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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.70

Tiling Random Regular Graphs Efficiently

Authors: Sahar Diskin, Ilay Hoshen, and Maksim Zhukovskii

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

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)

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@InProceedings{diskin_et_al:LIPIcs.ICALP.2025.70,
  author =	{Diskin, Sahar and Hoshen, Ilay and Zhukovskii, Maksim},
  title =	{{Tiling Random Regular Graphs Efficiently}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{70:1--70:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.70},
  URN =		{urn:nbn:de:0030-drops-234477},
  doi =		{10.4230/LIPIcs.ICALP.2025.70},
  annote =	{Keywords: Random regular graphs, Tree tilings}
}

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Tiling Random Regular Graphs Efficiently

Abstract

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