VIZING'S CONJECTURE AND THE ONE-HALF ARGUMENT
Saint Mary's University, Halifax,
Douglas F. Rall
Furman University, Greenville,
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  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 . 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.
Keywords: domination number, Cartesian product, Vizing's conjecture, clique.
1991 Mathematics Subject Classification: 05C70, 05C99.
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