===== k-partite graph =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $k\in\mathrm N$ |
| @#55CCEE: context     | @#55CCEE: $V$ ... set |

| @#FFBB00: definiendum | @#FFBB00: $\langle V,E\rangle \in \mathrm{it}(E,V) $ |

| @#AAFFAA: inclusion   | @#AAFFAA: $ \langle V,E\rangle $ ... undirected graph |

| @#DDDDDD: range       | @#DDDDDD: $ i,j\in\{1,\dots,k\} $ |
| @#DDDDDD: range       | @#DDDDDD: $ \bigcup_i X_i=V $ |
| @#DDDDDD: range       | @#DDDDDD: $ \forall i,j.\ X_i\cap X_j=\emptyset $ |
| @#DDDDDD: range       | @#DDDDDD: $ v,w\in V $ |

| @#55EE55: postulate   | @#55EE55: $\exists X_1,\dots,X_k.\ \forall u,v.\ \{u,v\}\in E\implies \forall i.\ \neg(v\in X_i\land w\in X_i) $ |

==== Discussion ====
The $X_i$ are the partitions of the graph and the condition says that there can be no edge within an $X_i$, i.e. there are only connection from one partition to another. One can also few the partitions as different coloring of their vertices.
=== Theorems ===
From any $v\in X_i$, there can be edges to only the other partitions, i.e. to at most $|V|-|X_i|$ different other vertices. If we sum up the edges for all partitions and divide the double-counting out, we find that for an $k$-partite graph

$|E|\le \frac{1}{2}\sum_{i=1}^k |X_i|\cdot (|V|-|X_i|)$
==== Parents ====
=== Subset of ===
[[Undirected graph]]