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.

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(n^3/2) 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 n^1+o(1) 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

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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. S2CID 17529604.
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. Vol. 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. Vol. 9. Springer-Verlag. pp. 17–34. ISBN 978-0-387-52685-0. MR 1083375.
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
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". 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.

External links

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• The panarborial formula, "Theorem of the Day" concerning universal graphs for trees