Powers of paths in tournaments

Draganić, Nemanja; Dross, François; Fox, Jacob; Girão, António; Havet, Frédéric; Korándi, Dániel; Lochet, William; Munhá, Correia, David; Scott, Alex; Sudakov, Benny

doi:10.1017/s0963548321000067

Cited by 8 publications

(13 citation statements)

References 5 publications

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“…He showed that any tournament on n vertices must contain a power of a directed path on at least n 0.295 vertices. Confirming a conjecture of Yuster, Draganić et al [5] showed that for every k there always exists a k-th power of a path of linear order. Theorem 1.1 (Draganić et al [5]).…”

Section: Introductionmentioning

confidence: 71%

Powers of paths and cycles in tournaments

Girão,

Korándi,

Scott

2021

Preprint

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We show that for every positive integer k, any tournament can be partitioned into at most 2 ck k-th powers of paths. This result is tight up to the exponential constant. Moreover, we prove that for every ε > 0 and every integer k, any tournament on n ≥ ε −Ck vertices which is ε-far from being transitive contains the k-th power of a cycle of length Ω(εn); both bounds are tight up to the implied constants.

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Section: Introductionmentioning

confidence: 71%

Powers of paths and cycles in tournaments

Girão,

Korándi,

Scott

2021

Preprint

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“…Previous work showing linear upper bounds on − → r 1 (H) when H is an oriented tree (e.g. [14]) or an acyclic digraph of bounded bandwidth [12] all depend on embedding H into a tournament T in some iterative manner according to its median ordering. We were not able to reproduce these upper bounds using greedy embedding arguments, which seem primarily suited for embedding digraphs H without long paths.…”

Section: − → G (N D) Andmentioning

confidence: 99%

“…Using the same argument, one can obtain a bound of − → r 1 (H) ≤ n O ℓ (1) for any n-vertex acyclic digraph of bandwidth at most ℓ, using the fact that the same binary-representation prefix coloring have dyadic complexity at most n O(log ℓ) . However, since the Ramsey number of bounded-bandwidth acyclic digraphs is known to be linear [12], we omit the proof of this weaker result.…”

Section: Upper Bounds For Random Digraphsmentioning

confidence: 99%

“…Yuster [35] recently initiated the study of the special case of Problem 1.1 when H = P ℓ n is the ℓ-th power of a directed path P n , which is the digraph on vertex set [n] whose edges are the ordered pairs (i, j) satisfying 1 ≤ j − i ≤ ℓ. This case was recently settled by Draganić et al [12], who showed that − → r 1 (P ℓ n ) = O ℓ (n). Letting the bandwidth of an acyclic digraph H on n vertices be the minimum ℓ such that P ℓ n contains a copy of H, this aforementioned result implies that − → r 1 (H) = O ℓ (n) if H has n vertices and bandwidth ℓ.…”

Section: Introductionmentioning

confidence: 99%

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Ramsey numbers of sparse digraphs

Fox¹,

He²,

Wigderson³

2021

Preprint

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Burr and Erdős in 1975 conjectured, and Chvátal, Rödl, Szemerédi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed analogue of the Burr-Erdős conjecture, answering a question of Bucić, Letzter, and Sudakov. If H is an acyclic digraph, the oriented Ramsey number of H, denoted − → r1 (H), is the least N such that every tournament on N vertices contains a copy of H. We show that for any ∆ ≥ 2 and any sufficiently large n, there exists an acyclic digraph H with n vertices and maximum degree ∆ such thatThis proves that − → r1 (H) is not always linear in the number of vertices for bounded-degree H. On the other hand, we show that − → r1 (H) is nearly linear in the number of vertices for typical bounded-degree acyclic digraphs H, and obtain linear or nearly linear bounds for several natural families of bounded-degree acyclic digraphs.For multiple colors, we prove a quasipolynomial upper boundfor all boundeddegree acyclic digraphs H on n vertices, where − → r k (H) is the least N such that every k-edge-colored tournament on N vertices contains a monochromatic copy of H. For k ≥ 2 and n ≥ 4, we exhibit an acyclic digraph H with n vertices and maximum degree 3 such that − → r k (H) ≥ n Ω(log n/ log log n) , showing that these Ramsey numbers can grow faster than any polynomial in the number of vertices.

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“…In order to find this partition into chains we use the recent result in [6], which shows that one can always find the k-th power of a long path in a tournament. We apply this iteratively, until a certain constant number of vertices is left.…”

Section: Cut-dense Tournamentsmentioning

confidence: 99%

Tight bound for powers of Hamilton cycles in tournaments

Draganić,

Correia,

Sudakov

2021

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A basic result in graph theory says that any n-vertex tournament with in-and out-degrees larger than n−2 4 contains a Hamilton cycle, and this is tight. In 1990, Bollobás and Häggkvist significantly extended this by showing that for any fixed k and ε > 0, and sufficiently large n, all tournaments with degrees at least n 4 + εn contain the k-th power of a Hamilton cycle. Given this, it is natural to ask for a more accurate error term in the degree condition, and also how large n should be in the Bollobás-Häggkvist theorem. We essentially resolve both questions. First, we show that if the degrees are at least n 4 + cn 1−1/⌈k/2⌉ for some constant c = c(k), then the tournament contains the k-th power of a Hamilton cycle. In particular, in order to guarantee the square of a Hamilton cycle, one only needs a constant additive term. We also present a construction which, modulo a well-known conjecture on Turán numbers for complete bipartite graphs, shows that the error term must be of order at least n 1−1/⌈(k−1)/2⌉ , which matches our upper bound for all even k. For odd k, we believe that the lower bound can be improved. Indeed, we show that for k = 3, there are tournaments with degrees n 4 +Ω(n 1/5 ) and no cube of a Hamilton cycle. In addition, our results imply that the Bollobás-Häggkvist theorem already holds for n = ε −Θ(k) , which is best possible.

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Powers of paths in tournaments

Cited by 8 publications

References 5 publications

Powers of paths and cycles in tournaments

Powers of paths and cycles in tournaments

Ramsey numbers of sparse digraphs

Tight bound for powers of Hamilton cycles in tournaments

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