# The Ramsey numbers of large cycles versus small 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 determine the Ramsey number $R(C_n,W_{m})$ for $m=4$ and $m=5$. We show that $R(C_{n},W_{4})=2n-1$ and $R(C_n,W_5)=3n-2$ for $n\geq 5$. For larger wheels it remains an open problem to determine $R(C_n,W_{m})$.
Original language English Enschede University of Twente, Department of Applied Mathematics 6 Published - 2002

### Publication series

Name Memorandum Department of Applied Mathematics, University of Twente 1634 0169-2690

### Fingerprint

Ramsey number
Wheel
Subgraph
Cycle
Graph in graph theory
Open Problems
Complement
Integer

• MSC-05C55
• IR-65821
• EWI-3454
• MSC-05D10
• METIS-208268

### Cite this

Surahmat, Baskoro, E. T., & Broersma, H. J. (2002). The Ramsey numbers of large cycles versus small wheels. (Memorandum; No. 1634). Enschede: University of Twente, Department of Applied Mathematics.
Surahmat ; Baskoro, E.T. ; Broersma, H.J. / The Ramsey numbers of large cycles versus small wheels. Enschede : University of Twente, Department of Applied Mathematics, 2002. 6 p. (Memorandum; 1634).
@book{fedffee1f5bd44b7bbd944621a1d305d,
title = "The Ramsey numbers of large cycles versus small 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 determine the Ramsey number $R(C_n,W_{m})$ for $m=4$ and $m=5$. We show that $R(C_{n},W_{4})=2n-1$ and $R(C_n,W_5)=3n-2$ for $n\geq 5$. For larger wheels it remains an open problem to determine $R(C_n,W_{m})$.",
keywords = "MSC-05C55, IR-65821, EWI-3454, MSC-05D10, METIS-208268",
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 = "1634",

}

Surahmat, Baskoro, ET & Broersma, HJ 2002, The Ramsey numbers of large cycles versus small wheels. Memorandum, no. 1634, University of Twente, Department of Applied Mathematics, Enschede.

The Ramsey numbers of large cycles versus small wheels. / Surahmat; Baskoro, E.T.; Broersma, H.J.

Enschede : University of Twente, Department of Applied Mathematics, 2002. 6 p. (Memorandum; No. 1634).

Research output: Book/ReportReportOther research output

TY - BOOK

T1 - The Ramsey numbers of large cycles versus small wheels

AU - Surahmat, null

AU - Broersma, H.J.

N1 - Imported from MEMORANDA

PY - 2002

Y1 - 2002

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 determine the Ramsey number $R(C_n,W_{m})$ for $m=4$ and $m=5$. We show that $R(C_{n},W_{4})=2n-1$ and $R(C_n,W_5)=3n-2$ for $n\geq 5$. For larger wheels it remains an open problem to determine $R(C_n,W_{m})$.

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 determine the Ramsey number $R(C_n,W_{m})$ for $m=4$ and $m=5$. We show that $R(C_{n},W_{4})=2n-1$ and $R(C_n,W_5)=3n-2$ for $n\geq 5$. For larger wheels it remains an open problem to determine $R(C_n,W_{m})$.

KW - MSC-05C55

KW - IR-65821

KW - EWI-3454

KW - MSC-05D10

KW - METIS-208268

M3 - Report

T3 - Memorandum

BT - The Ramsey numbers of large cycles versus small wheels

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -

Surahmat, Baskoro ET, Broersma HJ. The Ramsey numbers of large cycles versus small wheels. Enschede: University of Twente, Department of Applied Mathematics, 2002. 6 p. (Memorandum; 1634).