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 G, then
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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).
Examples
A 4-cycle C4 has domination number two: any single vertex only dominates itself and its two neighbors, but any pair of vertices dominates the whole graph. The product
It is possible for the domination number of a product to be much larger than the bound given by Vizing's conjecture. For instance, for a star K1,n, its domination number γ(K1,n) is one: it is possible to dominate the entire star with a single vertex at its hub. Therefore, for the graph
There exist infinite families of graph products for which the bound of Vizing's conjecture is exactly met. For instance, if G and H are both connected graphs, each having at least four vertices and having exactly twice as many total vertices as their domination numbers, then
Partial results
Clearly, the conjecture holds when either G or H has domination number one: for, the product contains an isomorphic copy of the other factor, dominating which requires at least γ(G)γ(H) vertices.
Vizing's conjecture is also known to hold for cycles and for graphs with domination number two.
Clark & Suen (2000) proved that the domination number of the product is at least half as large as the conjectured bound, for all G and H.
Upper bounds
Vizing (1968) observed that
A dominating set meeting this bound may be formed as the cartesian product of a dominating set in one of G or H with the set of all vertices in the other graph.