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Graphs and Combinatorics

, Volume 35, Issue 6, pp 1401–1404 | Cite as

An Improved Bound in Vizing’s Conjecture

  • Shira ZerbibEmail author
Original Paper
  • 77 Downloads

Abstract

A well-known conjecture of Vizing (Vyčisl Sistemy 9:30–43, 1963) is that \(\gamma (G \square H) \ge \gamma (G)\gamma (H)\) for any pair of graphs GH, where \(\gamma \) is the domination number and \(G \square H\) is the Cartesian product of G and H. Suen and Tarr [7], improving a result of Clark and Suen [5], showed \(\gamma (G \square H) \ge \frac{1}{2}\gamma (G)\gamma (H) + \frac{1}{2}\min (\gamma (G),\gamma (H))\). We further improve their result by showing \(\gamma (G \square H) \ge \frac{1}{2}\gamma (G)\gamma (H) + \frac{1}{2}\max (\gamma (G),\gamma (H))\).

Keywords

Vizing’s conjecture Cartesian product Graph domination 

Notes

Acknowledgements

The author is grateful to Ron Aharoni for helpful discussions.

References

  1. 1.
    Asplund, J., Davila, R., Krop, E.: A Vizing-type result for semi-total domination. Discret. Appl. Math. 258, 8–12 (2019)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barcalkin, A.M., German, L.F.: The external stability number of the Cartesian product of graphs. Bul. Akad. Stiinte RSS Moldoven 94, 5–8 (1979)MathSciNetGoogle Scholar
  3. 3.
    Brešar, B.: Improving the Clark–Suen bound on the domination number of the Cartesian product of graphs. Discret. Math. 340, 2398–2401 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brešar, B., Dorbec, P., Goddard, W., Hartnell, B.L., Henning, M.A., Klavžar, S., Rall, D.F.: Vizing’s conjecture: a survey and recent results. J. Graph Theory 69, 46–76 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Clark, W.E., Suen, S.: An inequality related to Vizing’s conjecture. Electron. J. Comb. 7, Note 4 (2000)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Hartnell, B., Rall, D.F.: Domination in Cartesian products: Vizing’s conjecture. In: Haynes, et al. (eds.) Domination in Graphs-Advanced Topics, pp. 163–189. Marcel Dekker Inc, New York (1998)zbMATHGoogle Scholar
  7. 7.
    Suen, S., Tarr, J.: An improved inequality related to Vizing’s conjecture. Electron. J. Comb. 19, 1 (2012)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Vizing, V.G.: The Cartesian product of graphs. Vyčisl. Sistemy 9, 30–43 (1963)MathSciNetGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA

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