L(p,q)L(p,q)-labeling of digraphs

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