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Oriented Trees and Paths in Digraphs

Published online by Cambridge University Press:  23 May 2024

By
Maya Stein
Edited by
Felix Fischer  and
Robert Johnson
Felix Fischer
Affiliation:
Queen Mary University of London
Robert Johnson
Affiliation:
Queen Mary University of London
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Summary

Which conditions ensure that a digraph contains all oriented paths of some given length, or even a all oriented trees of some given size, as a subgraph? One possible condition could be that the host digraph is a tournament of a certain order. In arbitrary digraphs and oriented graphs, conditions on the chromatic number, on the edge density, on the minimum outdegree and on the minimum semidegree have been proposed. In this survey, we review the known results, and highlight some open questions in the area.

Type
Chapter
Information
Surveys in Combinatorics 2024 , pp. 271 - 295
Publisher: Cambridge University Press
Print publication year: 2024

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References

Aboulker, P., Cohen, N., Havet, F., Lochet, W., F. S. Moura, P., and S. Thomassé, Subdivisions in digraphs of large out-degree or large dichromatic number, Electronic Journal of Combinatorics 26 (2019), P3.19.CrossRefGoogle Scholar
Addario-Berry, L., Havet, F., Linhares Sales, C., Reed, B., and Thomassé, S., Oriented trees in digraphs, Discrete Mathematics 313 (2013), 967–974.CrossRefGoogle Scholar
Addario-Berry, L., Havet, F., and Thomassé, S., Paths with two blocks in kchromatic digraphs, Journal of Combinatorial Theory, Series B 97 (2007), 620–626.CrossRefGoogle Scholar
Alon, N., The maximum number of hamiltonian paths in tournaments, Combinatorica 10 (1990), 319–324.CrossRefGoogle Scholar
Alspach, B. and Rosenfeld, M., Realisation of certain generalized paths in tournaments, Discrete Mathematics 34 (1981), 199–202.CrossRefGoogle Scholar
Bang-Jensen, J., Problems and conjectures concerning connectivity, paths, trees and cycles in tournament-like digraphs, Discrete Mathematics 309 (2009), 5655–5667.CrossRefGoogle Scholar
Bang-Jensen and G. Gutin, J., Paths, trees and cycles in tournaments, Congressus Numerantium (1996), 131–170.Google Scholar
Bang-Jensen and F. Havet, J., Tournaments and semicomplete digraphs, In: Classes of Directed Graphs, Springer International Publishing, 2018, pp. 35–124.CrossRefGoogle Scholar
Bender, E.A. and Wormald, N.C., Random trees in random graphs, Proceedings of the American Mathematical Society 103 (1988), no. 1, 314–320.CrossRefGoogle Scholar
Benford, A. and Montgomery, R., Trees with few leaves in tournaments, Journal of Combinatorial Theory, Series B 155 (2022), 141–170.CrossRefGoogle Scholar
Benford, A. and Montgomery, R., Trees with many leaves in tournaments, Preprint arXiv:2207.06384, 2022.Google Scholar
Bermond, J. C., Germa, A., Heydemann, M. C., and Sotteau, D., Longest paths in digraphs, Combinatorica 1 (1981), 337–341.CrossRefGoogle Scholar
Besomi, G., Pavez-Signé, M., and Stein, M., Degree conditions for embedding trees, SIAM Journal on Discrete Mathematics 33 (2019), 1521–1555.CrossRefGoogle Scholar
Besomi, G., Pavez-Signé, M., and Stein, M., On the Erdős–Sós conjecture for trees with bounded degree, Combinatorics, Probability and Computing 30 (2021), no. 5, 741–761.CrossRefGoogle Scholar
Burr, S. A., Subtrees of directed graphs and hypergraphs, Proceedings of the Eleventh Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, Fla., 1982), vol. 28, 1980, pp. 227–239.Google Scholar
Ceroi, S. and Havet, F., Trees with three leaves are (n+1)-unavoidable, Discrete Applied Mathematics 141 (2004), no. 1, 19–39, Brazilian Symposium on Graphs, Algorithms and Combinatorics.CrossRefGoogle Scholar
Chung, F.R.K., A note on subtrees in tournaments, Internal Memorandum, Bell Laboratories, 1981.Google Scholar
Csaba, B., Levitt, I., Nagy-Gÿorgy, J., and Szemerédi, E., Tight bounds for embedding bounded degree trees, Katona G.O.H., Schrijver A., Szenyi T., Sági G. (eds) Fête of Combinatorics and Computer Science, vol. 20, 2010, pp. 95–137.CrossRefGoogle Scholar
Davoodi, A., Piguet, D., R̆ada, H., and Sanhueza-Matamala, N., Beyond the ErdősSo’s conjecture, Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications EUROCOMB’23.Google Scholar
