The Radio Number of $C_n\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{C}_n$

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 $c:V(G)\to {\mathbb{Z}}_{+}$ such that $$d(u,v)+|c(u)–c(v)|\ge 1+\text{diam}(G)$$ for every two distinct vertices $u$ and $v$ of $G$, where $d(u,v)$ is the distance between $u$ and $v$. The \emph{span} of a radio labeling is the maximum integer assigned to a vertex. The \emph{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 $C_n\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{C}_n$).