The L(2,1)-labeling and operations of graphs

Motivated by a variation of the channel assignment problem, a graph labeling analogous to the graph vertex coloring has been presented and is called an L(2,1)-labeling. More precisely, 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. 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. A conjecture states that λ(G) ≤ Δ² for any simple graph with the maximum degree Δ≥2. This paper considers the graphs formed by the Cartesian product and the composition of two graphs. The new graph satisfies the conjecture above in both cases(with minor exceptions).