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.

翻译：图$G$的无线电标号是一个映射$f : V(G) \rightarrow \{0, 1, 2,\ldots\}$，满足对$G$的任意两个不同顶点$u,v$有$|f(u)-f(v)|\geq diam(G) + 1 - d(u,v)$，其中$diam(G)$是$G$的直径，$d(u,v)$是$u$与$v$在$G$中的距离。图$G$的无线电数$rn(G)$是使得$G$存在最大标号为$\max\{f(v):v \in V(G)\} = k$的无线电标号$f$的最小整数$k$。本文给出树与完全图笛卡尔积的无线电数下界，并给出达到该下界的两个充要条件。我们还给出三个达到该下界的充分条件。我们确定了达到该下界的逐层正则树与完全图笛卡尔积的无线电数。利用本文结果，可简洁地推导出文献[Radio number for the product of a path and a complete graph, J. Comb. Optim., 30 (2015), 139-149]中路径与完全图笛卡尔积的无线电数结果。