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Optimal induced universal graphs for bounded-degree graphs

Published online by Cambridge University Press:  11 October 2017

NOGA ALON  and
RAJKO NENADOV
NOGA ALON
Affiliation:
Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. e-mail: [email protected]
RAJKO NENADOV
Affiliation:
School of Mathematical Sciences, Monash University, VIC 3800, Australia. e-mail: [email protected]
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Abstract

We show that for any constant Δ ≥ 2, there exists a graph Γ with O(nΔ / 2) vertices which contains every n-vertex graph with maximum degree Δ as an induced subgraph. For odd Δ this significantly improves the best-known earlier bound and is optimal up to a constant factor, as it is known that any such graph must have at least Ω(nΔ/2) vertices.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 1 , January 2019 , pp. 61 - 74
Copyright
Copyright © Cambridge Philosophical Society 2017 

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Footnotes

†

Research supported in part by a USA-Israeli BSF grant 2012/107, by an ISF grant 620/13 and by the Israeli I-Core program.

References

REFERENCES

[1] Abrahamsen, M., Alstrup, S., Holm, J., Knudsen, M. B. T. and Stöckel, M. Near-optimal induced universal graphs for bounded degree graphs. Preprint. arXiv:1607.04911.Google Scholar
[2] Adjiashvili, D. and Rotbart, N. Labeling schemes for bounded degree graphs. In Proc. 41st Int. Colloq. on Automata, Languages and Programming, Part II (Springer, Berlin Heidelberg, 2014), pp. 375–386.Google Scholar
[3] Alon, N. Asymptotically optimal induced universal graphs. Preprint.Google Scholar
[4] Alon, N. and Capalbo, M. Sparse universal graphs for bounded-degree graphs. Random Structures Algorithms 31 (2) (2007), 123–133.Google Scholar
[5] Alon, N. and Capalbo, M. Optimal universal graphs with deterministic embedding. In Proc. 19th Annu. ACM-SIAM Symp. Discrete Algorithms (Society for Industrial and Applied Mathematics, 2008), pp. 373–378.Google Scholar
[6] Alon, N., Capalbo, M., Kohayakawa, Y., Rödl, V., Ruciński, A. and Szemerédi, E. Near-optimum universal graphs for graphs with bounded degrees. In Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques (Springer, 2001), pp. 170–180.Google Scholar
[7] Alon, N. and Spencer, J. H. The Probabilistic Method (Fourth Edition) (John Wiley and Sons, 2008).Google Scholar
[8] Alstrup, S., Dahlgaard, S. and Knudsen, M. B. T. Optimal induced universal graphs and adjacency labeling for trees. In Proc. 56th Annu. IEEE Symp. on Foundations of Computer Science (IEEE, 2015), pp. 1311–1326.Google Scholar
[9] Alstrup, S., Kaplan, H., Thorup, M. and Zwick, U. Adjacency labeling schemes and induced-universal graphs. In Proc. 47th Annu. ACM Symp. on Theory of Computing (ACM, 2015), pp. 625–634.Google Scholar
[10] Alstrup, S. and Rauhe, T. Small induced-universal graphs and compact implicit graph representations. In Proc. 43rd Annu. IEEE Symp. on Foundations of Computer Science (IEEE, 2002), pp. 53–62.Google Scholar
[11] Butler, S. Induced-universal graphs for graphs with bounded maximum degree. Graphs Combin. 25 (4) (2009), 461–468.Google Scholar
[12] Chung, F. R. Universal graphs and induced-universal graphs. J. Graph Theory 14 (4) (1990), 443–454.Google Scholar
[13] Davidoff, G., Sarnak, P. and Valette, A. Elementary Number Theory, Group Theory and Ramanujan Graphs, vol. 55 (Cambridge University Press, 2003).Google Scholar
[14] Esperet, L., Labourel, A. and Ochem, P. On induced-universal graphs for the class of bounded-degree graphs. Inform. Process. Lett. 108 (5) (2008), 255–260.Google Scholar
[15] Gavoille, C. and Labourel, A. Shorter implicit representation for planar graphs and bounded treewidth graphs. In Proc. European Symp. on Algorithms (Springer, 2007), pp. 582–593.Google Scholar
[16] Kannan, S., Naor, M. and Rudich, S. Implicit representation of graphs. SIAM J. Discrete Math. 5 (4) (1992), 596–603.Google Scholar
[17] Lovász, L. and Plummer, M. D. Matching theory, vol. 367 (American Mathematical Soc., 2009).Google Scholar
[18] Lubotzky, A., Phillips, R. and Sarnak, P. Ramanujan graphs. Combinatorica 8 (3) (1988), 261–277.Google Scholar
[19] Margulis, G. A. Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Problemy peredachi informatsii 24 (1) (1988), 51–60.Google Scholar
[20] Moon, J. W. On minimal n-universal graphs. Proc. Glasgow Mathematical Assoc. 7 (1) (1965), 32–33.Google Scholar
[21] Rado, R. Universal graphs and universal functions. Acta Arith. 4 (9) (1964), 331–340.Google Scholar
[22] Vizing, V. G. Some unsolved problems in graph theory. Russian Math. Surveys 23 (6) (1988), 125–141.Google Scholar