A lower bound for the radio number of graphs

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].