# 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}
}
• Published 2020
• Mathematics, Computer Science
• Eur. J. Comb.
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 $n-O… Expand #### Figures and Topics from this paper Embedding rainbow trees with applications to graph labelling and decomposition • Mathematics • 2018 A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on LatinExpand Linearly many rainbow trees in properly edge-coloured complete graphs • Computer Science, Mathematics • J. Comb. Theory, Ser. B • 2018 It is shown that in every proper edge-colouring of Kn there are 10^{−6}n edge-disjoint spanning isomorphic rainbow trees, giving further improvement on the Brualdi-Hollingsworth Conjecture. Expand Hamilton transversals in random Latin squares • Mathematics • 2021 Gyárfás and Sárközy conjectured that every n × n Latin square has a ‘cycle-free’ partial transversal of size n−2. We confirm this conjecture in a strong sense for almost all Latin squares, by showingExpand A rainbow blow‐up lemma • Mathematics • 2018 We prove a rainbow version of the blow-up lemma of Koml\'os, S\'ark\"ozy and Szemer\'edi for$\mu n$-bounded edge colourings. This enables the systematic study of rainbow embeddings of bounded degreeExpand Repeated Patterns in Proper Colorings • Mathematics • SIAM Journal on Discrete Mathematics • 2021 For a fixed graph H, what is the smallest number of colours C such that there is a proper edge-colouring of the complete graph K_n with C colours containing no two vertex-disjoint colour-isomorphicExpand #### References SHOWING 1-10 OF 42 REFERENCES Long rainbow cycles and Hamiltonian cycles using many colors in properly edge-colored complete graphs • Mathematics, Computer Science • Eur. J. Comb. • 2019 It is proved that every properly colored complete graph has a Hamilton cycle in which at least$n - O((\log n)^2)$different colors appear. Expand Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles • Mathematics • 2016 A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph Kn has a rainbowExpand Properly coloured copies and rainbow copies of large graphs with small maximum degree • Computer Science, Mathematics • Random Struct. Algorithms • 2012 The Lovasz local lemma is used to show the following two results about colourings χ of the edges of the complete graph Kn: if for each vertex v of Kn the colouring χ assigns each colour to at most (n - 2)/(22.4Δ2) edges emanating from v, then there is a copy of G in Kn which is properly edge-coloured by χ. Expand Properly colored and rainbow copies of graphs with few cherries • Mathematics • 2017 Abstract Let G be an n-vertex graph that contains linearly many cherries (i.e., paths on 3 vertices), and let c be a coloring of the edges of the complete graph K n such that at each vertex everyExpand Hamilton circuits with many colours in properly edge-coloured complete graphs. We prove that a properly edge-coloured complete graph K n has a Hamilton circuit with edges of at least n−√2n distinct colours. This is proved with a method inspired by work on long partialExpand Rainbow matchings in Dirac bipartite graphs • Mathematics, Computer Science • Random Struct. Algorithms • 2019 We show the existence of rainbow perfect matchings in$\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
• Computer Science
• Random Struct. Algorithms
• 1997
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
• Computer Science, Mathematics
• J. Comb. Theory, Ser. B
• 2005
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
• Computer Science, Mathematics
• Discret. Math.
• 2014
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