Labeling graphs with two distance constraints

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