The Radio Number of $C_n \square C_n$

TL;DR

This paper determines the radio number, the minimal maximum label, for the Cartesian product of a cycle graph with itself, advancing understanding of radio labelings in grid-like graph structures.

Contribution

It establishes the exact radio number for the Cartesian product of two cycle graphs, a problem previously unresolved.

Findings

Derived the radio number for $C_n imes C_n$.

Provided formulas for different values of n.

Enhanced understanding of radio labelings in grid graphs.

Abstract

Radio labeling is a variation of Hale's channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph $G$ subject to certain constraints involving the distances between the vertices. Specifically, a radio labeling of a connected graph $G$ is a function $c:V(G) \rightarrow \mathbb Z_+$ such that $$d(u,v)+|c(u)-c(v)|\geq 1+\text{diam}(G)$$ for every two distinct vertices $u$ and $v$ of $G$ (where $d(u,v)$ is the distance between $u$ and $v$). The span of a radio labeling is the maximum integer assigned to a vertex. The radio number of a graph $G$ is the minimum span, taken over all radio labelings of $G$. This paper establishes the radio number of the Cartesian product of a cycle graph with itself (i.e., of $C_n\square C_n$.)