Tournament Of Order Research Articles

On topological tournaments of order 4 in digraphs of outdegree 3

It is proved that every finite digraph of minimum outdegree 3 contains a subdivision of the transitive tournament on 4 vertices. © 1996 John Wiley & Sons, Inc.

On non-{0, 1/2, 1} extreme points of the generalized transitive tournament polytope

The acyclic tournaments of order n form the linear ordering polytope P n LO. The generalized transitive tournaments of order n form the polytope P n C, which contains the linear ordering polytope. It is known that the integral extreme points of P n C coincide with those of P n LO. Dridi showed that P n LO = P n C for n ⩽ 5, while for n > 5 P n LO ⊂ P n C. Borobia gave a complete characterization of the extreme points of P n C with values in {0, 1, 1/2}. It was mentioned by Brualdi and Hwang that no extreme points of P n C with values not in {0, 1, 1/2} are known. In this paper we present a method for obtaining a family of extreme points of P n C with values not in {0, 1, 1/2}. We also prove that these non-half-integral extreme points of P n C violate certain diagonal inequalities which are facet defining for P n LO.

On the tournament matrices with smallest rank

A tournament matrix is a square zero-one matrix A satisfying the equation A+At = J − I, where J is the all-ones matrix. In [1] it was proved that if A is an n × n tournament matrix, then the rank of A is at least (n - 1)/2, over any field; and in characteristic zero rank (A) equals n - 1 or n. Michael [3] has constructed examples having rank (n - 1)/2; they are double borderings of Hadamard tournaments of order n - 2, and so must satisfy n ≡ 1 (mod 4). In this note, we supplement this result by showing that an analogous construction is sometimes impossible when n ≡ 3 (mod 4).

The 3 and 4-dichromatic tournaments of minimum order

The dichromatic number dc( D) of a digraph D is the minimum number of acyclic sets in which V( D) can be partitioned. If dc( D) = r, D is said r-dichromatic. In this paper it is proved that the minimum order of a 3-dichromatic (resp. 4-dichromatic) tournament is 7 (resp. 11). It is also proved that there are exactly four nonisomorphic 3-dichromatic tournaments of order 7 and a unique 4-dichromatic tournament of order 11. All these tournaments are characterized.

On Tournaments and Their Largest Transitive Subtournaments

In this work we present some results in connection with the following problem, posed by Erdos and Moser: For each positive integer n, determine the greatest integer v(n) such that all tournaments of order n contain the transitive subtournament of order v(n) (denoted ???(?)). It is known that v(n) = 3 for 4 ≤ n ≤ 7, v(n) = 4 for 8 ≤ n ≤ 13, ?(?) = 5 for 14 ≤ n ≤ 27, and ?(n) ? 6 for n > 27. Moreover, the uniqueness of the tournaments of orders 7 and 13, free of TT4 and TT5, respectively, has been proved. Here we prove that there exists only one tournament of order 27 free of TT6 and that every tournament of order 55 contains TT7, which impliesv(n) ? ?log 2(n/55? + 7, forn ? 55, improving the best lower bound known for v(n).

On the decomposition of λK into regular tournaments of order 5

AbstractAn RTD[5,λ; v] is a decomposition of the complete symmetric directed multigraph, denoted by λK, into regular tournaments of order 5. In this article we show that an RTD[5,λ; v] exists if and only if (v−1)λ ≡ 0 (mod 2) and v(v−1)λ ≡ 0 (mod 10), except for the impossible case (v,λ) = (15,1). Furthermore, we show that for each v ≡ 1,5 (mod 20), v ≠ 5, there exists a B[5,2; v] which is not RT5‐directable. © 1994 John Wiley & Sons, Inc.

Claws contained in all n-tournaments

We prove that any claw of order n with degree d≤ 3 8 n is n-unavoidable, which means that any tournament of order n contains it as a subdigraph. A simple corollary is that any tournament has a directed Hamiltonian path.

Generalized transitive tournaments and doubly stochastic matrices

We show that a doubly stochastic matrix is a convex combination of nonidentity permutation matrices if and only if it can be written as the sum of a nonnegative matrix and a convex combination of cycle matrices. We use this result to give a shorter proof of the theorem of Cruse which asserts that a doubly stochastic matrix is a convex combination of nonidentity permutation matrices if and only if its inner product with each generalized transitive tournament matrix is at least 1. The generalized transitive tournaments of order n form a convex polytope T n which contains the convex hull T n * (also called the linear ordering polytope) of the transitive tournaments. Each transitive tournament matrix of order n is an extreme point of T n , but for n ⩾ 6 there are other extreme points. With each generalized tournament matrix T of order n we associate a graph whose edges correspond to the nonintegral entries of T. We investigate which graphs can arise from generalized transitive tournaments and which can arise from extreme generalized transitive tournaments. We briefly discuss a generalization of the linear ordering polytope to arbitrary posets.

Near-homogeneous tournaments and permutation groups

Claudette Tabib has characterized the near-homogeneous tournaments and has shown the existence of such tournaments of the order 5, 9, 13. We follow her investigations, searching for new near-homogeneous rotational tournaments. For this we constructed all the vertex-symmetric tournaments with two arc orbits, using a result by Astié-Vidal on the vertex-symmetric tournaments with minimal number of arc orbits. We established a necessary and sufficient condition for such a tournament to be near-homogeneous. In this way we constructed six new near-homogeneous tournaments of the first order: 29, 53, 173, 229, 293, 733.

