Multilevel Distance Labelings for Paths and Cycles

For a graph $G$, let $\diam(G)$ denote the diameter of $G$. For any two vertices $u$ and $v$ in $G$, let $d(u, v)$ denote the distance between $u$ and $v$. A multilevel distance labeling (or distance labeling) for $G$ is a function $f$ that assigns to each vertex of $G$ a nonnegative integer such that for any vertices $u$ and $v$, $|f(u)-f(v)| \geq \diam(G) - d_G(u, v) +1$. The span of $f$ is the largest number in $f(V)$. The radio number of $G$, denoted by $rn(G)$, is the minimum span of a distance labeling for $G$. In this paper, we completely determine the radio numbers for paths and cycles.