L(p,q)-labeling of digraphs

Given a graph G and two positive integers p,q with p>q an L(p,q)-labeling of G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)-f(y)|>=p if d"G(x,y)=1 and |f(x)-f(y)|>=q if d"G(x,y)=2. A k-L(p,q)-labeling is an L(p,q)-labeling such that no label is greater than k. The L(p,q)-labeling number of G, denoted by @l"p","q(G) is the smallest number k such that G has a k-L(p,q)-labeling. When considering the digraph D, we use @l"p","q^*(D) in place of @l"p","q(D). We study the L(p,q)-labeling number of a digraph D in this paper. We find some relations between the L(p,q)-labeling number of a graph G and an orientation D of G, and give some results for the L(p,q)-labeling numbers of k-partite digraphs. We also study the L(p,q)-labeling numbers for those graphs D for which the underlying graphs are paths, cycles or trees.