DeBiasio, L., Kühn, D., Molla, T., Osthus, D., and Taylor, A., Arbitrary orientations of hamilton cycles in digraphs, SIAM Journal on Discrete Mathematics 29 (2015), no. 3, 1553–1584.CrossRefGoogle Scholar
DeBiasio, L. and Molla, T., Semi-degree threshold for anti-directed hamiltonian cycles, Electronic Journal of Combinatorics 22 (2015), no. 4, P4.34.CrossRefGoogle Scholar
Dirac, G. A., Some theorems on abstract graphs, Proceedings of the London Mathematical Society 2 (1952), 69–81.CrossRefGoogle Scholar
Dross, F. and Havet, F., On the unavoidability of oriented trees, Journal of Combinatorial Theory, Series B 151 (2021), 83–110.CrossRefGoogle Scholar
El Sahili, A., Paths with two blocks in k-chromatic digraphs, Discrete Mathematics 287 (2004), 151–153.CrossRefGoogle Scholar
El Sahili, A., Trees in tournaments, Journal of Combinatorial Theory, Series B 92 (2004), 183–187.CrossRefGoogle Scholar
Sahili, A. El and Ghazo Hanna, Z., About the number of oriented hamiltonian paths and cycles in tournaments, Journal of Graph Theory 102 (2023), no. 4, 684–701.CrossRefGoogle Scholar
Erdős, P., Extremal problems in graph theory, Theory of graphs and its applications, Proc. Sympos. Smolenice, 1964, pp. 29–36.Google Scholar
Erdős, P. and Gallai, T., On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar 10 (1959), 337–356.CrossRefGoogle Scholar
Forcade, R., Parity of paths and circuits in tournaments, Discrete Mathematics 6 (1973), 115–118.CrossRefGoogle Scholar
Ghouila-Houri, A., Une condition suffisante d’existence d’un circuit Hamiltonien, Comptes rendus de lácadémie des Sciences 251 (1960), 495–497.Google Scholar
Graham, R. L., On subtrees of directed graphs with no path of length exceeding one, Canadian Mathematical Bulletin 13 (1970), 329–332.CrossRefGoogle Scholar
Grünbaum, B., Antidirected Hamiltonian paths in tournaments, Journal of Combinatorial Theory, Series B 11 (1971), 249–257.CrossRefGoogle Scholar
Häggkvist, R. and Thomason, A., Trees in tournaments, Combinatorica 11 (1991), 123–130.CrossRefGoogle Scholar
Hanna, C. B., Paths in tournaments: a simple proof of Rosenfeld’s conjecture, Preprint arXiv:2011.14394, 2020.Google Scholar
Havet, F., Directed path of length twice the minimum outdegree, Open Problem Garden, http://www.openproblemgarden.org/op/directed cycle of length twice the minimum outdegree, retrieved 20-06-2023.Google Scholar
Havet, F., Oriented trees in n-chromatic digraphs, Open Problem Garden, http://www.openproblemgarden.org/op/oriented trees in n chromatic digraphs, retrieved 20-06-2023.Google Scholar
Havet, F., Trees in tournaments, Discrete Mathematics 243 (2002), 121–134.CrossRefGoogle Scholar
Havet, F., On unavoidability of trees with k leaves, Graphs and Combinatorics 19 (2003), 101–110.CrossRefGoogle Scholar
Havet, F., Reed, B., Stein, M., and Wood, D. R., A variant of the Erdős–Sós conjecture, Journal of Graph Theory 94 (2020), 131–158.CrossRefGoogle Scholar
Havet, F. and S. Thomassé, Median orders of tournaments: a tool for the second neighbourhood problem and Sumner’s conjecture, Journal of Graph Theory 35 (2000), 244–256.3.0.CO;2-H>CrossRefGoogle Scholar
Havet, F. and S. Thomassé, Oriented Hamiltonian paths in tournaments: a proof of Rosenfeld’s conjecture, Journal of Combinatorial Theory, Series B 78 (2000), 243–273.CrossRefGoogle Scholar
Jackson, B., Long paths and cycles in oriented graphs, Journal of Graph Theory 5 (1981), no. 2, 145–157.CrossRefGoogle Scholar
Kathapurkar, A. and Montgomery, R., Spanning trees in dense directed graphs, Journal of Combinatorial Theory, Series B 156 (2022), 223–249.CrossRefGoogle Scholar
Keevash, P., Kühn, D., and Osthus, O., An exact minimum degree condition for hamilton cycles in oriented graphs, Journal of the London Mathematical Society 79 (2009), 144–166.CrossRefGoogle Scholar
Kelly, L., Arbitrary orientations of hamilton cycles in oriented graphs, Electronic Journal of Combinatorics 18 (2011), P.44.CrossRefGoogle Scholar
Kelly, L., Kühn, D., and Osthus, D., Cycles of given length in oriented graphs, Journal of Combinatorial Theory Series B 100 (2010), 251–264.CrossRefGoogle Scholar
Klimos̆ová, T. and Stein, M., Antipaths in oriented graphs, Discrete Mathematics 346 (2023), 113515.CrossRefGoogle Scholar
Komlós, J., Sárközy, G. N., and Szemerédi, E., Spanning trees in dense graphs, Combinatorics, Probability and Computing 10 (2001), no. 5, 397–416.CrossRefGoogle Scholar
