A complete bipartite graph, G := (V1 + V2, E), is a bipartite graph such that for any two vertices, v1 ∈ V1 and v2 ∈ V2, v1v2 is an edge in G. The complete bipartite graph with partitions of size |V1|=m and |V2|=n, is denoted Km,n.
The star graphs S3, S4, S5 and S6.
The utility graph K3,3
For any k, K1,k is called a star. All complete bipartite graphs which are trees are stars.
The graph K3,3 is called the utility graph. This usage comes from a standard mathematical puzzle in which three utilities must each be connected to three buildings; it is impossible to solve without crossings due to the nonplanarity of K3,3.
Given a bipartite graph, finding its complete bipartite subgraph Km,n with maximal number of edges mn is an NP-complete problem.
A planar graph cannot contain K3,3 as a minor; an outerplanar graph cannot contain K3,2 as a minor (These are not sufficient conditions for planarity and outerplanarity, but necessary).
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