A counterexample to Stein’s Equi-$n$-square Conjecture

Pokrovskiy, Alexey; Sudakov, Benny

doi:10.1090/proc/14220

Cited by 17 publications

(22 citation statements)

References 14 publications

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“…The best known positive result in this direction is due to Aharoni, Berger, Kotlar, and Ziv , who, using a topological approach, showed that any such array has a partial transversal of size at least

2 n false/ 3

. On the other hand, Pokrovskiy and Sudakov recently disproved Stein's conjecture: in fact, they showed that there are such arrays with largest transversal of size

n - normalΩ false(normallog n false)

.…”

Section: Introduction and Our Resultsmentioning

confidence: 99%

“…As noted above, even for proper colorings, the corollary is best possible up to the value of the final error term, that is, we cannot guarantee a Hamilton cycle. Moreover, a slight modification of the construction of Pokrovskiy and Sudakov , shows that there are locally

o false(n false)

‐bounded,

false(n - 1 false) false/ 2

‐bounded edge‐colorings of

K_{n}

with no rainbow cycle longer than

n - normalΩ false(normallog n false)

. For a more detailed discussion, see Section 5.…”

Section: Introduction and Our Resultsmentioning

confidence: 99%

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Rainbow structures in locally bounded colorings of graphs

Kim

Kühn

Kupavskii

et al. 2020

Random Struct Algorithms

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We study approximate decompositions of edge-colored quasirandom graphs into rainbow spanning structures: an edge-coloring of a graph is locally -bounded if every vertex is incident to at most edges of each color, and is (globally) g-bounded if every color appears at most g times. Our results imply the existence of: (1) approximate decompositions of properly edge-colored K n into rainbow almost-spanning cycles;(2) approximate decompositions of edge-colored K n into rainbow Hamilton cycles, provided that the coloring is (1 − o(1)) n 2 -bounded and locally o ( n log 4 n ) -bounded; and (3) an approximate decomposition into full transversals of any n × n array, provided each symbol appears (1 − o(1))n times in total and only o ( n log 2 n ) times in each row or column. Apart from the logarithmic factors, these bounds are essentially best possible. We also prove analogues for rainbow F-factors, where F is any fixed graph. Both (1) and (2) imply approximate versions of the Brualdi-Hollingsworth conjecture on decompositions into rainbow spanning trees.

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2 n false/ 3

. On the other hand, Pokrovskiy and Sudakov recently disproved Stein's conjecture: in fact, they showed that there are such arrays with largest transversal of size

n - normalΩ false(normallog n false)

.…”

Section: Introduction and Our Resultsmentioning

confidence: 99%

o false(n false)

‐bounded,

false(n - 1 false) false/ 2

‐bounded edge‐colorings of

K_{n}

with no rainbow cycle longer than

n - normalΩ false(normallog n false)

. For a more detailed discussion, see Section 5.…”

Section: Introduction and Our Resultsmentioning

confidence: 99%

Rainbow structures in locally bounded colorings of graphs

Kim

Kühn

Kupavskii

et al. 2020

Random Struct Algorithms

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“…Hahn conjectured even more that such a path can be found in any (not necessarily properly) coloured complete graph with at most

n / 2 - 1

edges of each colour (see ). Hahn's conjecture was recently disproved by the second and third author , who showed that without the ‘proper colouring’ assumption the graph might not have rainbow paths longer than

n - Ω (log n)

. Thus, it makes sense to restrict ourselves to colourings which are proper.…”

Section: Introductionmentioning

confidence: 99%

Decompositions into spanning rainbow structures

Montgomery

Pokrovskiy

Sudakov

2019

Proc. London Math. Soc.

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A subgraph of an edge‐coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than 200 years to the work of Euler on Latin squares and has been the focus of extensive research ever since. Euler posed a problem equivalent to finding properly n‐edge‐coloured complete bipartite graphs Kn,n which can be decomposed into rainbow perfect matchings. While there are proper edge‐colourings of Kn,n without even a single rainbow perfect matching, the theme of this paper is to show that with some very weak additional constraints one can find many disjoint rainbow perfect matchings. In particular, we prove that if some fraction of the colour classes have at most (1−o(1))n edges, then one can nearly‐decompose the edges of Kn,n into edge‐disjoint perfect rainbow matchings. As an application of this, we establish in a very strong form a conjecture of Akbari and Alipour and asymptotically prove a conjecture of Barat and Nagy. Both these conjectures concern rainbow perfect matchings in edge‐colourings of Kn,n with quadratically many colours. The above result also has implications to some conjectures of Snevily about subsquares of multiplication tables of groups. Finally, using our techniques, we also prove a number of results on near‐decompositions of graphs into other rainbow structures like Hamiltonian cycles and spanning trees. Most notably, we prove that any properly coloured complete graph can be nearly‐decomposed into spanning rainbow trees. This asymptotically proves the Brualdi–Hollingsworth and Kaneko–Kano–Suzuki conjectures which predict that a perfect decomposition should exist under the same assumptions.

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“…To focus the ideas, one well-known longstanding open problem is Ryser's Conjecture [97] that every Latin square of odd order has a transversal. (A generalised form of this conjecture by Stein [100] was recently disproved by Pokrovskiy and Sudakov [87].) To see the connection with hypergraph matchings, we associate to any Latin square a tripartite 3-graph in which the parts correspond to rows, columns and symbols, and each cell of the square corresponds to an edge consisting of its own row, column and symbol.…”

Section: Discussionmentioning

confidence: 99%