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k-partite graph
Set
| $k\in\mathrm N$ |
| $V$ … set |
| $\langle V,E\rangle \in \mathrm{it}(E,V) $ |
| $ \langle V,E\rangle $ … undirected graph |
| $ i,j\in\{1,\dots,k\} $ |
| $ X_1,\dots,X_k\subset V $ |
| $ \bigcup_i X_i=V $ |
| $ \forall i,j.\ X_i\cap X_j=\emptyset $ |
| $ v,w\in V $ |
| $\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
