Radio number for the Cartesian product of a tree and a complete graph

A radio labelling of a graph $G$ is a mapping $f : V(G) \rightarrow \{0, 1, 2,\ldots\}$ such that $|f(u)-f(v)|\geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u,v$ of $G$, where $diam(G)$ is the diameter of $G$ and $d(u,v)$ is the distance between $u$ and $v$ in $G$. The radio number $rn(G)$ of $G$ is the smallest integer $k$ such that $G$ admits a radio labelling $f$ with $\max\{f(v):v \in V(G)\} = k$. In this paper, we give a lower bound for the radio number of the Cartesian product of a tree and a complete graph and give two necessary and sufficient conditions to achieve the lower bound. We also give three sufficient conditions to achieve the lower bound. We determine the radio number for the Cartesian product of a level-wise regular trees and a complete graph which attains the lower bound. The radio number for the Cartesian product of a path and a complete graph derived in [Radio number for the product of a path and a complete graph, J. Comb. Optim., 30 (2015), 139-149] can be obtained using our results in a short way.