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A complete bipartite graph K m,n has m n−1 n m−1 spanning trees. [13] 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 ...
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 .
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 ...
For each fixed choice of m, the truth of the conjecture for all K m,n can be verified by testing only a finite number of choices of n. [15] More generally, it has been proven that every complete bipartite graph requires a number of crossings that is (for sufficiently large graphs) at least 83% of the number given by the Zarankiewicz bound.
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 ...
A bipartite graph with 4 vertices on each side, 13 edges, and no , subgraph, and an equivalent set of 13 points in a 4 × 4 grid, showing that (;).. The number (;) asks for the maximum number of edges in a bipartite graph with vertices on each side that has no 4-cycle (its girth is six or more).
A strengthened version of the conjecture states that each such graph has an equitable coloring with exactly Δ colors, with one additional exception, a complete bipartite graph in which both sides of the bipartition have the same odd number of vertices. [1]
The matching complex of a complete bipartite graph K m,n is known as a chessboard complex. It is the clique graph of the complement graph of a rook's graph, [5] and each of its simplices represents a placement of rooks on an m × n chess board such that no two of the rooks attack each other.