The domination number of a graph G is the smallest order, γ(G), of a dominating set for G. A conjecture of V. G. Vizing [5] states that for every pair of graphs G and H, γ(G☐H) ≥ γ(G)γ(H), where G☐H denotes the Cartesian product of G and H. We show that if the vertex set of G can be partitioned in a certain way then the above inequality holds for every graph H. The class of graphs G which have this type of partitioning includes those whose 2-packing number is no smaller than γ(G)-1 as well as the collection of graphs considered by Barcalkin and German in [1]. A crucial part of the proof depends on the well-known fact that the domination number of any connected graph of order at least two is no more than half its order.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1018,
author = {Bert Hartnell and Douglas F. Rall},
title = {Vizing's conjecture and the one-half argument},
journal = {Discussiones Mathematicae Graph Theory},
volume = {15},
year = {1995},
pages = {205-216},
zbl = {0845.05074},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1018}
}
Bert Hartnell; Douglas F. Rall. Vizing's conjecture and the one-half argument. Discussiones Mathematicae Graph Theory, Tome 15 (1995) pp. 205-216. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1018/
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