Improved upper bounds on the L(2,1) -labeling of the skew and converse skew product graphs

An L (2, 1)-labeling of a graph G is a function f from the vertex set V (G) to the set of all 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, where d (x, y) denotes the distance between x and y in G. The L (2, 1)-labeling number λ (G) of G is the smallest number k such that G has an L (2, 1)-labeling with max {f (v) : v ∈ V (G)} = k. Griggs and Yeh conjecture that λ (G) ≤ Δ2 for any simple graph with maximum degree Δ ≥ 2. This paper considers the graph formed by the skew product and the converse skew product of two graphs with a new approach on the analysis of adjacency matrices of the graphs as in [W.C. Shiu, Z. Shao, K.K. Poon, D. Zhang, A new approach to the L (2, 1)-labeling of some products of graphs, IEEE Trans. Circuits Syst. II: Express Briefs (to appear)] and improves the previous upper bounds significantly.