Colorings of the graph $K^m_2+K_n$

In this paper, we characterize chromatically unique, determine list-chromatic number and characterize uniquely list colorability of the graph $G=K^m_2+K_n$. We shall prove that $G$ is $\chi$-unique, $\mathrm{ch}(G)=m+n$, $G$ is uniquely $3$-list colorable graph if and only if $2m+n\geqslant 7$ and $m \geqslant 2$.