Such problems occur, for example, in the theory of scheduling (partitioning of the edges of a bipartite graph into a minimal number of disjoint matchings), in the problem of assignment (finding the maximum number of elements in a matching), etc. the elements of a given set and a subset of it yield the relation of "membership of an element to a subset", for executors and types of jobs one has the relation "a given executor can carry out a given job", etc.Īn important problem concerning bipartite graphs is the study of matchings, that is, families of pairwise non-adjacent edges. Bipartite graphs are convenient for the representation of binary relations between elements of two different types - e.g. Another frequently used definition of a bipartite graph is a graph in which two subsets $V'$ and $V''$ of vertices (parts) are given in advance. A graph is bipartite if and only if all its simple cycles have even length. $V=V'\cup V''$, $V'\cap V''=\emptyset$) so that each edge connects some vertex of $V'$ with some vertex of $V''$. 2010 Mathematics Subject Classification: Primary: 05C Ī graph whose set $V$ of vertices can be partitioned into two disjoint sets $V'$ and $V''$ (i.e. Toft, Bjarne (2011), Graph Coloring Problems, Wiley Series in Discrete Mathematics and Optimization, vol. 39, Wiley, p. 16, ISBN 9781118030745. ^ Jungnickel, Dieter (2012), Graphs, Networks and Algorithms, Algorithms and Computation in Mathematic, vol. 5, Springer, p. 557, ISBN 9783642322785.^ Biggs, Norman (1993), Algebraic Graph Theory, Cambridge University Press, p. 181, ISBN 9780521458979.(1979), " Balanced complete bipartite subgraph", Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, p. ^ Coxeter, Regular Complex Polytopes, second edition, p.114.(eds.), Combinatorics: Ancient and Modern, Oxford University Press, pp. 7–37, ISBN 0191630624. (2013), "Two thousand years of combinatorics", in Wilson, Robin Watkins, John J. ^ a b c Diestel, Reinhard (2005), Graph Theory (3rd ed.), Springer, ISBN 2-6.(1976), Graph Theory with Applications, North-Holland, p. Complete multipartite graph, a generalization of complete bipartite graphs to more than two sets of vertices.Crown graph, a graph formed by removing a perfect matching from a complete bipartite graph.Biclique-free graph, a class of sparse graphs defined by avoidance of complete bipartite subgraphs.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.A complete bipartite graph K n, n has a proper n-edge-coloring corresponding to a Latin square.A complete bipartite graph with m = 5 and n = 3
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