Abstract
How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given r-edge-coloured graph G? These problems were introduced in the 1960s and were intensively studied by various researchers over the last 50 years. In this paper, we establish a connection between this problem and the following natural Helly-type question in hypergraphs. Roughly speaking, this question asks for the maximum number of vertices needed to cover all the edges of a hypergraph H if it is known that any collection of a few edges of H has a small cover. We obtain quite accurate bounds for the hypergraph problem and use them to give some unexpected answers to several questions about covering graphs by monochromatic trees raised and studied by Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Cirão, Letzter and Sahasrabudhe.
Similar content being viewed by others
References
A. Abu-Khazneh, J. Barát, A. Pokrovskiy and T. Szabó: A family of extremal hypergraphs for Ryser’s conjecture, J. Combin. Theory Ser. A 161 (2019), 164–177.
A. Abu-Khazneh and A. Pokrovskiy: Intersecting extremal constructions in Ryser’s conjecture for r-partite hypergraphs, J. Combin. Math. Combin. Comput. 103 (2017), 81–104.
R. Aharoni: Ryser’s conjecture for tripartite 3-graphs, Combinatorica 21 (2001), 1–4.
R. Aharoni, J. Barát and I. M. Wanless: Multipartite hypergraphs achieving equality in Ryser’s conjecture, Graphs Combin. 32 (2016), 1–15.
N. Alon: An extremal problem for sets with applications to graph theory, J. Combin. Theory Ser. A 40 (1985), 82–89.
N. Alon and J. H. Spencer: The probabilistic method, fourth ed., Wiley Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2016.
D. Bal and L. DeBiasio: Partitioning random graphs into monochromatic components, Electron. J. Combin. 24 (2017), P1.18.
P. Bennett, L. DeBiasio, A. Dudek and S. English: Large monochromatic components and long monochromatic cycles in random hypergraphs, European J. Combin. 76 (2019), 123–137.
B. Bollobás: On generalized graphs, Acta Math. Acad. Sci. Hungar. 16 (1965), 447–452.
B. Bollobás: Combinatorics: set systems, hypergraphs, families of vectors, and combinatorial probability, Cambridge University Press, 1986.
D. Conlon and W. T. Gowers: Combinatorial theorems in sparse random sets, Ann. of Math. 2 (2016), 367–454.
L. Danzer, B. Grünbaum and V. Klee: Helly’s theorem and its relatives, in: Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 101–180, 1963.
P. Erdős, D. G. Fon-Der-Flaass, A. V. Kostochka and Zs. Tuza: Small transversals in uniform hypergraphs, Siberian Advances in Math. 2 (1992), 82–88.
P. Erdős, A. Gyárfás and L. Pyber: Vertex coverings by monochromatic cycles and trees, J. Combin. Theory Ser. B 51 (1991), 90–95.
P. Erdős, A. Hajnal and J. W. Moon: A problem in graph theory, Amer. Math. Monthly 71 (1964), 1107–1110.
P. Erdős, A. Hajnal and Zs. Tuza: Local constraints ensuring small representing sets, J. Combin. Theory, Ser. A 58 (1991), 78–84.
D. G. Fon-Der-Flaass, A. V. Kostochka and D. R. Woodall: Transversals in uniform hypergraphs with property (7,2), Discrete Math. 207 (1999), 277–284.
N. Francetić, S. Herke, B. D. McKay and I. M. Wanless: On Ryser’s conjecture for linear intersecting multipartite hypergraphs, European J. Combin. 61 (2017), 91–105.
Z. Füredi: Matchings and covers in hypergraphs, Graphs Combin. 4 (1988), 115–206.
L. Gerencsér and A. Gyárfás: On Ramsey-type problems, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 10 (1967), 167–170.
A. Girão, S. Letzter and J. Sahasrabudhe: Partitioning a graph into monochromatic connected subgraphs, J. Graph Theory 91 (2019), 353–364.
