Labeling Trees with a Condition at Distance Two

For positive integers j⩾k, an L(j,k)-labeling of graph G is an integer labeling of V(G) such that adjacent vertices receive labels which differ by at least j, and vertices that are distance two apart receive labels which differ by at least k. The λj,k-number of G is the minimum span over the L(j,k)-labelings of G. In this paper, we derive the λj,k-number of the infinite regular tree. For x=j/k, we also introduce a rational variation λx(G) of λj,k(G), and provide a proof that λx(G) is continuous.