  The radio number problem uses a graph-theoretical model to simulate optimal
frequency assignments on wireless networks. A radio labeling of a connected
graph $G$ is a function $f:V(G) \to \mathbb Z_{0}^+$ such that for every pair
of vertices $u,v \in V(G)$, we have $\lvert f(u)-f(v)\rvert \ge \text{diam}(G)
+ 1 - d(u,v)$ where $\text{diam}(G)$ denotes the diameter of $G$ and $d(u,v)$
the distance between vertices $u$ and $v$. Let $\text{span}(f)$ be the
difference between the greatest label and least label assigned to $V(G)$. Then,
the \textit{radio number} of a graph $\text{rn}(G)$ is defined as the minimum
value of $\text{span}(f)$ over all radio labelings of $G$. So far, there have
been few results on the radio number of the grid graph: In 2009 Calles and
Gomez gave an upper and lower bound for square grids, and in 2008 Flores and
Lewis were unable to completely determine the radio number of the ladder graph
(a 2 by $n$ grid). In this paper, we completely determine the radio number of
the grid graph $G_{a,b}$ for $a,b>2$, characterizing three subcases of the
problem and providing a closed-form solution to each. These results have
implications in the optimization of radio frequency assignment in wireless
networks such as cell towers and environmental sensors.

Combinatorics