Labeling graphs with two distance constraints

Given a graph GG and integers p,q,d1p,q,d1 and d2d2, with p>qp>q, d2>d1⩾1d2>d1⩾1, an L(d1,d2;p,q)L(d1,d2;p,q)-labeling of GG is a function f:V(G)→{0,1,2,…,n}f:V(G)→{0,1,2,…,n} such that |f(u)−f(v)|⩾p|f(u)−f(v)|⩾p if dG(u,v)⩽d1dG(u,v)⩽d1 and |f(u)−f(v)|⩾q|f(u)−f(v)|⩾q if dG(u,v)⩽d2dG(u,v)⩽d2. A k-L(d1,d2;p,q)k-L(d1,d2;p,q)-labeling   is an L(d1,d2;p,q)L(d1,d2;p,q)-labeling ff such that maxv∈V(G)f(v)⩽kmaxv∈V(G)f(v)⩽k. The L(d1,d2;p,q)L(d1,d2;p,q)-labeling number of  GG, denoted by λd1,d2p,q(G), is the smallest number kk such that GG has a kk-L(d1,d2;p,q)L(d1,d2;p,q)-labeling. In this paper, we give upper bounds and lower bounds of the L(d1,d2;p,q)L(d1,d2;p,q)-labeling number for general graphs and some special graphs. We also discuss the L(d1,d2;p,q)L(d1,d2;p,q)-labeling number of GG, when GG is a path, a power of a path, or Cartesian product of two paths.