Radio labelings of distance graphs

Motivated by the Channel Assignment Problem, we study radio kk-labelings of graphs. A radio kk-labeling of a connected graph GG is an assignment cc of non-negative integers to the vertices of GG such that |c(x)−c(y)|≥k+1−d(x,y),|c(x)−c(y)|≥k+1−d(x,y), for any two vertices xx and yy, x≠yx≠y, where d(x,y)d(x,y) is the distance between xx and yy in GG.In this paper, we study radio kk-labelings of distance graphs, i.e., graphs with the set ZZ of integers as vertex set and in which two distinct vertices i,j∈Zi,j∈Z are adjacent if and only if |i−j|∈D|i−j|∈D. We give some lower and upper bounds for radio kk-labelings of distance graphs with distance sets D(1,2,…,t)D(1,2,…,t), D(1,t)D(1,t) and D(t−1,t)D(t−1,t) for any positive integer t>1t>1.