The Ramsey numbers of large cycles versus small wheels

Surahmat, E.T. Baskoro, H.J. Broersma

Research output: Book/ReportReportOther research output

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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 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 languageEnglish
Place of PublicationEnschede
PublisherUniversity of Twente, Department of Applied Mathematics
Number of pages6
Publication statusPublished - 2002

Publication series

NameMemorandum
PublisherDepartment of Applied Mathematics, University of Twente
No.1634
ISSN (Print)0169-2690

Fingerprint

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

Keywords

  • 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).
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author = "Surahmat and E.T. Baskoro and H.J. Broersma",
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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

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AU - Baskoro, E.T.

AU - Broersma, H.J.

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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 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 $R(C_n,W_{m})$.

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KW - IR-65821

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KW - MSC-05D10

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PB - University of Twente, Department of Applied Mathematics

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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).