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Near-perfect clique-factors in sparse pseudorandom graphs

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  11 December 2020

Jie Han ,
Yoshiharu Kohayakawa  and
Yury Person
Jie Han
Affiliation:
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China
Yoshiharu Kohayakawa
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil
Yury Person*
Affiliation:
Institut für Mathematik, Technische Universität Ilmenau, 98684 Ilmenau, Germany
*
*Corresponding author. Email: yury.person@tu-ilmenau.de
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Abstract

We prove that, for any $t \ge 3$, there exists a constant c = c(t) > 0 such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying $\lambda \le c{d^{t - 1}}/{n^{t - 2}}$ contains vertex-disjoint copies of kt covering all but at most ${n^{1 - 1/(8{t^4})}}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo (Combinatorica24 (2004), pp. 403–426) that (n, d, λ)-graphs with n ∈ 3ℕ and $\lambda \le c{d^2}/n$ for a suitably small absolute constant c > 0 contain triangle-factors. Our arguments combine tools from linear programming with probabilistic techniques, and apply them in a certain weighted setting. We expect this method will be applicable to other problems in the field.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05D40: Probabilistic methods 90C35: Programming involving graphs or networks
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 4 , July 2021 , pp. 570 - 590
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

JH was supported by FAPESP (2014/18641-5, 2013/03447-6).

‡

YK was partially supported by CNPq (311412/2018-1, 423833/2018-9) and FAPESP (2018/04876-1).

§

YP was supported by DFG grant PE 2299/1-1.

The cooperation of the authors was supported by a joint CAPES-DAAD PROBRAL project (project 430/15, 57350402, 57391197). This research was partially supported by CAPES (Finance Code 001). FAPESP is the São Paulo Research Foundation. CNPq is the National Council for Scientific and Technological Development of Brazil.

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