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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]
A complete bipartite graph with m = 5 and n = 3 The Heawood graph is bipartite. In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets and , that is, every edge connects a vertex in to one in .
Complete bipartite graph (or biclique), a special bipartite graph where every vertex on one side of the bipartition is connected to every vertex on the other side The simplex , which is identical to a complete graph of n + 1 {\displaystyle n+1} vertices, where n {\displaystyle n} is the dimension of the simplex.
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 ...
ch(G) cannot be bounded in terms of chromatic number in general, that is, there is no function f such that ch(G) ≤ f(χ(G)) holds for every graph G. In particular, as the complete bipartite graph examples show, there exist graphs with χ(G) = 2 but with ch(G) arbitrarily large. [2] ch(G) ≤ χ(G) ln(n) where n is the number of vertices of G ...
It is #P-complete to compute this quantity, even for bipartite graphs. [11] It is also #P-complete to count perfect matchings, even in bipartite graphs, because computing the permanent of an arbitrary 0–1 matrix (another #P-complete problem) is the same as computing the number of perfect matchings in the bipartite graph having the given ...
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.
Closely related concepts to complete subgraphs are subdivisions of complete graphs and complete graph minors. In particular, Kuratowski's theorem and Wagner's theorem characterize planar graphs by forbidden complete and complete bipartite subdivisions and minors, respectively.