Cycles In Tournaments Research Articles

Tight bounds for powers of Hamilton cycles in tournaments

A basic result in graph theory says that any n-vertex tournament with in- and out-degrees larger than n−24 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 n4+εn contain the k-th power of a Hamilton cycle. Up until now, there has not been any progress on determining a more accurate error term in the degree condition, neither in understanding how large n should be in the Bollobás-Häggkvist theorem. We essentially resolve both of these questions. First, we show that if the degrees are at least n4+cn1−1/⌈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 n1−1/⌈(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 n4+Ω(n1/5) 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.

Minimizing cycles in tournaments and normalized \(q\)-norms

Akin to the Erd\H{o}s-Rademacher problem, Linial and Morgenstern made the following conjecture in tournaments: for any $d\in (0,1]$, among all $n$-vertex tournaments with $d\binom{n}{3}$ many 3-cycles, the number of 4-cycles is asymptotically minimized by a special random blow-up of a transitive tournament. Recently, Chan, Grzesik, Kr\'al' and Noel introduced spectrum analysis of adjacency matrices of tournaments in this study, and confirmed this for $d\geq 1/36$. In this paper, we investigate the analogous problem of minimizing the number of cycles of a given length. We prove that for integers $\ell\not\equiv 2\mod 4$, there exists some constant $c_\ell>0$ such that if $d\geq 1-c_\ell$, then the number of $\ell$-cycles is also asymptotically minimized by the same family of extremal examples for $4$-cycles. In doing so, we answer a question of Linial and Morgenstern about minimizing the $q$-norm of a probabilistic vector with given $p$-norm for any integers $q>p>1$. For integers $\ell\equiv 2\mod 4$, however the same phenomena do not hold for $\ell$-cycles, for which we can construct an explicit family of tournaments containing fewer $\ell$-cycles for any given number of $3$-cycles. We conclude by proposing two conjectures on the minimization problem for general cycles in tournaments.

About the number of oriented Hamiltonian paths and cycles in tournaments

AbstractWe prove that a tournament and its converse contain the same number of oriented Hamiltonian paths and the same number of oriented Hamiltonian cycles, of any given type, as a generalization of Rosenfeld's result proved for antidirected paths.

The shifted Turán sieve method on tournaments II

In a previous work [5], we developed the shifted Turán sieve method on a bipartite graph and applied it to problems on cycles in tournaments. More precisely, we obtained upper bounds for the number of tournaments which contain a small number of r-cycles. In this paper, we improve our sieve inequality and apply it to obtain an upper bound for the number of bipartite tournaments which contain a number of 2r-cycles far from the average. We also provide the exact bound for the number of tournaments which contain few 3-cycles, using other combinatorial arguments.

Packing Arc-Disjoint Cycles in Tournaments

A tournament is a directed graph in which there is a single arc between every pair of distinct vertices. Given a tournament T on n vertices, we explore the classical and parameterized complexity of the problems of determining if T has a cycle packing (a set of pairwise arc-disjoint cycles) of size k and a triangle packing (a set of pairwise arc-disjoint triangles) of size k. We refer to these problems as Arc-disjoint Cycles in Tournaments (ACT) and Arc-disjoint Triangles in Tournaments (ATT), respectively. Although the maximization version of ACT can be seen as the dual of the well-studied problem of finding a minimum feedback arc set (a set of arcs whose deletion results in an acyclic graph) in tournaments, surprisingly no algorithmic results seem to exist for ACT. We first show that ACT and ATT are both NP-complete. Then, we show that the problem of determining if a tournament has a cycle packing and a feedback arc set of the same size is NP-complete. Next, we prove that ACT is fixed-parameter tractable via a $$2^{\mathcal {O}(k \log k)} n^{\mathcal {O}(1)}$$ -time algorithm and admits a kernel with $$\mathcal {O}(k)$$ vertices. Then, we show that ATT too has a kernel with $$\mathcal {O}(k)$$ vertices and can be solved in $$2^{\mathcal {O}(k)} n^{\mathcal {O}(1)}$$ time. Afterwards, we describe polynomial-time algorithms for ACT and ATT when the input tournament has a feedback arc set that is a matching. We also prove that ACT and ATT cannot be solved in $$2^{o(\sqrt{n})} n^{\mathcal {O}(1)}$$ time under the exponential-time hypothesis.

Spanning eulerian subdigraphs avoiding [formula omitted] prescribed arcs in tournaments

Fraise and Thomassen (1987) proved that every (k+1)-strong tournament has a hamiltonian cycle which avoids any prescribed set of k arcs. Bang-Jensen, Havet and Yeo showed in Bang-Jensen et al. (2019) that a number of results concerning vertex-connectivity and hamiltonian cycles in tournaments have analogues when we replace vertex connectivity by arc-connectivity and hamiltonian cycles by spanning eulerian subdigraphs. They proved the existence of a smallest function f(k) with the property that every f(k)-arc-strong semicomplete digraph has a spanning eulerian subdigraph which avoids any prescribed set of k arcs by showing that f(k)≤(k+1)2+44. They conjectured that every (k+1)-arc-strong semicomplete digraph has a spanning eulerian subdigraph which avoids any prescribed set of k arcs and verified this for k=2,3. In this paper we prove that f(k)≤⌈6k+15⌉. In particular, the conjecture holds for k≤4

Lichiardopol’s Conjecture on Disjoint Cycles in Tournaments

In 2010, Lichiardopol conjectured for $q \geqslant 3$ and $k \geqslant 1$ that any tournament with minimum out-degree at least $(q-1)k-1$ contains $k$ disjoint cycles of length $q$. Previously the conjecture was known to hold for $q\leqslant 4$. We prove that it holds for $q \geqslant 5$, thereby completing the proof of the conjecture.

