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Graphs and Combinatorics

, Volume 32, Issue 2, pp 511–519 | Cite as

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

  • Csilla Bujtás
  • Sandi KlavžarEmail author
Original Paper

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 \(\gamma < 0.2927n\).

Keywords

Domination number Minimum degree Greedy algorithm 

Mathematics Subject Classfication

05C69 05C35 

Notes

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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Copyright information

© Springer Japan 2015

Authors and Affiliations

  1. 1.Department of Computer Science and Systems TechnologyUniversity of PannoniaVeszprémHungary
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia
  4. 4.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia

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