Long directed rainbow cycles and rainbow spanning trees

@article{Benzing2020LongDR,
  title={Long directed rainbow cycles and rainbow spanning trees},
  author={Frederik Benzing and Alexey Pokrovskiy and Benny Sudakov},
  journal={Eur. J. Comb.},
  year={2020},
  volume={88},
  pages={103102}
}
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. The problem of finding rainbow subgraphs goes back to the work of Euler on transversals in Latin squares and was extensively studied since then. In this paper we consider two related questions concerning rainbow subgraphs of complete, edge-coloured graphs and digraphs. In the first part, we show that every properly edge-coloured complete directed graph contains a directed rainbow cycle of length $n-O… Expand
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References

SHOWING 1-10 OF 42 REFERENCES
Long rainbow cycles and Hamiltonian cycles using many colors in properly edge-colored complete graphs
TLDR
It is proved that every properly colored complete graph has a Hamilton cycle in which at least $n - O((\log n)^2)$ different colors appear. Expand
Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph Kn has a rainbowExpand
Properly coloured copies and rainbow copies of large graphs with small maximum degree
TLDR
The Lovasz local lemma is used to show the following two results about colourings χ of the edges of the complete graph Kn: if for each vertex v of Kn the colouring χ assigns each colour to at most (n - 2)/(22.4Δ2) edges emanating from v, then there is a copy of G in Kn which is properly edge-coloured by χ. Expand
Properly colored and rainbow copies of graphs with few cherries
Abstract Let G be an n-vertex graph that contains linearly many cherries (i.e., paths on 3 vertices), and let c be a coloring of the edges of the complete graph K n such that at each vertex everyExpand
Hamilton circuits with many colours in properly edge-coloured complete graphs.
We prove that a properly edge-coloured complete graph K n has a Hamilton circuit with edges of at least n−√2n distinct colours. This is proved with a method inspired by work on long partialExpand
Rainbow matchings in Dirac bipartite graphs
We show the existence of rainbow perfect matchings in $\mu n$-bounded edge colourings of Dirac bipartite graphs, for a sufficiently small $\mu>0$. As an application of our results, we obtain severalExpand
Properly colored Hamilton cycles in edge-colored complete graphs
It is shown that for every 2 > 0 and n > n0(2), any complete graph K on n vertices whose edges are colored so that no vertex is incident with more than (1 − 1 √ 2 − 2)n edges of the same color,Expand
On packing Hamilton cycles in ?-regular graphs
TLDR
It is proved that if α ≫ e > 0 are not too small, then every (α, e)-regular graph on n vertices contains a family of (α/2 - O(e))n edge-disjoint Hamilton cycles, and it is derived that for every constant 0 < p < 1, with high probability in the random graph G(n, p), almost all edges can be packed into edge-redundant Hamilton cycles. Expand
Rainbow matchings and cycle-free partial transversals of Latin squares
TLDR
It is shown that properly edge-colored graphs G with |V (G)| ≥ 4δ(G) − 3 have rainbow matchings of size δ( G); this gives the best known bound for a recent question of Wang. Expand
Problems and Results on 3-chromatic Hypergraphs and Some Related Questions
A hypergraphi is a collection of sets. This paper deals with finite hy-pergraphs only. The sets in the hypergraph are called edges, the elements of these edges are points. The degree of a point isExpand
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