Unique list colorability of the graph $K^n_2+K_r$

Given a list $L(v)$ for each vertex $v$, we say that the graph $G$ is $L$-colorable if there is a proper vertex coloring of $G$, where each vertex $v$ takes its color from $L(v)$. The graph is uniquely $k$-list colorable if there is a list assignment $L$ such that $|L(v)| = k$ for every vertex $v$ and the graph has exactly one $L$-coloring with these lists. If a graph $G$ is not uniquely $k$-list colorable, we also say that $G$ has property $M(k)$. The least integer $k$ such that $G$ has the property $M(k)$ is called the $m$-number of $G$, denoted by $m(G)$. In this paper, we characterize the unique list colorability of the graph $G=K^n_2+K_r$. In particular, we determine the number $m(G)$ of the graph $G=K^n_2+K_r$.