Notation

For $K_{n_1,n_2,...,n_m}$ the complete m-partite graph, there exist a set of labels $\cal F$, some nonnegative integer $r$, and positive integer $\cal Z$, such that $K_{n_1,n_2,...,n_m} \cup \overline {K_r}$ is a mod sum graph (MSG) modulo $Z$. Denote $\hbox{I\kern-.25em\hbox{K}}_{n_1,n_2,...,n_m}=K_{n_1,n_2,...,n_m} \cup \overline {K_r}$. Let $V(K_{n_1,n_2,...,n_m})=V$, $V(\overline {K_r})=R$, $E(K_{n_1,n_2,...,n_m})=E$. Denote partite sets $V_{i}$ for $1 \le i \le m$. We refer to vertices by their label, so $V(\hbox{I\kern-.25em\hbox{K}}_{n_1,n_2,...,n_m})= \cal F$. For easier notation, if $G=(V,E)$ is a MSG, then $v \in V$ means $v$ $(mod$ ${\cal Z}) \in V$, and $\{u,v\} \in E$ means $u+v$ $(mod$ ${\cal Z}) \in \cal F$. Define $x \equiv y$ if and only if $x=y$ $(mod$ $\cal Z)$.