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Tiling Random Regular Graphs Efficiently

Authors Sahar Diskin, Ilay Hoshen, Maksim Zhukovskii

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  • Mathematics of computing → Random graphs
Keywords
  • Random regular graphs
  • Tree tilings

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Abstract

We show that for every ε > 0 there exists a sufficiently large d₀ ∈ ℕ such that for every d ≥ d₀, whp the random d-regular graph G(n,d) contains a T-factor for every tree T on at most (1-ε)d/log d vertices. This is best possible since, for large enough integer d, whp G(n,d) does not contain a ((1+ε)d)/(log d)-star-factor. Our method gives a randomised algorithm which whp finds said T-factor and whose expected running time is O(n^{1+o(1)}), as well as an efficient deterministic counterpart.

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Sahar Diskin, Ilay Hoshen, and Maksim Zhukovskii. Tiling Random Regular Graphs Efficiently. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 70:1-70:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.70

Author Details

Sahar Diskin
  • School of Mathematical Sciences, Tel Aviv University, Israel
Ilay Hoshen
  • School of Mathematical Sciences, Tel Aviv University, Israel
Maksim Zhukovskii
  • School of Computer Science, The University of Sheffield, UK

Acknowledgements

The authors thank Itai Benjamini for bringing the question to our attention. The authors wish to thank Michael Krivelevich and Noga Alon for fruitful discussions. In particular, we thank Michael Krivelevich for showing us the alternative argument from [Draganić and Krivelevich, 2025], after the first version of this paper was uploaded. It allowed us to improve the presentation significantly.

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