New 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 defined as a function f from the vertex set V(G) into the nonnegative integers such that for any two vertices x, y, |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) is the distance between x and y in G. The L(2,1)-labeling number λ2,1(G) of G is the smallest number k such that G has an L(2,1)-labeling with k=max{f(x)|x∈V(G)}. In this paper, we consider the graph formed by the skew product and converse skew product of two graphs, and give new upper bounds of the L(2,1)-labeling number, which improves the upper bounds obtained by Shao and Zhang [Z.D. Shao, D. Zhang, Improved upper bounds on the L(2,1)-labeling of the skew and converse skew product graphs, Theoret. Comput. Sci. 400 (2008) 230–233] in many cases.