An Improved Inequality Related to Vizing's Conjecture

Vizing's conjecture and the one-half argument

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 ...

Fair reception and Vizing's conjecture

In this paper we introduce the concept of fair reception of a graph which is related to its domination number. We prove that all graphs G with a fair reception of size γ(G) satisfy Vizing’s conjecture on the domination number of Cartesian product graphs, by which we extend the well-known result of Barcalkin and German concerning decomposable graphs. Combining our concept with a result of Aharon...

Vizing's conjecture: A two-thirds bound for claw-free graphs

Vizing's conjecture: a survey and recent results

Vizing’s conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained an...

On domination numbers of graph bundles

Let γ(G) be the domination number of a graph G. It is shown that for any k ≥ 0 there exists a Cartesian graph bundle B φF such that γ(B φF ) = γ(B)γ(F )−2k. The domination numbers of Cartesian bundles of two cycles are determined exactly when the fibre graph is a triangle or a square. A statement similar to Vizing’s conjecture on strong graph bundles is shown not to be true by proving the inequ...