
series = "Memorandum",
publisher = "University of Twente, Department of Applied Mathematics",
number = "1621",

}

Surahmat, Baskoro, ET & Broersma, HJ 2002, The Ramsey numbers of large star-like trees versus large odd wheels. Memorandum, no. 1621, University of Twente, Department of Applied Mathematics, Enschede.

The Ramsey numbers of large star-like trees versus large odd wheels. / Surahmat; Baskoro, E.T.; Broersma, H.J.

Enschede : University of Twente, Department of Applied Mathematics, 2002. 9 p. (Memorandum; No. 1621).

Research output: Book/ReportReportOther research output

TY - BOOK

T1 - The Ramsey numbers of large star-like trees versus large odd wheels

AU - Surahmat, null

AU - Broersma, H.J.

N1 - Imported from MEMORANDA

PY - 2002

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N2 - 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.

AB - 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.

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KW - IR-65808

KW - EWI-3441

KW - MSC-05D10

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

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -

Surahmat, Baskoro ET, Broersma HJ. The Ramsey numbers of large star-like trees versus large odd wheels. Enschede: University of Twente, Department of Applied Mathematics, 2002. 9 p. (Memorandum; 1621).