Abstract
Kim, Kühn, Osthus and Tyomkyn (Trans. Amer. Math. Soc. 371 (2019), 4655–4742) greatly extended the well-known blow-up lemma of Komlós, Sárközy and Szemerédi by proving a ‘blow-up lemma for approximate decompositions’ which states that multipartite quasirandom graphs can be almost decomposed into any collection of bounded degree graphs with the same multipartite structure and slightly fewer edges. This result has already been used by Joos, Kim, Kühn and Osthus to prove the tree packing conjecture due to Gyárfás and Lehel from 1976 and Ringel’s conjecture from 1963 for bounded degree trees as well as implicitly in the recent resolution of the Oberwolfach problem (asked by Ringel in 1967) by Glock, Joos, Kim, Kühn and Osthus.
Here we present a new and significantly shorter proof of the blow-up lemma for approximate decompositions. In fact, we prove a more general theorem that yields packings with stronger quasirandom properties which is useful for potential applications.
Similar content being viewed by others
References
P. Allen, J. Böttcher, D. Clemens, J. Hladký, D. Piguet and A. Taraz: The tree packing conjecture for trees of almost linear maximum degree, arXiv:2106.11720 (2021).
P. Allen, J. Böttcher, D. Clemens and A. Taraz: Perfectly packing graphs with bounded degeneracy and many leaves, Israel J. Math., to appear
P. Allen, J. Böttcher, H. Hàn, Y. Kohayakawa and Y. Person: Blow-up lemmas for sparse graphs, arXiv:1612.00622 (2016).
P. Allen, J. Böttcher, J. Hladký and D. Piguet: Packing degenerate graphs, Adv. Math. 354 (2019), 106739.
J. Böttcher, J. Hladký, D. Piguet and A. Taraz: An approximate version of the tree packing conjecture, Israel J. Math. 211 (2016), 391–446.
P. Condon, J. Kim, D. Kühn and D. Osthus: A bandwidth theorem for approximate decompositions, Proc. Lond. Math. Soc. 118 (2019), 1393–1449.
R. A. Duke, H. Lefmann and V. Rödl: A fast approximation algorithm for computing the frequencies of subgraphs in a given graph, SIAM J. Comput. 24 (1995), 598–620.
S. Ehard, S. Glock and F. Joos: Pseudorandom hypergraph matchings, Combin. Probab. Comput. 29 (2020), 868–885.
S. Ehard, S. Glock and F. Joos: A rainbow blow-up lemma for almost optimally bounded edge-colourings, Forum Math. Sigma 8 (2020), Paper No. e37, 32.
S. Ehard and F. Joos: Decompositions of quasirandom hypergraphs into hypergraphs of bounded degree, arXiv:2011.05359 (2020).
A. Ferber, C. Lee and F. Mousset: Packing spanning graphs from separable families, Israel J. Math. 219 (2017), 959–982.
A. Ferber and W. Samotij: Packing trees of unbounded degrees in random graphs, J. Lond. Math. Soc. 99 (2019), 653–677.
S. Glock and F. Joos: A rainbow blow-up lemma, Random Structures Algorithms 56 (2020), 1031–1069.
S. Glock, F. Joos, J. Kim, D. Kühn and D. Osthus: Resolution of the Oberwolfach problem, J. Eur. Math. Soc. 23 (2021), 2511–2547.
S. Glock, D. Kühn, A. Lo and D. Osthus: The existence of designs via iterative absorption: hypergraph F-designs for arbitrary F, Mem. Amer. Math. Soc. (to appear).
F. Joos, J. Kim, D. Kühn and D. Osthus: Optimal packings of bounded degree trees, J. Eur. Math. Soc. 21 (2019), 3573–3647.
P. Keevash: A hypergraph blow-up lemma, Random Structures Algorithms 39 (2011), no. 3, 275–376.
P. Keevash: The existence of designs, arXiv:1401.3665 (2014).
P. Keevash: The existence of designs II, arXiv:1802.05900 (2018).
P. Keevash and K. Staden: The generalised Oberwolfach problem, J. Combin. Theory Ser. B 152 (2022), 281–318.
P. Keevash and K. Staden: Ringel’s tree packing conjecture in quasirandom graphs, arXiv:2004.09947 (2020).
J. Kim, Y. Kim and H. Lui: Tree decompositions of graphs without large bipartite holes, Random Structures Algorithms 57 (2020), 150–168.
J. Kim, D. Kühn, A. Kupavskii and D. Osthus: Rainbow structures in locally bounded colourings of graphs, Random Structures Algorithms 56 (2020), 1171–1204.
J. Kim, D. Kühn, D. Osthus and M. Tyomkyn: A blow-up lemma for approximate decompositions, Trans. Amer. Math. Soc. 371 (2019), 4655–4742.
J. Komlós, G. N. Sárközy and E. Szemerédi: Blow-up lemma, Combinatorica 17 (1997), 109–123.
D. Král’, B. Lidický, T. Martins and Y. Pehova: Decomposing graphs into edges and triangles, Combin. Probab. Comput. 28 (2019), 465–472.
D. Kühn and D. Osthus: Hamilton decompositions of regular expanders: A proof of Kelly’s conjecture for large tournaments, Adv. Math. 237 (2013), 62–146.
C. McDiarmid: On the method of bounded differences, Surveys in combinatorics, 1989 (Norwich, 1989), London Math. Soc. Lecture Note Ser., vol. 141, Cambridge Univ. Press, 1989, 148–188.
S. Messuti, V. Rödl and M. Schacht: Packing minor-closed families of graphs into complete graphs, J. Combin. Theory Ser. B 119 (2016), 245–265.
R. Montgomery, A. Pokrovskiy and B. Sudakov: Embedding rainbow trees with applications to graph labelling and decomposition, J. Eur. Math. Soc. 22 (2020), 3101–3132.
R. Montgomery, A. Pokrovskiy and B. Sudakov: A proof of Ringel’s Conjecture, Geom. Funct. Anal. 31 (2021), 663–720.
V. Rödl and A. Ruciński; Perfect matchings in ϵ-regular graphs and the blow-up lemma, Combinatorica 19 (1999), 437–452.
R. M. Wilson: An existence theory for pairwise balanced designs I. Composition theorems and morphisms, J. Combin. Theory Ser. A 13 (1972), 220–245.
R. M. Wilson: An existence theory for pairwise balanced designs II. The structure of PBD-closed sets and the existence conjectures, J. Combin. Theory Ser. A 13 (1972), 246–273.
R. M. Wilson: An existence theory for pairwise balanced designs III. Proof of the existence conjectures, J. Combin. Theory Ser. A 18 (1975), 71–79.
Acknowledgement
We thank the referees for useful comments on our manuscript and the second author thanks Jaehoon Kim for stimulating discussions at early stages of the project.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research leading to these results was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 339933727 (F. Joos).
Rights and permissions
About this article
Cite this article
Ehard, S., Joos, F. A Short Proof of the Blow-Up Lemma for Approximate Decompositions. Combinatorica 42, 771–819 (2022). https://doi.org/10.1007/s00493-020-4640-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-020-4640-9