Improving the Clark-Suen bound on the domination number of the Cartesian product of graphs

A long-standing Vizing's conjecture asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers; one of the most significant results related to the conjecture is the bound of Clark and Suen, γ(G□H)≥γ(G)γ(H)∕2, where γ stands for the domination number, and G□H is the Cartesian product of graphs G and H. In this note, we improve this bound by employing the 2-packing number ρ(G) of a graph G into the formula, asserting that γ(G□H)≥(2γ(G)−ρ(G))γ(H)∕3. The resulting bound is better than that of Clark and Suen whenever G is a graph with ρ(G)<γ(G)∕2, and in the case G has diameter 2 reads as γ(G□H)≥(2γ(G)−1)γ(H)∕3.