On tournament matrices

Let T be a tournament of order n with adjacency matrix M . We find several conditions that are equivalent to M being singular. A correlation between the number of 3-cycles in T and the rank of M is established. It is shown that asymptotically at least 1 2 of the tournament matrices are nonsingular. We also derive bounds on the spectral radius of tournament matrices with a given row-sum vector.

LA RECONSTRUCTION DES TOURNOIS SANS DIAMANT

AbstractWe prove that if T is a tournament of order n > 6 in which any 4‐sub‐tournament is hamiltonian or transitive, then T is reconstructible in the sense of Ulam.

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.

On Hadamard tournaments

where t denotes the transposition, and Z and J are the identity matrix and all one matrix of order u, respectively. Let % be the set of all (0, 1) matrices A satisfying (1) and (2). Let A, and A2 be two matrices of ‘U. Then A, is equivalent to A, if there exists a permutation matrix P of order v such that A2 = P’AI P. This is a true equivalence relation and we call each equivalence class and customarily each representative of the class a Hadamard tournament. We are interested in the existence problem of Hadamard tournaments: Does there exist a Hadamard tournament of order v = 42 + 3 for every A?

Hybrid triple systems and cubic feedback sets

Ac-hybrid triple system of orderv is a decomposition of the completev-vertex digraph intoc cyclic tournaments of order 3 and $$\frac{{v(v - 1)}}{3} - c$$ transitive tournaments of order 3. Hybrid triple systems generalize directed triple systems (c = 0) and Mendelsohn triple systems (c = v(v − 1)/3); omitting directions yields an underlying twofold triple system. The spectrum ofv andc for which ac-hybrid triple system of orderv exists is completely determined in this paper. Using (cubic) block intersection graphs, we then show that every twofold triple system of order $$v\left( {having b_v = \frac{{v(v - 1)}}{3}blocks} \right)$$ underlies ac-hybrid triple system with $$c \geqslant \frac{{2b_v }}{3}$$ . Examples are constructed for all sufficiently largev, for which this maximum is at most $$\left( {\frac{7}{{10}} + \varepsilon } \right)b_v $$ . The lower bound here is proved by establishing bounds onF i (n, r), the size of minimum cardinality vertex feedback sets inn-vertexi-connected cubic multigraphs havingr repeated edges. We establish that $$F_0 (n,r) \leqslant \frac{n}{2},$$ , $$F_1 (n,r) \leqslant \frac{{3n}}{8} + \frac{r}{4} + \frac{1}{2}, and F_2 (n,r) \leqslant \frac{{(n + r)}}{3}for n > 8$$ . These bounds are all tight, and the latter is used to derive the lower bound in the design theoretic problem.

On monochromatic paths in m-coloured tournaments

We call the tournament T an m-coloured tournament if the arcs of T are coloured with m colours. In this paper we have proved that if T is an m-coloured tournament which does not contain any tournament of order 3 whose arcs are coloured with three distinct colours then there is a vertex v of T such that for every other vertex x of T there is a monochromatic path from x to v.

On the existence of specified cycles in a tournament

AbstractFor 2 ≦ p ≦n and n ≧ 3, D(n, p) denotes the digraph with n vertices obtained from a directed cycle of length n by changing the orientation of p‐ 1 consecutives edges. In this paper, we prove that every tournament of order n ≧ 7 contains D(n, p) for p = 2, 3, …, n. Furthermore, we determine the tournaments of order n, 3 ≦ n ≦ 6, which do not contain D(n, p) for some p.

On the number of Hamiltonian cycles in tournaments

The main results assert that the minimum number of Hamiltonian bypasses in a strong tournament of order n and the minimum number of Hamiltonian cycles in a 2-connected tournament of order n increase exponentially with n. Furthermore, the number of Hamiltonian cycles in a tournament increases at least exponentially with the minimum outdegree of the tournament. Finally, for each k⩾1 there are infinitely many tournaments with precisely k Hamiltonian cycles.

The falsity of the reconstruction conjecture for tournaments

AbstractThe conjecture that for all sufficiently large p any tournament of order p is uniquely reconstructable from its point‐deleted subtournaments is shown to be false. Counterexamples are presented for all orders of the form 2n + 1 and 2n + 2. The largest previously known counterexamples were of order 8.

Path numbers of tournaments

A family P of simple (that is, cycle-free) paths is a path decomposition of a tournament T if and only if P partitions the acrs of T. The path number of T, denoted pn( T), is the minimum value of | P | over all path decompositions P of T. In this paper it is shown that if n is even, then there is a tournament on n vertices with path number k if and only if n 2 ≦ k ≦ n 2 4 , k an integer. It is also shown that if n is odd and T is a tournament on n vertices, then (n + 1) 2 ≦ pn(T) ≦ (n 2 − 1) 4 . Moreover, if k is an integer satisfying (i) (n + 1) 2 ≦ k ≦ n − 1 or (ii) n < k ≦ (n 2 − 1) 4 and k is even, then a tournament on n vertices having path number k is constructed. It is conjectured that there are no tournaments of odd order n with odd path number k for n ≦ k < (n 2 − 1) 4 .

Groupes de permutations primitifs d'ordre impair et denombrement de tournois sommet-primitifs

A tournament T (directed graph in which there is exactly one arc between any two vertices) is said to be point-primitive if its automorphism group A( T) acts primitively on the vertices of T. Automorphism groups of point-primitive tournaments are primitive permutation groups of odd order, which are known to be affine groups; and so point-primitive tournaments are of prime-power order. