Abstract
A construction of Alon and Krivelevich gives highly pseudorandom Kk-free graphs on n vertices with edge density equal to Θ(n−1=(k−2)). In this short note we improve their result by constructing an infinite family of highly pseudorandom Kk-free graphs with a higher edge density of Θ(n−1=(k−1)).
Similar content being viewed by others
References
N. Alon: Explicit Ramsey graphs and orthonormal labelings, Electronic J. Combin.1 #R12, 1994.
N. Alon: Lovász, vectors, graphs and codes, Manuscript https://www.tau.ac.il/~nogaa/PDFS/ll70.pdf, 2018.
N. Alon and M. Krivelevich: Constructive Bounds for a Ramsey-Type Problem, Graphs Combin.13 (1997), 217–225.
S. Ball: Finite Geometry and Combinatorial Applications, Cambridge University Press, 2015.
E. Bannai, S. Hao and S.-Y. Song: Character tables of the association schemes of finite orthogonal groups acting on the nonisotropic points, J. Combin. Theory Ser. A54 (1990), 164–200.
T. Bohman and P. Keevash: The early evolution of the H-free process, Invent. Math.181 (2010), 291–336.
A. E. Brouwer and J. H. van Lint: Strongly regular and graphs partial geometries, in: D. H. Jackson, S. A. Vanstone (Eds.), Enumeration and Design, 85–122, London: Academic Press, 1984.
D. Conlon: A sequence of triangle-free pseudorandom graphs, Comb. Prob. Comp.26 (2017), 195–200.
D. Conlon and J. Lee: On the extremal number of subdivisions, arXiv:1807.05008 [math.CO], 2019.
J. Fox, C. Lee and B. Sudakov: Chromatic number, clique subdivisions, and the conjectures of Hajós and Erdős-Fajtlowicz, Combinatorica33 (2013), 181–197.
W. H. Haemers: Interlacing eigenvalues and graphs, Linear Algebra Appl. 226-228 (1995), 593–616.
X. L. Hubaut: Strongly regular graphs, Disc. Math.13 (1975), 357–381.
F. Ihringer and A. Munemasa: New Strongly Regular Graphs from Finite Geometries via Switching, Linear Algebra Appl.580 (2019), 464–474.
C. Jordan: Traité des substitutions et des équations algébriques, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Reprint of the 1870 original, Éditions Jacques Gabay, Sceaux, 1989.
S. Kopparty: Cayley Graphs, Lecture Notes, http://sites.math.rutgers.edu/~sk1233/courses/graphtheory-F11/cayley.pdf.
M. Krivelevich and B. Sudakov: Pseudo-random graphs, in: Bolyai Soc. Math. Stud., vol. 15, Springer, Berlin: 199–262, 2006.
M. Krivelevich, B. Sudakov and T. Szabó: Triangle factors in sparse pseudorandom graphs, Combinatorica24 (2004), 304–426.
D. Mubayi and J. Verstraëte: A note on pseudorandom Ramsey graphs, arXiv:1909.01461, [math.CO], 2019.
A. Munemasa: The Geometry of Orthogonal Groups over Finite Fields, JSPS-DOST Lecture Notes in Mathematics 3, http://www.math.is.tohoku.ac.jp/~munemasa/documents/polar.pdf, 1996.
J. Soto-Andrade: Harmoniques sphériques sur un corps fini, C. R. Acad. Sci. Paris Sr. A-B272 (1971), A1642–A1645.
J. Soto-Andrade: Concerning spherical functions (the finite case), Workshop on the geometry and theory of group representations (Spanish) (Santiago, 1985), Notas Soc. Mat. Chile1 (1971), 71–93.
B. Sudakov, T. Szabó and V. H. Vu: A generalization of Turán's theorem, J. Graph Theory49 (2005), 187–195.
E. Witt: Theorie der quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math.176 (1937), 31–44.
Acknowledgments
We would like to thank David Conlon for his helpful remarks on an earlier draft of this paper. We would like to thank Akihiro Munemasa whose work together with the second author in [13, §6] on cospectral graphs inspired the current result. Finally, we would like to thank the referees for their careful reading and helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by a Humboldt Research Fellowship for Postdoctoral Researchers and by Discovery Early Career Award of the Australian Research Council (No. DE190100666).
The author is supported by a postdoctoral fellowship of the Research Foundation — Flanders (FWO).
The author is supported by INDAM (Istituto Nazionale Di Alta Matemetica)
Rights and permissions
About this article
Cite this article
Bishnoi, A., Ihringer, F. & Pepe, V. A Construction for Clique-Free Pseudorandom Graphs. Combinatorica 40, 307–314 (2020). https://doi.org/10.1007/s00493-020-4226-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-020-4226-6