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An Intermediate Value Theorem for the Decycling Numbers of Toeplitz Graphs

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Abstract

In this paper we obtain an intermediate value theorem for the decycling numbers of Toeplitz graphs: if \(n \ge 3\), and \(0 \le r \le n - 2\), then there exists a Toeplitz graph of order \(n\) with decycling number \(r\). We also prove that the decycling numbers of connected Cayley graphs of order \(n\) satisfy the intermediate value property if and only if \(n = 4\) or \(6\).

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Acknowledgments

The authors thank the referees whose comments improved the presentation of the paper.

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Authors and Affiliations

  1. School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa

    Sheng Bau, Benjamin van Niekerk & David White

  2. Institute of Discrete Mathematics, Inner Mongolia University of Nationalities, Tongliao, China

    Sheng Bau

Authors
  1. Sheng Bau
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  2. Benjamin van Niekerk
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  3. David White
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Correspondence to Sheng Bau.

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Research supported by NRF South Africa.

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Bau, S., van Niekerk, B. & White, D. An Intermediate Value Theorem for the Decycling Numbers of Toeplitz Graphs. Graphs and Combinatorics 31, 2037–2042 (2015). https://doi.org/10.1007/s00373-014-1492-3

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  • DOI: https://doi.org/10.1007/s00373-014-1492-3

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