Radio Number for Even Square Cycles

RM 207

We are investigating optimal radio labelings of radio stations that avoid radio interference between them. We present this issue using graph theory, comprising a mathematical model where each vertex represents a station and the edges represent the closeness of the stations. Let \textit{G} be a connected graph. The \textit{distance} between two vertices $u$ and $v$ in G is defined by the length of the shortest path in \textit{G} between $u$ and $v$, which is denoted by $d_G(u,v)$. The \textit{diameter} of \ textit{G}, denoted by diam$(G) $, is the maximum distance between two veritices in \textit{G}. The \ textit{radio labeling} of $G$ is a function $f$ that assigns each vertex a non-negative integer such that $|f (u)- f(v)|$ $\geq$ diam$ (G)-d_G(u,v)+1$ holds for any two distinct vertices $u$ and $v$ of $G$. The $span$ of $f$ is the difference of the largest and the smallest channels used. The \textit{radio number} of \textit{G}, denoted by all radio labelings of \textit{G}. \textit{f} is said to be a \ textit{optimal radio labeling} of G if \textit{span} $f= $ rn$(G) $. A \textit{cycle} with \textit{n} vertices, denoted by $C_n$, is a graph whose vertex set can be reordered as $\ {v_1, v_2,..., v_n\}$ such that $E(C_n) = \{v_1 v_2, v_2 v_3,...,v_{n-1}v_n, v_n v_1\}$. The $square$ of a graph \textit{G} has the same vertex set as \textit{G}, but the edge set is now $E(G^2) =$ $E(G)\cup \{uv : d_G(u,v) = 2\}.$ In this presentation, we will discuss the progress we made on the unsloved case for rn$(C_n^2)$, where $n=4k+3$ with $k=4m+3$, for some some $m$ in the integer.