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 @l(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 @l(G)@?@D^2 for any simple graph with maximum degree @D>=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.