Radio-k-Labeling of Cycles for Large k

Let $G$ be a simple connected graph. For any two vertices $u$ and $v$, let $d(u,v)$ denote the distance between $u$ and $v$ in $G$. A radio-$k$-labeling of $G$ for a fixed positive integer $k$ is a function $f$ which assigns to each vertex a non-negative integer label such that for every two vertices $u$ and $v$ in $G$, $\mathrm{\mid }f(u)-f(v)\mathrm{\mid }\ge k-d(u,v)+1$. The span of $f$ is the difference between the largest and smallest labels of $f(V)$. The radio-$k$-number of a graph $G$, denoted by $r{n}_k(G)$, is the smallest span among all radio-$k$-labelings admitted by $G$. A cycle $C_n$ has diameter $d=\lfloor n/2 \rfloor$. In this paper, we combine a lower bound approach with cyclic group structure to determine the value of $r{n}_k(C_n)$ for $k\ge n-3$. For $d \leq k < n-3$,weobtainthevaluesof$rn_k(C_n)$when$n$and$k$ have the same parity, and prove partial results when $n$ and $k$ have different parities. Our results extend the known values of $r{n}_d(C_n)$ and $r{n}_{d+1}(C_n)$ shown by Liu and Zhu, and by Karst, Langowitz, Oehrlein and Troxell, respectively.