Long rainbow cycles and Hamiltonian cycles using many colors in properly edge-colored complete graphs

Balogh, József; Molla, Theodore

doi:10.1016/j.ejc.2019.02.008

Cited by 15 publications

(25 citation statements)

References 15 publications

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“…Despite considerable research, even the existence of an almost spanning path or cycle was a major open question until recently. Alon, Pokrovskiy, and Sudakov were able to settle this by showing that any properly edge‐colored

K_{n}

contains a rainbow cycle of length

n - O false(n^{3 false/ 4} false)

(the error term was subsequently improved in ). A corollary of our second main theorem (Theorem ) states that we can arrive at a stronger conclusion (i.e., we obtain many edge‐disjoint almost‐spanning rainbow cycles) under much weaker assumptions (though with a larger error term).…”

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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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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K_{n}

contains a rainbow cycle of length

n - O false(n^{3 false/ 4} false)

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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show abstract

“…Call a vertex b ∈ Kn 'bad' if ov(b) > γn. Observe that by the bound given above, we can have at most δn bad vertices since δn · γn ≥ 2αn 2 by (6). We now define random variables X1, .…”

Section: Proof Of Lemma 31mentioning

confidence: 99%

“…By this coupling argument and the classic Chernoff bound (see e.g. [3]) we can use α2 ≥ 2γ from (6) to obtain:…”

Section: Proof Of Lemma 31mentioning

confidence: 99%

Long directed rainbow cycles and rainbow spanning trees

Benzing

Pokrovskiy

Sudakov

2020

European Journal of Combinatorics

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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(n 4/5 ). This is motivated by an old problem of Hahn and improves a result of Gyarfas and Sarkozy. In the second part, we show that any tree T on n vertices with maximum degree ∆T ≤ βn/ log n has a rainbow embedding into a properly edge-coloured Kn provided that every colour appears at most αn times and α, β are sufficiently small constants.

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“…A wellknown conjecture in this context due to Andersen [And89] asserts that every properly edge-coloured complete graph on n vertices has a rainbow path of length n − 2, i.e., a path that has distinct colors along each of its edges. Progress towards resolving this conjecture was recently made by Alon, Pokrovskiy and Sudakov [APS16], and Balogh and Molla [BM17].…”

Section: Introductionmentioning

confidence: 99%

Rainbow Cycles in Flip Graphs

Felsner¹,

Kleist²,

Mütze³

et al. 2020

SIAM J. Discrete Math.

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The flip graph of triangulations has as vertices all triangulations of a convex n-gon, and an edge between any two triangulations that differ in exactly one edge. An r-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly r times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of r-rainbow cycles for three different flip graphs on classes of geometric objects: the aforementioned flip graph of triangulations of a convex n-gon, the flip graph of plane trees on an arbitrary set of n points, and the flip graph of non-crossing perfect matchings on a set of n points in convex position. In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of {1, 2, . . . , n} and the flip graph of k-element subsets of {1, 2, . . . , n}. In each of the five settings, we prove the existence and nonexistence of rainbow cycles for different values of r, n and k.

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Long rainbow cycles and Hamiltonian cycles using many colors in properly edge-colored complete graphs

Cited by 15 publications

References 15 publications

Rainbow structures in locally bounded colorings of graphs

Rainbow structures in locally bounded colorings of graphs

Long directed rainbow cycles and rainbow spanning trees

Rainbow Cycles in Flip Graphs

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