Complete bipartite graph

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Complete bipartite graph
Biclique K 3 5.svg
A complete bipartite graph with m = 5 and n = 3
Vertices n + m
Edges mn
Radius \left\{\begin{array}{ll}1 & m = 1 \vee n = 1\\ 2 & \text{otherwise}\end{array}\right.
Diameter \left\{\begin{array}{ll}1 & m = n = 1\\ 2 & \text{otherwise}\end{array}\right.
Girth \left\{\begin{array}{ll}\infty & m = 1 \vee n = 1\\ 4 & \text{otherwise}\end{array}\right.
Automorphisms \left\{\begin{array}{ll}2 m! n! & n = m\\ m! n! & \text{otherwise}\end{array}\right.
Chromatic number 2
Chromatic index max{m, n}
Spectrum \{0^{n + m - 2}, (\pm \sqrt{n m})^1\}
Notation K_{m,n}

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 Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher.[3][4] Llull himself had made similar drawings of complete graphs three centuries earlier.[3]

Definition[edit]

A complete bipartite graph is a graph whose vertices can be partitioned into two subsets V1 and V2 such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph (V1, V2, E) such that for every two vertices v1V1 and v2V2, v1v2 is an edge in E. A complete bipartite graph with partitions of size |V1|=m and |V2|=n, is denoted Km,n;[1][2] every two graphs with the same notation are isomorphic.

Examples[edit]

The star graphs S3, S4, S5 and S6.
The utility graph K3,3
  • For any k, K1,k is called a star.[2] All complete bipartite graphs which are trees are stars.
  • The graph K1,3 is called a claw, and is used to define the claw-free graphs.[5]
  • The graph K3,3 is called the utility graph. This usage comes from a standard mathematical puzzle in which three utilities must each be connected to three buildings; it is impossible to solve without crossings due to the nonplanarity of K3,3.[6]

Properties[edit]

See also[edit]

References[edit]

  1. ^ a b Bondy, John Adrian; Murty, U. S. R. (1976), Graph Theory with Applications, North-Holland, p. 5, ISBN 0-444-19451-7 .
  2. ^ a b c Diestel, Reinhard (2005), Graph Theory (3rd ed.), Springer, ISBN 3-540-26182-6 . Electronic edition, page 17.
  3. ^ a b Knuth, Donald E. (2013), "Two thousand years of combinatorics", in Wilson, Robin; Watkins, John J., Combinatorics: Ancient and Modern, Oxford University Press, pp. 7–37 .
  4. ^ Read, Ronald C.; Wilson, Robin J. (1998), An Atlas of Graphs, Clarendon Press, p. ii, ISBN 9780198532897 .
  5. ^ Lovász, László; Plummer, Michael D. (2009), Matching theory, AMS Chelsea Publishing, Providence, RI, p. 109, ISBN 978-0-8218-4759-6, MR 2536865 . Corrected reprint of the 1986 original.
  6. ^ Gries, David; Schneider, Fred B. (1993), A Logical Approach to Discrete Math, Springer, p. 437, ISBN 9780387941158 .
  7. ^ Garey, Michael R.; Johnson, David S. (1979), "[GT24] Balanced complete bipartite subgraph", Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, p. 196, ISBN 0-7167-1045-5 .
  8. ^ Diestel, elect. ed. p. 105.
  9. ^ Biggs, Norman (1993), Algebraic Graph Theory, Cambridge University Press, p. 181, ISBN 9780521458979 .
  10. ^ Bollobás, Béla (1998), Modern Graph Theory, Graduate Texts in Mathematics 184, Springer, p. 104, ISBN 9780387984889 .
  11. ^ Bollobás (1998), p. 266.
  12. ^ Jungnickel, Dieter (2012), Graphs, Networks and Algorithms, Algorithms and Computation in Mathematic 5, Springer, p. 557, ISBN 9783642322785 .
  13. ^ Jensen, Tommy R.; Toft, Bjarne (2011), Graph Coloring Problems, Wiley Series in Discrete Mathematics and Optimization 39, John Wiley & Sons, p. 16, ISBN 9781118030745 .