Radio Labelings of Distance Graphs

A radio $k$-labeling of a connected graph $G$ is an assignment $c$ of non negative integers to the vertices of $G$ such that $$\mathrm{\mid }c(x)-c(y)\mathrm{\mid }\ge k+1-d(x,y),$$ for any two vertices $x$ and $y$, $x\mathrm{\ne }y$, where $d(x,y)$ is the distance between $x$ and $y$ in $G$. In this paper, we study radio labelings of distance graphs, i.e., graphs with the set $\mathbb{Z}$ of integers as vertex set and in which two distinct vertices $i,j\in \mathbb{Z}$ are adjacent if and only if $|i - j| \in D$.