331 research outputs found
Linearly many rainbow trees in properly edge-coloured complete graphs
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different
colours. The study of rainbow decompositions has a long history, going back to the work of
Euler on Latin squares. In this paper we discuss three problems about decomposing complete
graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine’s Conjecture, and
the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of Kn there
are 10−6n edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the
best known bounds on all these conjectures. Using our method we also show that every properly
(n − 1)-edge-coloured Kn has n/9 − 6 edge-disjoint rainbow trees, giving further improvement on
the Brualdi-Hollingsworth Conjectur
Linearly many rainbow trees in properly edge-coloured complete graphs
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different
colours. The study of rainbow decompositions has a long history, going back to the work of
Euler on Latin squares. In this paper we discuss three problems about decomposing complete
graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine’s Conjecture, and
the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of Kn there
are 10−6n edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the
best known bounds on all these conjectures. Using our method we also show that every properly
(n − 1)-edge-coloured Kn has n/9 − 6 edge-disjoint rainbow trees, giving further improvement on
the Brualdi-Hollingsworth Conjecture
Linearly many rainbow trees in properly edge-coloured complete graphs
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. In this paper we discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine's Conjecture, and the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of Kn there are 10^{−6}n edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the best known bounds on all these conjectures. Using our method we also show that every properly (n−1)-edge-coloured Kn has n/9 edge-disjoint rainbow trees, giving further improvement on the Brualdi-Hollingsworth Conjecture
Properly coloured copies and rainbow copies of large graphs with small maximum degree
Let G be a graph on n vertices with maximum degree D. We use the Lov\'asz
local lemma to show the following two results about colourings c of the edges
of the complete graph K_n. If for each vertex v of K_n the colouring c assigns
each colour to at most (n-2)/22.4D^2 edges emanating from v, then there is a
copy of G in K_n which is properly edge-coloured by c. This improves on a
result of Alon, Jiang, Miller, and Pritikin [Random Struct. Algorithms 23(4),
409-433, 2003]. On the other hand, if c assigns each colour to at most n/51D^2
edges of K_n, then there is a copy of G in K_n such that each edge of G
receives a different colour from c. This proves a conjecture of Frieze and
Krivelevich [Electron. J. Comb. 15(1), R59, 2008]. Our proofs rely on a
framework developed by Lu and Sz\'ekely [Electron. J. Comb. 14(1), R63, 2007]
for applying the local lemma to random injections. In order to improve the
constants in our results we use a version of the local lemma due to Bissacot,
Fern\'andez, Procacci, and Scoppola [preprint, arXiv:0910.1824].Comment: 9 page
Edge-disjoint rainbow trees in properly coloured complete graphs
A subgraph of an edge-coloured complete graph is called rainbow if all its edges
have different colours. The study of rainbow decompositions has a long history,
going back to the work of Euler on Latin squares. We discuss three problems
about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth
Conjecture, Constantine’s Conjecture, and the Kaneko-Kano-Suzuki Conjecture.
The main result which we discuss is that in every proper edge-colouring of Kn there
are 10−6n edge-disjoint isomorphic spanning rainbow trees. This simultaneously
improves the best known bounds on all these conjectures. Using our method it is also
possible to show that every properly (n−1)-edge-coloured Kn has n/9 edge-disjoint
spanning rainbow trees, giving a further improvement on the Brualdi-Hollingsworth
Conjectur
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