Kühn, D., Mycroft, R., and Osthus, D., An approximate version of Sumner’s universal tournament conjecture, J. Combin. Theory Ser. B 101 (2011), no. 6, 415–447.CrossRefGoogle Scholar
Kühn, D., Mycroft, R., and Osthus, D., A proof of Sumner’s universal tournament conjecture for large tournaments, Proc. Lond. Math. Soc. (3) 102 (2011), no. 4, 731–766.CrossRefGoogle Scholar
Lu, X., On claws belonging to every tournament, Combinatorica 11 (1991), 173–179.CrossRefGoogle Scholar
Lu, X., Claws contained in all n-tournaments, Discrete Mathematics 119 (1993), 107–111.CrossRefGoogle Scholar
Lu, X., Wang, D.-W., and Wong, C.-K., On avoidable and unavoidable claws, Discrete Mathematics 184 (1998), 259–265.CrossRefGoogle Scholar
Mader, W., Connectivity keeping trees in k-connected graphs, Journal of Graph Theory 69 (2012), no. 3, 324–329.CrossRefGoogle Scholar
Mycroft, R. and Naia, T., Unavoidable trees in tournaments, Random Structures & Algorithms 53 (2018), no. 2, 352–385.CrossRefGoogle Scholar
Naia, T., Large structures in dense directed graphs, PhD thesis, University of Birmingham, 2018.Google Scholar
Naia, T., Trees contained in every orientation of a graph, Electronic Journal of Combinatorics 29 (2022), no. 2, P2.26.CrossRefGoogle Scholar
Rédei, L., Ein kombinatorischer Satz, Acta Universitatis Szegediensis 7 (1934), 39–43.Google Scholar
Reed, B. and Stein, M., Spanning trees in graphs of high minimum degree with a universal vertex I: An asymptotic result, Journal of Graph Theory 102 (2023), no. 4, 737–783.CrossRefGoogle Scholar
Reed, B. and Stein, M., Spanning trees in graphs of high minimum degree with a universal vertex II: A tight result, Journal of Graph Theory 102 (2023), no. 4, 797–821.CrossRefGoogle Scholar
Reid, K. and Wormald, N., Embedding oriented n-trees in tournaments, Studis Scientiarum Mathematicarum Hungarica 18 (1983), 377–387.Google Scholar
Rosenfeld, M., Antidirected Hamiltonian paths in tournaments, Journal of Combinatorial Theory, Series B 12 (1971), 93–99.CrossRefGoogle Scholar
Saks, M.E. and Sós, V. T., On avoidable subgraphs of tournaments, Colloquia Mathematica Societatis Janós Bolyai (L. Lovász A. Hajnal and V.T.Sós, eds.), vol. 2, North Holland, 1981, pp. 663–674.Google Scholar
Stein, M., Tree containment and degree conditions, Discrete Mathematics and Applications (Raigorodskii, A. M. and Rassias, M. Th., eds.), Springer International Publishing, Cham, 2020, pp. 459–486.CrossRefGoogle Scholar
Stein, M. and Zárate-Guerén, C., Antidirected subgraphs of oriented graphs, Preprint 2022, arXiv 2212.00769.Google Scholar
Straight, H. J., The existence of certain type of semi-walks in tournaments, Proceedings, XI Southeastern Conference on Combinatorics, Graph Theory and Computing, Cong. Numer. 29 (1980), 901–908.Google Scholar
Sullivan, B. D., A summary of results and problems related to the Caccetta-Ḧaggkvist conjecture, Technical Report 2006-13. American Institute of Mathematics, Palo Alto, CA, 2006.Google Scholar
Szele, T., Kombinatorikai vizsgalatok az iranyitott teljes graffal kapcsolatban, Matematikai és fizikai lapok 50 (1943), no. 1943, 223–256.Google Scholar
Thomason, A., Antidirected Hamiltonian paths in tournaments, Transactions of the American Mathematical Society 296 (1986), 167–180.CrossRefGoogle Scholar
Thomassen, C., Hamiltonian connected tournaments, Journal of Combinatorial Theory, Series B 28 (1980), 142–163.CrossRefGoogle Scholar
Thomassen, C., Edge-disjoint hamiltonian paths and cycles in tournaments, Proceedings of the London Mathematical Society 45 (1982), 151–168.CrossRefGoogle Scholar
Thomassen, C., Connectivity in tournaments, Graph Theory and Combinatorics (Bollobás, Bela, ed.), Acad. Press, 1984, pp. 305–313.Google Scholar
Tian, Y., Lai, H.-J., Xu, L., and Meng, J., Nonseparating trees in 2-connected graphs and oriented trees in strongly connected digraphs, Discrete Mathematics 342 (2019), no. 2, 344–351.CrossRefGoogle Scholar
Wormald, N. C., Subtrees of large tournaments, Combinatorial Mathematics X ( Reynolds Antoine Casse, Louis, ed.), Springer Berlin Heidelberg, 1983, pp. 417–419.CrossRefGoogle Scholar
Zhang, C.-Q., Some results on tournaments, Journal of Qufu Teachers College 1 (1985), 51–53.Google Scholar

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