In 1985, Mader conjectured that for every acyclic digraph F there exists K = K(F) such that every digraph D with minimum out-degree at least K contains a subdivision of F. This conjecture remains widely open, even for digraphs F on five vertices. Recently, Aboulker, Cohen, Havet, Lochet, Moura and Thomassé studied special cases of Mader’s problem and made the following conjecture: for every ℓ ≥ 2 there exists K = K(ℓ) such that every digraph D with minimum out-degree at least K contains a subdivision of every orientation of a cycle of length ℓ.
We prove this conjecture and answer further open questions raised by Aboulker et al.
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P. Aboulker, N. Cohen, F. Havet, W. Lochet, P. S. Moura and S. Thomassé: Subdivisions in digraphs of large out-degree or large dichromatic number, Electronic Journal of Combinatorics 26 (2019), Article Number P3.19.
N. Alon: Disjoint directed cycles, Journal of Combinatorial Theory, Series B 68 (1996), 167–178.
J. Bang-Jensen and G. Z. Gutin: Digraphs: theory, algorithms and applications, Springer Science & Business Media, 2008.
B. Bollobás and A. Thomason: Proof of a conjecture of Mader, Erdős and Hajnal on topological complete subgraphs, European Journal of Combinatorics 19 (1998), 883–887.
B. Bollobás and A. Thomason: Highly linked graphs. Combinatorica, 16(3), 313–320.
M. Bucić: An improved bound for disjoint directed cycles, Discrete Mathematics 341 (2018), 2231–2236.
S. Burr: Antidirected subtrees of directed graphs, Canadian Mathematical Bulletin 25 (1982), 119–120.
M. DeVos, J. McDonald, B. Mohar and D. Scheide: Immersing complete digraphs. European Journal of Combinatorics 33 (2012), 1294–1302.
D. Dellamonica Jr, V. Koubek, D. M. Martin and V. Rödl: On a conjecture of Thomassen concerning subgraphs of large girth. Journal of Graph Theory 67 (2011), 316–331.
A. Girão, K. Popielarz and R. Snyder: Subdivisions of digraphs in tournaments. arXiv:1908.03733, 2019.
J. Komlós and E. Szemerédi: Topological cliques in graphs II, Combinatorics, Probability and Computing 5 (1996), 79–90.
W. Mader: Homomorphieeigenschaften und mittlere Kantendichte von Graphen, Mathematische Annalen 174 (1967), 265–268.
W. Mader: Degree and local connectivity in digraphs, Combinatorica 5 (1985), 161–165.
W. Mader: Existence of vertices of local connectivity k in digraphs of large outdegree, Combinatorica 15 (1995), 533–539.
W. Mader: On topological tournaments of order 4 in digraphs of outdegree 3, Journal of Graph Theory 21 (1996), 371–376.
K. Mader: Zur allgemeinen Kurventheorie, Fundamenta Mathematicae 10 (1927), 96–115.
N. Robertson, P. Seymour and R. Thomas: Permanents, Pfaffian orientations and even directed circuits, Annals of Mathematics 150 (1999), 929–975.
P. Seymour and C. Thomassen: Characterization of even directed graphs, Journal of Combinatorial Theory, Series B 42 (1987), 36–45.
C. Thomassen: Disjoint cycles in digraphs, Combinatorica 3 (1983), 393–396.
C. Thomassen: The 2-linkage problem for acyclic digraphs, Discrete Mathematics 55 (1985), 73–87.
C. Thomassen: Even cycles in directed graphs, European Journal of Combinatorics 6 (1985), 85–89.
C. Thomassen: Sign-nonsingular matrices and even cycles in directed graphs, Linear Algebra and its Applications 75 (1986), 27–41.
C. Thomassen: Highly connected non-2-linked digraphs, Combinatorica 11 (1991), 393–395.
C. Thomassen: The even cycle problem for directed graphs, Journal of the American Mathematical Society 5 (1992), 217–229.
The research on this project was initiated during a joint research workshop of Tel Aviv University and the Freie Universität Berlin on Ramsey Theory, held in Tel Aviv in March 2020, and partially supported by GIF grant G-1347-304.6/2016. We would like to thank the German-Israeli Foundation (GIF) and both institutions for their support.
During the work on this project, the author was supported by ERC Starting Grant 633509.
Funded by DFG-GRK 2434 Facets of Complexity.
Research supported in part by GIF grant No. G-1347-304.6/2016 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).
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Gishboliner, L., Steiner, R. & Szabó, T. Oriented Cycles in Digraphs of Large Outdegree. Combinatorica 42 (Suppl 1), 1145–1187 (2022). https://doi.org/10.1007/s00493-021-4750-z
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