Vizing's Conjecture - Wikipedia

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 C4 □ C4 is a four-dimensional hypercube graph; it has 16 vertices, and any single vertex can only dominate itself and four neighbors, so three vertices could only dominate 15 of the 16 vertices. Therefore, at least four vertices are required to dominate the entire graph, the bound given by Vizing's conjecture.

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 G = K1,n □ K1,n formed as the product of two stars, Vizing's conjecture states only that the domination number should be at least 1 × 1 = 1. However, the domination number of this graph is actually much higher. It has n2 + 2n + 1 vertices: n2 formed from the product of a leaf in both factors, 2n from the product of a leaf in one factor and the hub in the other factor, and one remaining vertex formed from the product of the two hubs. Each leaf-hub product vertex in G dominates exactly n of the leaf-leaf vertices, so n leaf-hub vertices are needed to dominate all of the leaf-leaf vertices. However, no leaf-hub vertex dominates any other such vertex, so even after n leaf-hub vertices are chosen to be included in the dominating set, there remain n more undominated leaf-hub vertices, which can be dominated by the single hub-hub vertex. Thus, the domination number of this graph is γ(K1,n □ K1,n) = n + 1 far higher than the trivial bound of one given by Vizing's conjecture.

There exist infinite families of graph products for which the bound of Vizing's conjecture is exactly met.[1] 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 γ(G □ H) = γ(G) γ(H).[2] The graphs G and H with this property consist of the four-vertex cycle C4 together with the rooted products of a connected graph and a single edge.[2]