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In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set.

- Given a bipartite graph, finding its complete bipartite subgraph $K\_\{m,n\}$ with maximal number of edges $mcdot\; n$ is a NP-complete problem.
- A planar graph cannot contain $K\_\{3,3\}$ as a minor; an outerplanar graph cannot contain $K\_\{3,2\}$ as a minor (These are not sufficient conditions for planarity and outerplanarity, but necessary).
- A complete bipartite graph. $K\_\{n,n\}$ is a Moore graph and a $(n,4)$-cage
- A complete bipartite graph $K\_\{n,n\}$ or $K\_\{n,n+1\}$ is a Turán graph
- A complete bipartite graph $K\_\{m,n\}$ has a vertex covering number of $min\; lbrace\; m,n\; rbrace$ and an edge covering number of $maxlbrace\; m,nrbrace$
- A complete bipartite graph $K\_\{m,n\}$ has a maximum independent set of size $maxlbrace\; m,nrbrace$
- A complete bipartite graph $K\_\{m,n\}$ has a maximum matching of size $minlbrace\; m,nrbrace$
- A complete bipartite graph $K\_\{n,n\}$ has a proper n-edge-coloring
- A complete bipartite graph $K\_\{m,n\}$ has m
^{n-1}n^{m-1}Spanning trees. - The last two results are corollaries of the Marriage Theorem as applied to a k-regular bipartite graph

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Last updated on Tuesday October 07, 2008 at 13:53:40 PDT (GMT -0700)

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Last updated on Tuesday October 07, 2008 at 13:53:40 PDT (GMT -0700)

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