Radio-k-Labeling of Cycles for Large k

TL;DR

This paper determines the radio-k-number for cycles when k is large relative to cycle length, extending previous results and using a combination of lower bounds and cyclic group structures.

Contribution

It provides exact values of radio-k-numbers for cycles for k ≥ n-3 and partial results for other cases, extending prior work on specific k values.

Findings

Exact values of rn_k(C_n) for k ≥ n-3.

Partial results for rn_k(C_n) when n and k have different parity.

Extension of known results for specific k values.

Abstract

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 $G$, $|f(u)-f(v)| \geq k - d(u,v) +1$. 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 $rn_k(G)$, 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 $ rn_k(C_n)$ for $k \geq n-3$. For $d \leq k < n-3$, we obtain the values of $rn_k(C_n)$ 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 $rn_d (C_n)$ and $rn_{d+1} (C_n)$ shown by Liu and Zhu, and by Karst, Langowitz, Oehrlein and Troxell, respectively.