Abstract
We give a polynomial time randomized algorithm that, on receiving as input a pair (H,G) of n-vertex graphs, searches for an embedding of H into G. If H has bounded maximum degree and G is suitably dense and pseudorandom, then the algorithm succeeds with high probability. Our algorithm proves that, for every integer d ≥ 3 and suitable constant C = C d , as n → ∞, asymptotically almost all graphs with n vertices and \(\lfloor Cn^{2-1/d}\log^{1/d}n\rfloor\) edges contain as subgraphs all graphs with n vertices and maximum degree at most d.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alon, N., Capalbo, M.: Sparse universal graphs for bounded-degree graphs. Random Structures Algorithms 31(2), 123–133 (2007)
Alon, N., Capalbo, M.: Optimal universal graphs with deterministic embedding. In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 373–378. ACM, New York (2008)
Alon, N., Capalbo, M., Kohayakawa, Y., Rödl, V., Ruciński, A., Szemerédi, E.: Universality and tolerance (extended abstract). In: 41st Annual Symposium on Foundations of Computer Science, Redondo Beach, CA, pp. 14–21. IEEE Comput. Soc. Press, Los Alamitos (2000)
Alon, N., Capalbo, M., Kohayakawa, Y., Rödl, V., Ruciński, A., Szemerédi, E.: Near-optimum Universal Graphs for Graphs with Bounded Degrees (Extended Abstract). In: Goemans, M.X., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) RANDOM 2001 and APPROX 2001. LNCS, vol. 2129, pp. 170–180. Springer, Heidelberg (2001)
Alon, N., Krivelevich, M., Sudakov, B.: Embedding nearly-spanning bounded degree trees. Combinatorica 27(6), 629–644 (2007)
Balogh, J., Csaba, B., Pei, M., Samotij, W.: Large bounded degree trees in expanding graphs. Electron. J. Combin. 17(1), Research Paper 6, 9 (2010)
Bhatt, S.N., Chung, F.R.K., Leighton, F.T., Rosenberg, A.L.: Universal graphs for bounded-degree trees and planar graphs. SIAM J. Discrete Math. 2(2), 145–155 (1989)
Capalbo, M.R., Kosaraju, S.R.: Small universal graphs. In: Annual ACM Symposium on Theory of Computing, Atlanta, GA, pp. 741–749 (electronic). ACM, New York (1999)
Dellamonica Jr., D., Kohayakawa, Y., Rödl, V., Ruciński, A.: Universality of random graphs. SIAM J. Discrete Math. (to appear)
Dellamonica Jr., D., Kohayakawa, Y.: An algorithmic Friedman–Pippenger theorem on tree embeddings and applications. Electron. J. Combin. 15(1), Research Paper 127, 14 (2008)
Dellamonica Jr., D., Kohayakawa, Y., Rödl, V., Ruciński, A.: Universality of random graphs. In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 782–788. ACM, New York (2008)
Hajnal, A., Szemerédi, E.: Proof of a conjecture of P. Erdős. In: Combinatorial Theory and its Applications II (Proc. Colloq., Balatonfüred, 1969), pp. 601–623. North-Holland, Amsterdam (1970)
Janson, S.: Poisson approximation for large deviations. Random Structures Algorithms 1(2), 221–229 (1990)
Janson, S., Łuczak, T., Rucinski, A.: Random graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York (2000)
Janson, S., Oleszkiewicz, K., Ruciński, A.: Upper tails for subgraph counts in random graphs. Israel J. Math. 142, 61–92 (2004)
Johansson, A., Kahn, J., Vu, V.H.: Factors in random graphs. Random Struct. Algorithms 33(1), 1–28 (2008)
Kierstead, H.A., Kostochka, A.V.: A short proof of the hajnal-szemerédi theorem on equitable colouring. Combinatorics, Probability & Computing 17(2), 265–270 (2008)
Kierstead, H.A., Kostochka, A.V., Mydlarz, M., Szemerédi, E.: A fast algorithm for equitable coloring. Combinatorica 30(2), 217–224 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dellamonica, D., Kohayakawa, Y., Rödl, V., Ruciński, A. (2012). An Improved Upper Bound on the Density of Universal Random Graphs. In: Fernández-Baca, D. (eds) LATIN 2012: Theoretical Informatics. LATIN 2012. Lecture Notes in Computer Science, vol 7256. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29344-3_20
Download citation
DOI: https://doi.org/10.1007/978-3-642-29344-3_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29343-6
Online ISBN: 978-3-642-29344-3
eBook Packages: Computer ScienceComputer Science (R0)