k-partite graph


context $k\in\mathrm N$
context $V$ … set
definiendum $\langle V,E\rangle \in \mathrm{it}(E,V) $
inclusion $ \langle V,E\rangle $ … undirected graph
range $ i,j\in\{1,\dots,k\} $
range $ \bigcup_i X_i=V $
range $ \forall i,j.\ X_i\cap X_j=\emptyset $
range $ v,w\in V $
postulate $\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) $


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.


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

$|E|\le \frac{1}{2}\sum_{i=1}^k |X_i|\cdot (|V|-|X_i|)$


Subset of

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