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A complete bipartite graph K m,n has a maximum matching of size min{m,n}. A complete bipartite graph K n,n has a proper n-edge-coloring corresponding to a Latin square. [14] Every complete bipartite graph is a modular graph: every triple of vertices has a median that belongs to shortest paths between each pair of vertices. [15]
Bipartite graphs may be recognized in polynomial time but, for any k > 2 it is NP-complete, given an uncolored graph, to test whether it is k-partite. [1] However, in some applications of graph theory, a k -partite graph may be given as input to a computation with its coloring already determined; this can happen when the sets of vertices in the ...
The degree sequence of a bipartite graph is the pair of lists each containing the degrees of the two parts and . For example, the complete bipartite graph K 3,5 has degree sequence (,,), (,,,,). Isomorphic bipartite graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a bipartite graph; in ...
However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph K 5 plays a key role in the characterizations of planar graphs: by Kuratowski's theorem, a graph is planar if and only if it contains neither K 5 nor the complete bipartite graph K 3,3 as a subdivision, and by ...
Mathematically, this problem can be formalized as asking for a graph drawing of a complete bipartite graph, whose vertices represent kilns and storage sites, and whose edges represent the tracks from each kiln to each storage site. The graph should be drawn in the plane with each vertex as a point, each edge as a curve connecting its two ...
On the other hand, it is easy to see that G is 3-choosable: picking arbitrary colors for the vertices A and B leaves at least one available color for each of the remaining vertices, and these colors may be chosen arbitrarily. A list coloring instance on the complete bipartite graph K 3,27 with three colors per vertex.
In graph theory, a star S k is the complete bipartite graph K 1,k : a tree with one internal node and k leaves (but no internal nodes and k + 1 leaves when k ≤ 1). Alternatively, some authors define S k to be the tree of order k with maximum diameter 2; in which case a star of k > 2 has k − 1 leaves. A star with 3 edges is called a claw.
The bipartite double cover of the Petersen graph is the Desargues graph: K 2 × G(5,2) = G(10,3). The bipartite double cover of a complete graph K n is a crown graph (a complete bipartite graph K n,n minus a perfect matching). The tensor product of a complete graph with itself is the complement of a Rook's graph.