Menu
Home Explore People Places Arts History Plants & Animals Science Life & Culture Technology
On this page
Universal graph

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 and is now called the Rado graph or random graph. More recent work has focused on universal graphs for a graph family F: that is, an infinite graph belonging to F that contains all finite graphs in F. For instance, the Henson graphs are universal in this sense for the i-clique-free graphs.

A universal graph for a family F of graphs can also refer to a member of a sequence of finite graphs that contains all graphs in F; for instance, every finite tree is a subgraph of a sufficiently large hypercube graph 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 n-vertex trees, with only n vertices and O(n log n) edges, and that this is optimal. A construction based on the planar separator theorem can be used to show that n-vertex planar graphs have universal graphs with O(n3/2) edges, and that bounded-degree planar graphs have universal graphs with O(n log n) edges. It is also possible to construct universal graphs for planar graphs that have n1+o(1) vertices. Sumner's conjecture states that tournaments are universal for polytrees, in the sense that every tournament with 2n − 2 vertices contains every polytree with n vertices as a subgraph.

A family F of graphs has a universal graph of polynomial size, containing every n-vertex graph as an induced subgraph, if and only if it has an adjacency labelling scheme in which vertices may be labeled by O(log n)-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 F 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.

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.

We don't have any images related to Universal graph yet.
We don't have any YouTube videos related to Universal graph yet.
We don't have any PDF documents related to Universal graph yet.
We don't have any Books related to Universal graph yet.
We don't have any archived web articles related to Universal graph yet.

References

  1. Rado, R. (1964). "Universal graphs and universal functions". Acta Arithmetica. 9 (4): 331–340. doi:10.4064/aa-9-4-331-340. MR 0172268. /wiki/Richard_Rado

  2. Rado, R. (1967). "Universal graphs". A Seminar in Graph Theory. Holt, Rinehart, and Winston. pp. 83–85. MR 0214507. /wiki/Richard_Rado

  3. Goldstern, Martin; Kojman, Menachem (1996). "Universal arrow-free graphs". Acta Mathematica Hungarica. 1973 (4): 319–326. arXiv:math.LO/9409206. doi:10.1007/BF00052907. MR 1428038. https://doi.org/10.1007%2FBF00052907

  4. Cherlin, Greg; Shelah, Saharon; Shi, Niandong (1999). "Universal graphs with forbidden subgraphs and algebraic closure". Advances in Applied Mathematics. 22 (4): 454–491. arXiv:math.LO/9809202. doi:10.1006/aama.1998.0641. MR 1683298. S2CID 17529604. /wiki/Saharon_Shelah

  5. Wu, A. Y. (1985). "Embedding of tree networks into hypercubes". Journal of Parallel and Distributed Computing. 2 (3): 238–249. doi:10.1016/0743-7315(85)90026-7. /wiki/Doi_(identifier)

  6. Chung, F. R. K.; Graham, R. L. (1983). "On universal graphs for spanning trees" (PDF). Journal of the London Mathematical Society. 27 (2): 203–211. CiteSeerX 10.1.1.108.3415. doi:10.1112/jlms/s2-27.2.203. MR 0692525.. /wiki/Fan_Chung

  7. Babai, L.; Chung, F. R. K.; Erdős, P.; Graham, R. L.; Spencer, J. H. (1982). "On graphs which contain all sparse graphs". In Rosa, Alexander; Sabidussi, Gert; Turgeon, Jean (eds.). Theory and practice of combinatorics: a collection of articles honoring Anton Kotzig on the occasion of his sixtieth birthday (PDF). Annals of Discrete Mathematics. Vol. 12. pp. 21–26. MR 0806964. /wiki/L%C3%A1szl%C3%B3_Babai

  8. Bhatt, Sandeep N.; Chung, Fan R. K.; Leighton, F. T.; Rosenberg, Arnold L. (1989). "Universal graphs for bounded-degree trees and planar graphs". SIAM Journal on Discrete Mathematics. 2 (2): 145–155. doi:10.1137/0402014. MR 0990447. /wiki/Fan_Chung

  9. Chung, Fan R. K. (1990). "Separator theorems and their applications". In Korte, Bernhard; Lovász, László; Prömel, Hans Jürgen; et al. (eds.). Paths, Flows, and VLSI-Layout. Algorithms and Combinatorics. Vol. 9. Springer-Verlag. pp. 17–34. ISBN 978-0-387-52685-0. MR 1083375. 978-0-387-52685-0

  10. Dujmović, Vida; Esperet, Louis; Joret, Gwenaël; Gavoille, Cyril; Micek, Piotr; Morin, Pat (2021), "Adjacency Labelling for Planar Graphs (And Beyond)", Journal of the ACM, 68 (6): 1–33, arXiv:2003.04280, doi:10.1145/3477542 /wiki/ArXiv_(identifier)

  11. Sumner's Universal Tournament Conjecture, Douglas B. West, retrieved 2010-09-17. http://www.math.uiuc.edu/~west/openp/univtourn.html

  12. Kannan, Sampath; Naor, Moni; Rudich, Steven (1992), "Implicit representation of graphs", SIAM Journal on Discrete Mathematics, 5 (4): 596–603, doi:10.1137/0405049, MR 1186827. /wiki/Moni_Naor

  13. Czerwiński, Wojciech; Daviaud, Laure; Fijalkow, Nathanaël; Jurdziński, Marcin; Lazić, Ranko; Parys, Paweł (2018-07-27). "Universal trees grow inside separating automata: Quasi-polynomial lower bounds for parity games". Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 2333–2349. arXiv:1807.10546. doi:10.1137/1.9781611975482.142. ISBN 978-1-61197-548-2. S2CID 51865783. 978-1-61197-548-2