  Let $\mathbb{N}$ be the set of positive integers. A radio labeling of a graph
$G$ is a mapping $\varphi : V(G) \rightarrow \mathbb{N} \cup \{0\}$ such that
the inequality $|\varphi(u)-\varphi(v)| \geq diam(G) + 1 - d(u,v)$ holds 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 $\varphi$ with $\max\{\varphi(v) : v \in V(G)\}$ = $k$. Das et al.
[Discrete Math. $\mathbf{340}$(2017) 855-861] gave a technique to find a lower
bound for the radio number of graphs. In [Algorithms and Discrete Applied
Mathematics: CALDAM 2019, Lecture Notes in Computer Science $\mathbf{11394}$,
springer, Cham, 2019, 161-173], Bantva modified this technique for finding an
improved lower bound on the radio number of graphs and gave a necessary and
sufficient condition to achieve the improved lower bound. In this paper, one
more useful necessary and sufficient condition to achieve the improved lower
bound for the radio number of graphs is given. Using this result, the radio
number of the Cartesian product of a path and a wheel graphs is determined.
