Edge-disjoint rainbow spanning trees in complete graphs

Carraher, James M.; Hartke, Stephen G.; Horn, Paul

doi:10.1016/j.ejc.2016.04.003

Cited by 32 publications

(32 citation statements)

References 11 publications

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“…These conjectures attracted a lot of attention from various researchers (see, for example, ) who showed how to find several disjoint spanning rainbow trees. The best known results for these problem guarantee the existence of

ε n

edge‐disjoint rainbow trees (see for Conjecture and for Conjecture ).…”

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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ε n

edge‐disjoint rainbow trees (see for Conjecture and for Conjecture ).…”

Section: Introductionmentioning

confidence: 99%

Decompositions into spanning rainbow structures

Montgomery

Pokrovskiy

Sudakov

2019

Proc. London Math. Soc.

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show abstract

“…The most naive way to attempt this would be to select each edge to be in independently with probability 1 ; thus each graph would have the law of the Erdős-Rényi ( , 1 ) random graph. This is precisely the approach of Carraher, Hartke, and the author in [4]. There is a natural bottleneck to this approach, however.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, they conjectured that every proper edge coloring (not necessarily arising from a one‐factorization) of

K_{n}

can have its edge set partitioned into rainbow spanning trees. Finding three such trees remained, until recently, the best known result towards either conjecture, but recently Carraher, Hartke, and the author proved that every coloring of

K_{n}

(not necessarily proper) where every color is used as most

n / 2

times contains

ε \frac{n}{log n}

edge disjoint rainbow spanning trees (for some absolute constant

ε > 0

). This work also improved earlier results of Akbari and Alipour from which showed that if

K_{n}

is edge colored so that no color appears more than

n / 2

times contains at least two rainbow spanning trees.…”

Section: Introductionmentioning

confidence: 99%

“…This is precisely the approach of Carraher, Hartke, and the author in . There is a natural bottleneck to this approach, however.…”

Section: Introductionmentioning

confidence: 99%

“…disconnected if

p < false(1 - ε false) \frac{log n}{n}

. This roadblock was the primary reason why, in , it was only possible to assert the existence of

ε \frac{n}{log n}

rainbow spanning trees in graphs where each color class has size at most

n / 2

.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Rainbow spanning trees in complete graphs colored by one‐factorizations

Horn

2017

Journal of Graph Theory

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Brualdi and Hollingsworth conjectured that, for even n, in a proper edge coloring of Kn using precisely n−1 colors, the edge set can be partitioned into n/2 spanning trees which are rainbow (and hence, precisely one edge from each color class is in each spanning tree). They proved that there always are two edge disjoint rainbow spanning trees. Krussel, Marshall, and Verrall improved this to three edge disjoint rainbow spanning trees. Recently, Carraher, Hartke and the author proved a theorem improving this to εnlogn rainbow spanning trees, even when more general edge colorings of Kn are considered. In this article, we show that if Kn is properly edge colored with n−1 colors, a positive fraction of the edges can be covered by edge disjoint rainbow spanning trees.

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Anti‐Ramsey Problems for t Edge‐Disjoint Rainbow Spanning Subgraphs: Cycles, Matchings, or Trees

Jahanbekam

West

2015

Journal of Graph Theory

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We seek the maximum number of colors in an edge-coloring of the complete graph K n not having t edge-disjoint rainbow spanning subgraphs of specified types. Let c(n, t), m(n, t), and r(n, t) denote the answers when the spanning subgraphs are cycles, matchings, or trees, respectively. We prove c(n, t) = n−1 2 + t for n ≥ 8t − 1 and m(n, t) = n−2 2 + t for n > 4t + 10. We prove r(n, t) = n−2 2 + t for n > 2t + 6t − and r(n, t) = n 2 − t for n = 2t. We also provide constructions for the more general problem in which colorings are restricted so that colors do not appear on more than q edges at a vertex.

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Edge-disjoint rainbow spanning trees in complete graphs

Cited by 32 publications

References 11 publications

Decompositions into spanning rainbow structures

Decompositions into spanning rainbow structures

Rainbow spanning trees in complete graphs colored by one‐factorizations

Anti‐Ramsey Problems for t Edge‐Disjoint Rainbow Spanning Subgraphs: Cycles, Matchings, or Trees

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