The chromaticity of complete split graphs

The join of null graph $O_m$ and complete graph $K_n$, $O_m+K_n=S(m,n)$, is called a complete split graph. In this paper, we characterize chromatically unique, determine list-chromatic number and characterize unique list colorability of the complete split graph $G=S(m,n)$. We shall prove that $G$ is chromatically unique if and only if $1\le m\le 2$, $ch(G)=n+1$, $G$ is uniquely $3$-list colorable graph if and only if $m\ge 4$, $n\ge 4$ and $m+n\ge 10$, $m(G)\le 4$ for every $1\le m\le 5$ and $n\ge 6$. Some the property of the graph $G=S(m,n)$ when it is $k$-list colorable graph also proved.