Properties
- 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).
- A complete bipartite graph. Kn,n is a Moore graph and a (n,4)-cage.
- A complete bipartite graph Kn,n or Kn,n+1 is a Turán graph.
- A complete bipartite graph Km,n has a vertex covering number of min{m,n} and an edge covering number of max{m,n}.
- A complete bipartite graph Km,n has a maximum independent set of size max{m,n}.
- The adjacency matrix of a complete bipartite graph Km,n has eigenvalues √(nm), −√(nm) and 0; with multiplicity 1, 1 and n+m−2 respectively.
- The laplacian matrix of a complete bipartite graph Km,n has eigenvalues n+m, n, m, and 0; with multiplicity 1, m−1, n−1 and 1 respectively.
- A complete bipartite graph Km,n has mn−1 nm−1 spanning trees.
- A complete bipartite graph Km,n has a maximum matching of size min{m,n}.
- A complete bipartite graph Kn,n has a proper n-edge-coloring corresponding to a Latin square.
- The last two results are corollaries of the Marriage Theorem as applied to a k-regular bipartite graph.
Read more about this topic: Complete Bipartite Graph
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—John Locke (16321704)
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