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[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 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[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 n-vertex trees, with only n vertices and O(n log n) edges, and that this is optimal.[6] 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.[7][8][9] It is also possible to construct universal graphs for planar graphs that have O(n4/3) vertices.[10] 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.[11]

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.[12]

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.[13]

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.
  2. Rado, R. (1967). "Universal graphs". A Seminar in Graph Theory. Holt, Rinehart, and Winston. pp. 83–85. MR 0214507.
  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.
  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.
  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.
  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..
  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. 12. pp. 21–26. MR 0806964.
  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.
  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. 9. Springer-Verlag. pp. 17–34. ISBN 978-0-387-52685-0. MR 1083375.
  10. Bonamy, Marthe; Gavoille, Cyril; Pilipczuk, Michal (2019), Shorter labeling schemes for planar graphs, arXiv:1908.03341
  11. Sumner's Universal Tournament Conjecture, Douglas B. West, retrieved 2010-09-17.
  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.
  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". arXiv:1807.10546 [cs.FL].
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