Vizing's conjecture

In graph theory, Vizing's conjecture concerns a relation between the domination number and the cartesian product of graphs. This conjecture was first stated by Vadim G. Vizing (1968), and states that, if γ(G) denotes the minimum number of vertices in a dominating set for the graph G, then

${\displaystyle \gamma (G\,\Box \,H)\geq \gamma (G)\gamma (H).\,}$

Gravier & Khelladi (1995) conjectured a similar bound for the domination number of the tensor product of graphs; however, a counterexample was found by Klavžar & Zmazek (1996). Since Vizing proposed his conjecture, many mathematicians have worked on it, with partial results described below. For a more detailed overview of these results, see Brešar et al. (2012).