A. Gyárfás: Covering complete graphs by monochromatic paths, in: Irregularities of partitions (Fertőd, 1986), Algorithms Combin. Study Res. Texts, vol. 8, Springer, Berlin, 89–91, 1989.
A. Gyárfás: Vertex covers by monochromatic pieces-a survey of results and problems, Discrete Mathematics 339 (2016), 1970–1977.
A. Gyárfás, M. Ruszinkó, G. N. Sárközy and E. Szemerédi: An improved bound for the monochromatic cycle partition number, J. Combin. Theory Ser. B 96 (2006), 855–873.
P. E. Haxell and A. D. Scott: On Ryser’s conjecture, Electron. J. Combin. 19 (2012), P23.
P. E. Haxell and A. D. Scott: A note on intersecting hypergraphs with large cover number, Electron. J. Combin. 24 (2017), P3.26.
E. Helly: Über mengen konvexer körper mit gemeinschaftlichen punkte., Jahresbericht der Deutschen Mathematiker-Vereinigung 32 (1923), 175–176.
J. R. Henderson: Permutation decomposition of (0, 1)-matrices and decomposition transversals, Ph.D. thesis, California Institute of Technology, 1971.
Y. Kohayakawa, G. O. Mota and M. Schacht: Monochromatic trees in random graphs, in: Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 1–18, 2018.
D. Korándi, R. Lang, S. Letzter and A. Pokrovskiy: Minimum degree conditions for monochromatic cycle partitioning, J. Combin. Theory Ser. B 146 (2021), 96–123.
D. Korandi, F. Mousset, R. Nenadov, N. Skoric and B. Sudakov: Monochromatic cycle covers in random graphs, Random Structures Algorithms 53 (2018), 667–691.
A. Kostochka: Transversals in uniform hypergraphs with property (p, 2), Combinatorica 22 (2002), 275–285.
R. Lang and A. Lo: Monochromatic cycle partitions in random graphs, Comb. Prob. Comput. 30 (2021), 136–152.
J. Lehel: τ-critical hypergraphs and the Helly property, in: Combinatorial mathematics (Marseille-Luminy, 1981), North-Holland Math. Stud., vol. 75, North-Holland, Amsterdam, 413–418, 1983.
L. Lovász: On minimax theorems of combinatorics, Mat. Lapok 26 (1975), 209–264 (1978).
L. Lovász: On the ratio of optimal integral and fractional covers, Discrete Math. 13 (1975), 383–390.
L. Lovász: Combinatorial problems and exercises, second ed., North-Holland Publishing Co., Amsterdam, 1993.
G. Moshkovitz and A. Shapira: Exact bounds for some hypergraph saturation problems, J. Combin. Theory Ser. B 111 (2015), 242–248.
A. Pokrovskiy: Partitioning edge-coloured complete graphs into monochromatic cycles and paths, J. Combin. Theory Ser. B 106 (2014), 70–97.
M. Schacht: Extremal results for random discrete structures, Ann. of Math. 2 (2016), 333–365.
Z. Tuza: On the order of vertex sets meeting all edges of a 3-partite hypergraph, in: Proceedings of the First Catania International Combinatorial Conference on Graphs, Steiner Systems, and their Applications, Vol. 1 (Catania, 1986), vol. 24 A, 59–63, 1987.
Acknowledgements
We thank Louis DeBiasio and Tuan Tran for helpful comments on an earlier version of this manuscript. We would like to thank the anonymous referees for their careful reading of the paper and many useful suggestions. In particular, we are grateful for a suggestion on how to rewrite the proof of Lemma 4.3 to make it easier to follow.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by SNSF grants 200020-162884 and 200021-175977.
Research supported in part by SNSF grant 200021_196965.
Rights and permissions
About this article
Cite this article
Bucić, M., Korándi, D. & Sudakov, B. Covering Graphs by Monochromatic Trees and Helly-Type Results for Hypergraphs. Combinatorica 41, 319–352 (2021). https://doi.org/10.1007/s00493-020-4292-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-020-4292-9