Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T10:51:52.729Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Erdős–Sós conjecture for trees with bounded degree

Part of: Graph theory

Published online by Cambridge University Press:  01 February 2021

Guido Besomi ,
Matías Pavez-Signé  and
Maya Stein
Guido Besomi
Affiliation:
Department of Mathematical Engineering, University of Chile
Matías Pavez-Signé
Affiliation:
Department of Mathematical Engineering, University of Chile
Maya Stein*
Affiliation:
Department of Mathematical Engineering, University of Chile
*
*Corresponding author. Email: mstein@dim.uchile.cl
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the Erdős–Sós conjecture for trees with bounded maximum degree and large dense host graphs. As a corollary, we obtain an upper bound on the multicolour Ramsey number of large trees whose maximum degree is bounded by a constant.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C07: Vertex degrees 05C05: Trees
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 5 , September 2021 , pp. 741 - 761
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

MPS was supported by ANID Doctoral scholarship ANID-PFCHA/Doctorado Nacional/2017-21171132.

‡

MS is also affiliated to Centro de Modelamiento Matemático, Universidad de Chile, UMI 2807 CNRS. MS acknowledges support by CONICYT + PIA/Apoyo a centros científicos y tecnológicos de excelencia con financiamiento Basal, Código AFB170001, and by Fondecyt Regular Grant 1183080.

References

Ajtai, M., Komlós, J. and Szemerédi, E. (1995) On a conjecture of Loebl. In Graph Theory, Combinatorics, and Algorithms (Kalamazoo, MI, 1992), pp. 1135–1146. Wiley-Interscience.Google Scholar
Besomi, G., Pavez-Signé, M. and Stein, M. (2019) Degree conditions for embedding trees. SIAM J. Discrete Math. 33 1521–1555.CrossRefGoogle Scholar
Brandt, S. and Dobson, E. (1996) The Erdős–Sós conjecture for graphs of girth 5. Discrete Math. 150 411–414.10.1016/0012-365X(95)00207-DCrossRefGoogle Scholar
Erdős, P. (1964) Extremal problems in graph theory. In Theory of Graphs and its Applications (Proc. Sympos. Smolenice) (Fiedler, M., ed.), pp. 29–36. Academic Press.Google Scholar
Erdős, P. and Gallai, T. (1959) On maximal paths and circuits of graphs. Acta Math. Acad. Sci. Hungar. 10 337–356.CrossRefGoogle Scholar
Erdős, P. and Graham, R. L. (1975) On partition theorems for finite graphs. In Infinite and Finite sets (Colloq. Keszthely, 1973), Vol. 10 of Colloquia Mathematica Societatis János Bolyai, pp. 515–527. North-Holland.Google Scholar
Goerlich, A. and Zak, A. (2016) On Erdős–Sós conjecture for trees of large size. Electron. J. Combin. 23 P1–52.10.37236/5405CrossRefGoogle Scholar
Havet, F., Reed, B., Stein, M. and Wood, D. R. (2020) A variant of the Erdős–Sós conjecture. J. Graph Theory 94 131–158.CrossRefGoogle Scholar
Hladký, J., Komlós, J., Piguet, D., Simonovits, M., Stein, M. and Szemerédi, E. (2017) The approximate Loebl–Komlós–Sós conjecture IV: Embedding techniques and the proof of the main result. SIAM J. Discrete Math. 31 1072–1148.10.1137/140982878CrossRefGoogle Scholar
Komlós, J., Shokoufandeh, A., Simonovits, M. and Szemerédi, E. (2000) The regularity lemma and its applications in graph theory. In Theoretical Aspects of Computer Science, Vol. 2292 of Lecture Notes in Computer Science, pp. 84–112. Springer.CrossRefGoogle Scholar
Krivelevich, M. (2010) Embedding spanning trees in random graphs. SIAM J. Discrete Math. 24 1495–1500.10.1137/100805753CrossRefGoogle Scholar
Montgomery, R. (2019) Spanning trees in random graphs. Adv. Math. 356 106793.10.1016/j.aim.2019.106793CrossRefGoogle Scholar
Reed, B. and Stein, M. (2019) Spanning trees in graphs of high minimum degree with a universal vertex I: An approximate asymptotic result. arXiv:1905.09801Google Scholar
Rozhoň, V. (2019) A local approach to the Erdős–Sós conjecture. SIAM J. Discrete Math. 33 643–664.CrossRefGoogle Scholar
Saclé, J.-F. and Woźniak, M. (1997) A note on the Erdős–Sós conjecture for graphs without C 4 . J. Combin. Theory Ser. B 70 229–234.10.1006/jctb.1997.1758CrossRefGoogle Scholar
Szemerédi, E. (1978) Regular partitions of graphs. In Problèmes Combinatoires et Théorie des Graphes, Vol. 260 of Colloq. Internat. CNRS, pp. 399–401. CNRS.Google Scholar