Universal graph explained

In mathematics, a universal graph is an infinite graph that contains every finite (or at-most-countable) graph as an induced subgraph. A universal graph of this type was first constructed by Richard Rado[1] [2] and is now called the Rado graph or random graph. More recent work[3] [4] has focused on universal graphs for a graph family : that is, an infinite graph belonging to F that contains all finite graphs in . For instance, the Henson graphs are universal in this sense for the -clique-free graphs.

A universal graph for a family of graphs can also refer to a member of a sequence of finite graphs that contains all graphs in ; for instance, every finite tree is a subgraph of a sufficiently large hypercube graph[5] so a hypercube can be said to be a universal graph for trees. However it is not the smallest such graph: it is known that there is a universal graph for -vertex trees, with only  vertices and edges, and that this is optimal.[6] A construction based on the planar separator theorem can be used to show that -vertex planar graphs have universal graphs with edges, and that bounded-degree planar graphs have universal graphs with edges.[7] [8] [9] It is also possible to construct universal graphs for planar graphs that have vertices. Sumner's conjecture states that tournaments are universal for polytrees, in the sense that every tournament with vertices contains every polytree with vertices as a subgraph.[10]

A family of graphs has a universal graph of polynomial size, containing every -vertex graph as an induced subgraph, if and only if it has an adjacency labelling scheme in which vertices may be labeled by -bit bitstrings such that an algorithm can determine whether two vertices are adjacent by examining their labels. For, if a universal graph of this type exists, the vertices of any graph in may be labeled by the identities of the corresponding vertices in the universal graph, and conversely if a labeling scheme exists then a universal graph may be constructed having a vertex for every possible label.[11]

In older mathematical terminology, the phrase "universal graph" was sometimes used to denote a complete graph.

The notion of universal graph has been adapted and used for solving mean payoff games.[12]

External links

Notes and References

  1. Rado . R. . Richard Rado . Universal graphs and universal functions . Acta Arithmetica . 9 . 4 . 1964 . 331–340 . 0172268 . 10.4064/aa-9-4-331-340. free .
  2. Rado . R. . Richard Rado . Universal graphs . 1967 . A Seminar in Graph Theory . Holt, Rinehart, and Winston . 83–85 . 0214507.
  3. Goldstern, Martin . Kojman, Menachem . Universal arrow-free graphs . 1996 . . 1973 . 319–326 . 10.1007/BF00052907 . free . math.LO/9409206 . 1428038 . 4.
  4. Cherlin . Greg . Shelah . Saharon . Saharon Shelah . Shi . Niandong . Universal graphs with forbidden subgraphs and algebraic closure . Advances in Applied Mathematics . 22 . 1999 . 454–491 . 10.1006/aama.1998.0641 . math.LO/9809202 . 1683298 . 4. 17529604 .
  5. Wu, A. Y. . Embedding of tree networks into hypercubes . Journal of Parallel and Distributed Computing . 2 . 1985 . 238–249 . 10.1016/0743-7315(85)90026-7 . 3 .
  6. F. R. K.. Chung. Fan Chung. R. L.. Graham. Ronald Graham. On universal graphs for spanning trees. Journal of the London Mathematical Society. 27. 1983. 203–211. 10.1112/jlms/s2-27.2.203. 2. 0692525. 10.1.1.108.3415. .
  7. Book: Babai . L. . László Babai . Chung . F. R. K. . Fan Chung . Erdős . P. . Paul Erdős . Graham . R. L. . Ronald Graham . Spencer . J. H. . Joel Spencer . On graphs which contain all sparse graphs . Rosa . Alexander . Sabidussi . Gert . Turgeon . Jean . 21–26 . Annals of Discrete Mathematics . Theory and practice of combinatorics: a collection of articles honoring Anton Kotzig on the occasion of his sixtieth birthday . 12 . 1982 . 0806964.
  8. Bhatt . Sandeep N. . Chung . Fan R. K. . Fan Chung . Leighton . F. T. . F. Thomson Leighton . Rosenberg . Arnold L. . Arnold L. Rosenberg . 10.1137/0402014 . . 145–155 . Universal graphs for bounded-degree trees and planar graphs . 2 . 1989 . 2 . 0990447.
  9. Book: Chung , Fan R. K. . Fan Chung . Separator theorems and their applications . Korte . Bernhard . Bernhard Korte . Lovász . László . László Lovász . Prömel . Hans Jürgen . Schrijver . Alexander . 3 . 978-0-387-52685-0 . 17–34 . Springer-Verlag . Algorithms and Combinatorics . Paths, Flows, and VLSI-Layout . 9 . 1990 . 1083375 .
  10. http://www.math.uiuc.edu/~west/openp/univtourn.html Sumner's Universal Tournament Conjecture
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  12. Book: Czerwiński. Wojciech. Daviaud. Laure. Fijalkow. Nathanaël. Jurdziński. Marcin. Lazić. Ranko. Parys. Paweł. 2018-07-27. Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms. Universal trees grow inside separating automata: Quasi-polynomial lower bounds for parity games. 2333–2349. 10.1137/1.9781611975482.142. 1807.10546. 978-1-61197-548-2. 51865783.

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