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Improved Upper Bounds on the Domination Number of Graphs With Minimum Degree at Least Five

Graphs and Combinatorics volume 32, pages 511–519 (2016)Cite this article

Abstract

An algorithmic upper bound on the domination number \(\gamma \) of graphs in terms of the order n and the minimum degree \(\delta \) is proved. It is demonstrated that the bound improves best previous bounds for any \(5\le \delta \le 50\). In particular, for \(\delta =5\), Xing et al. (Graphs Comb. 22:127–143, 2006) proved that \(\gamma \le 5n/14 < 0.3572 n\). This bound is improved to 0.3440 n. For \(\delta =6\), Clark et al. (Congr. Numer. 132:99–123, 1998) established \(\gamma <0.3377 n\), while Biró et al. (Bull. Inst. Comb. Appl. 64:73–83, 2012) recently improved it to \(\gamma <0.3340 n\). Here the bound is further improved to \(\gamma < 0.3159n\). For \(\delta =7\), the best earlier bound 0.3088n is improved to \(\gamma < 0.2927n\).

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Acknowledgments

Research of the first author was supported by the European Union and Hungary through the projects TÁMOP-4.2.2.C-11/1/KONV-2012-0004 and the Campus Hungary B2/4H/12640. The second author was supported by the Ministry of Science of Slovenia under the grant P1-0297.

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Authors and Affiliations

  1. Department of Computer Science and Systems Technology, University of Pannonia, Veszprém, Hungary

    Csilla Bujtás

  2. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Sandi Klavžar

  3. Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia

    Sandi Klavžar

  4. Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

    Sandi Klavžar

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  1. Csilla Bujtás
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  2. Sandi Klavžar
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Correspondence to Sandi Klavžar.

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Bujtás, C., Klavžar, S. Improved Upper Bounds on the Domination Number of Graphs With Minimum Degree at Least Five. Graphs and Combinatorics 32, 511–519 (2016). https://doi.org/10.1007/s00373-015-1585-7

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  • DOI: https://doi.org/10.1007/s00373-015-1585-7

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