An improvement of Lichiardopol’s theorem on disjoint cycles in tournaments

Let k ≥ 1 and q ≥ 3 be integers and let f(q)=(6q2−16q+10)/(3q2−3q−4). In this paper, we prove that if q ≥ 4, then every tournament T with both minimum out-degree and in-degree at least (q−1)k−1 contains at least f(q)k−2q disjoint cycles of length q. We also prove that if q=3 and k ≥ 6, then T contains at least 16k/15−5 disjoint triangles. Our results improve Lichiardopol’s theorem ([Discrete Math. 310 (19) (2010) 2567–2570]): for given integers q ≥ 3 and k ≥ 1, a tournament T with both minimum out-degree and in-degree at least (q−1)k−1 contains at least k disjoint cycles of length q.

Antidirected Hamiltonian paths and directed cycles in tournaments

We prove that in a tournament of odd order n≥5, the number of antidirected Hamiltonian paths starting with a forward arc and the number of Hamiltonian circuits have the same parity.

What Moser Could Have Asked: Counting Hamilton Cycles in Tournaments

Moser asked for a construction of explicit tournaments on n vertices having at least ()n Hamilton cycles. We show that he could have asked for rather more.

A note on arc-disjoint cycles in tournaments

A note on arc-disjoint cycles in tournaments

Componentwise complementary cycles in multipartite tournaments

The problem of complementary cycles in tournaments and bipartite tournaments was completely solved. However, the problem of complementary cycles in semicomplete n-partite digraphs with n ≥ 3 is still open. Based on the definition of componentwise complementary cycles, we get the following result. Let D be a 2-strong n-partite (n ≥ 6) tournament that is not a tournament. Let C be a 3-cycle of D and D \ V (C) be nonstrong. For the unique acyclic sequence D1,D2, ...,Dα of DV (C), where α ≥ 2, let Dc = {Di\Di contains cycles, i = 1, 2, ..., α}, \(D_{\bar c} = \{ D_1 ,D_2 , \cdots ,D_\alpha \} \backslash D_c\). If Dc ≠ ∅, then D contains a pair of componentwise complementary cycles.

Problems and conjectures concerning connectivity, paths, trees and cycles in tournament-like digraphs

In this paper we collect a substantial number of challenging open problems and conjectures on connectivity, paths, trees and cycles in tournaments and classes of digraphs which contain tournaments as a subclass. The list is by no means exhaustive but is meant to show that the area has a large number of interesting open problems. We also mention problems for general digraphs when they are relevant in the context.

Oriented Hamiltonian Cycles in Tournaments

We prove that every tournament of order n⩾68 contains every oriented Hamiltonian cycle except possibly the directed one when the tournament is reducible.

A Parallel Reduction of Hamiltonian Cycle to Hamiltonian Path in Tournaments

We propose a parallel algorithm which reduces the problem of computing Hamiltonian cycles in tournaments to the problem of computing Hamiltonian paths. The running time of our algorithm is O(log n) using O( n 2/log n) processors on a CRCW PRAM, and O(log n log log n) on an EREW PRAM using O( n 2/log n log log n) processors. As a corollary, we obtain a new parallel algorithm for computing Hamiltonian cycles in tournaments. This algorithm can be implemented in time O(log n) using O( n 2/log n) processors in the CRCW model and in time O(log 2 n) with O( n 2/log n log log n) processors in the EREW model.

A linear-time algorithm for finding Hamiltonian cycles in tournaments

We give a simple algorithm to transform a Hamiltonian path in a Hamiltonian cycle, if one exists, in a tournament T of order n. Our algorithm is linear in the number of arcs, i.e., of complexity O( m)=O( n 2) and when combined with the O( n log n) algorithm of [2] to find a Hamiltonian path in T, it yields an O( n 2) algorithm for searching a Hamiltonian cycle in a tournament. Up to now, algorithms for searching Hamiltonian cycles in tournaments were of order O( n 3) [3], or O( n 2 log n) [5].

Powers of Hamilton cycles in tournaments

Our main aim is to show that for every ε > 0 and k ∈ N there is an n( ε, k) such that if T is a tournament of order n ≥ n( ε, k) and in T every vertex has indegree at least ( 1 4 +ε)n and at most ( 3 4 −ε)n then T contains the kth power of a Hamilton cycle.

Paths and Cycles in Tournaments

Paths and Cycles in Tournaments

Paths and cycles in tournaments

Sufficient conditions are given for the existence of an oriented path with given end vertices in a tournament. As a consequence a conjecture of Rosenfeld is established. This states that if n n is large enough, then every non-strongly oriented cycle of order n n is contained in every tournament of order n n .

Edge-Disjoint Hamiltonian Paths and Cycles in Tournaments

We describe sufficient conditions for the existence of Hamiltonian paths in oriented graphs and use these to provide a complete description of the tournaments with no two edge-disjoint Hamiltonian paths. We prove that tournaments with small irregularity have many edge-disjoint Hamiltonian cycles in support of Kelly's conjecture.