Improved Upper Bounds on the Domination Number of Graphs With Minimum Degree at Least Five
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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 \(\gamma < 0.2927n\).
KeywordsDomination number Minimum degree Greedy algorithm
Mathematics Subject Classfication05C69 05C35
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.
- 7.Bujtás, Cs.: Domination game on trees without leaves at distance four. In: Frank, A., Recski, A., Wiener, G. (eds.) Proceedings of the 8th Japanese–Hungarian Symposium on Discrete Mathematics and Its Applications. Veszprém, Hungary, pp. 73–78. 4–7 June 2013Google Scholar
- 8.Bujtás, Cs.: On the game domination number of graphs with given minimum degree. Also: Electron. J. Combin., to appear (2014) arXiv:1406.7372 [math.CO]
- 10.Clark, W.E., Dunning, L.A.: Tight upper bounds for the domination numbers of graphs with given order and minimum degree. Electron. J. Comb. 4(1), #R26 (1997)Google Scholar
- 12.Imrich, W., Klavžar, S., Rall, D.F.: Topics in Graph Theory: Graphs and Their Cartesian Product. A K Peters, Wellesley (2008)Google Scholar