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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 ...
The graph theoretic formulation of Marshal Hall's extension of the marriage theorem can be stated as follows: Given a bipartite graph with sides A and B, we say that a subset C of B is smaller than or equal in size to a subset D of A in the graph if there exists an injection in the graph (namely, using only edges of the graph) from C to D, and ...
K{m,n} Table of graphs and parameters. In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set. [1][2] Graph theory itself is typically dated as beginning with Leonhard Euler 's 1736 work on the Seven ...
An optimal drawing of K4,7, with 18 crossings (red dots) In the mathematics of graph drawing, Turán's brick factory problem asks for the minimum number of crossings in a drawing of a complete bipartite graph. The problem is named after Pál Turán, who formulated it while being forced to work in a brick factory during World War II.
The assignment problem consists of finding, in a weighted bipartite graph, a matching of a given size, in which the sum of weights of the edges is minimum. If the numbers of agents and tasks are equal, then the problem is called balanced assignment. Otherwise, it is called unbalanced assignment. [1] If the total cost of the assignment for all ...
Bipartite realization problem. The bipartite realization problem is a classical decision problem in graph theory, a branch of combinatorics. Given two finite sequences and of natural numbers with , the problem asks whether there is a labeled simple bipartite graph such that is the degree sequence of this bipartite graph.
The Hamiltonian path problem is a topic discussed in the fields of complexity theory and graph theory. It decides if a directed or undirected graph, G, contains a Hamiltonian path, a path that visits every vertex in the graph exactly once. The problem may specify the start and end of the path, in which case the starting vertex s and ending ...
Conversely, Kőnig's theorem proves the perfection of the complements of bipartite graphs, a result proven in a more explicit form by Gallai (1958). One can also connect Kőnig's line coloring theorem to a different class of perfect graphs, the line graphs of bipartite graphs. If G is a graph, the line graph L (G) has a vertex for each edge of ...