Discussiones
Mathematicae Graph Theory 21(1) (2001) 5-11
DOI: https://doi.org/10.7151/dmgt.1129
ON VIZING'S CONJECTURE
Bostjan Bresar
University of Maribor, FK, Vrbanska 30
2000 Maribor, Slovenia
e-mail: Bostjan.Bresar@uni-mb.si
Abstract
A dominating set D for a graph G is a subset of V(G) such that any vertex in V(G)−D has a neighbor in D, and a domination number γ(G) is the size of a minimum dominating set for G. For the Cartesian product Gࢀ H Vizing's conjecture [10] states that γ(GࢀH) ≥ γ(G)γ(H) for every pair of graphs G,H. In this paper we introduce a new concept which extends the ordinary domination of graphs, and prove that the conjecture holds when γ(G) = γ(H) = 3.
Keywords: graph, Cartesian product, domination number.
2000 Mathematics Subject Classification: 05C69, 05C12.
References
| [1] | A.M. Barcalkin and L.F. German, The external stability number of the Cartesian product of graphs, Bul. Acad. Stiinte RSS Moldovenesti 1 (1979) 5-8. |
| [2] | T.Y. Chang and W.Y. Clark, The domination number of the 5×n and 6×n grid graphs, J. Graph Theory 17 (1993) 81-107, doi: 10.1002/jgt.3190170110. |
| [3] | M. El-Zahar and C.M. Pareek, Domination number of products of graphs, Ars Combin. 31 (1991) 223-227. |
| [4] | R.J. Faudree, R.H. Schelp, W.E. Shreve, The domination number for the product of graphs, Congr. Numer. 79 (1990) 29-33. |
| [5] | D.C. Fisher, Domination, fractional domination, 2-packing, and graph products, SIAM J. Discrete Math. 7 (1994) 493-498, doi: 10.1137/S0895480191217806. |
| [6] | B. Hartnell and D.F. Rall, Vizing's conjecture and the one-half argument, Discuss. Math. Graph Theory 15 (1995) 205-216, doi: 10.7151/dmgt.1018. |
| [7] | M.S. Jacobson and L.F. Kinch, On the domination number of products of graphs I, Ars Combin. 18 (1983) 33-44. |
| [8] | M.S. Jacobson and L.F. Kinch, On the domination number of products of graphs II: Trees, J. Graph Theory 10 (1986) 97-106, doi: 10.1002/jgt.3190100112. |
| [9] | S. Klavžar and N. Seifter, Dominating Cartesian product of cycles, Discrete Appl. Math. 59 (1995) 129-136, doi: 10.1016/0166-218X(93)E0167-W. |
| [10] | V.G. Vizing, The Cartesian product of graphs, Vycisl. Sist. 9 (1963) 30-43. |
Received 9 December 1999
Revised 22 January 2001
Close