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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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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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