# A lower bound for the radio number of graphs

## TL;DR

This paper improves the lower bound for the radio number of graphs, characterizes when the bound is tight, and determines the radio number for specific graph classes like Cartesian products of paths, Peterson graph, and complete graphs.

## Contribution

It provides a refined lower bound for the radio number, necessary and sufficient conditions for achieving this bound, and explicit calculations for certain graph products.

## Key findings

- Improved lower bound for the radio number of graphs.
- Necessary and sufficient conditions for the bound to be tight.
- Exact radio numbers for Cartesian products of paths, Peterson graph, and complete graphs.

## Abstract

A radio labeling of a graph $G$ is a mapping $\vp : V(G) \rightarrow \{0, 1, 2,...\}$ such that $|\vp(u)-\vp(v)|\geq \diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u,v$ of $G$, where $\diam(G)$ and $d(u,v)$ are the diameter of $G$ and distance between $u$ and $v$ in $G$, respectively. The radio number $\rn(G)$ of $G$ is the smallest number $k$ such that $G$ has radio labeling with $\max\{\vp(v):v \in V(G)\}$ = $k$. In this paper, we slightly improve the lower bound for the radio number of graphs given by Das \emph{et al.} in [5] and, give necessary and sufficient condition to achieve the lower bound. Using this result, we determine the radio number for cartesian product of paths $P_{n}$ and the Peterson graph $P$. We give a short proof for the radio number of cartesian product of paths $P_{n}$ and complete graphs $K_{m}$ given by Kim \emph{et al.} in [6].