complete k-partite graph

A complete k-partite graph is a maximal k-partitegraph. In other words, a graph is complete k-partite providedthat its vertex set admits a partition $V=V_1\bigsqcup \mathrm{\dots }\bigsqcup V_k$such that for any $u\in V_i$ and $v\in V_j$, uv is an edge if andonly if $i\ne j$.

For any tuple $(a_1,\mathrm{\dots },a_k)$, there is up to graph isomorphism asingle complete k-partite graph with maximal stable sets$V_1,\mathrm{\dots },V_k$ such that for each $i$, there are exactly $a_i$vertices in $V_i$. This graph is denoted by $K_{a_1,\mathrm{\dots },a_k}$.

Below we display the 3-partite complete graph $K_{2,3,4}$: