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 ∆ 2 . We prove that the conjecture holds for dot-Cartesian products and dot-lexicographic products of two graphs
... cts of two graphs with possible minor exceptions in some special cases. The bounds obtained are in general much better than the ∆ 2 -bound. a k-L(2, 1)-labeling. The L(2, 1)-labeling number of G, denoted by λ(G), is the smallest number k such that G has a k-L(2, 1)-labeling. The theory of L(2, 1)-labeling is now already very extensive, see the 2006 survey of Yeh  and the 2011 updated survey and annotated bibliography by Calamoneri  containing 184 references. From recent results we point to two appealing algorithmic achievements: A linear time algorithm for L(2, 1)-labeling of trees  and a polynomial space algorithm to determine the L(2, 1)-span in the general case . Griggs and Yeh  proved that it is NP-complete to decide whether a given graph G allows an L(2, 1)-labeling of span at most n. Thus, it is important to obtain good lower and upper bounds for λ. For a diameter two graph G, it is known that λ(G) ≤ ∆ 2 , where ∆ = ∆(G) is the maximum degree of G, and the upper bound can be attained by Moore graphs, that is, diameter 2 graphs of order ∆ 2 + 1  . Based on the previous research, Griggs and Yeh  conjectured that λ(G) ≤ ∆ 2 holds for any graph G with ∆ ≥ 2. The conjecture is known as the ∆ 2 -conjecture and considered as the most important open problem in the area. The best general bound ∆ 2 + ∆ − 2 so far is due to Gonçalves . Havet, Reed and Sereni  proved that the ∆ 2 -conjecture holds for sufficiently large ∆. A lot of research regarding L(2, 1)-labelings (and, more generally of L(j, k)-labelings) was done on standard graph products, cf. recent investigations on the Cartesian product [10, 11, 12, 13, 14] , the direct product [15, 16] , the lexicographic product  , and the strong product ; cf. also references therein. A special emphasize was put on the ∆ 2 -conjecture. In  the conjecture was verified for lexicographic products as well as for Cartesian products with factors of minimum degree at least 2. In  the ∆ 2 -conjecture was confirmed for the strong and the direct product of graphs. The obtained upper bounds on these two products were later improved in  . Shiu et al.  used an analysis of the adjacency matrices of the graphs to obtain improvements of the previous bounds on all the above four (standard) graph products. Finally, in  the ∆ 2 -conjecture was verified for modular products of two graphs with minor exceptions. Now, modular product is obtained from the strong product by superimposing edges that come from non-edges in both facts. This construction is not really interesting for the direct product. Hence, as there are four standard graph products, there are two natural additional products (w.r.t. the superimposition of the edges that come from non-edges) to consider-the products obtained from the Cartesian product and the lexicographic product, named the dot-Cartesian and the dot-lexicographic (see the next section for formal definitions). In this paper we prove that the ∆ 2 -conjecture is true also for these products with possible minor exceptions. The bounds obtained are typically much better than the ∆ 2 -bound.