Labeling Dot-Cartesian and Dot-Lexicographic Product Graphs with a Condition at Distance Two

If d(x, y) denotes the distance between vertices x and y in a graph G, then an L(2, 1)-labeling of a graph G is a function f from vertices of G to nonnegative integers such that |f(x)−f(y)| ≥ 2 if d(x, y) = 1, and |f(x)−f(y)| ≥ 1 if d(x, y) = 2. Griggs and Yeh conjectured that for any graph with maximum degree ∆ ≥ 2, there is an L(2, 1)-labeling with all labels not greater than ∆. We prove that the conjecture holds for dot-Cartesian products and dot-lexicographic products of two graphs with possible minor exceptions in some special cases. The bounds obtained are in general much better than the ∆-bound.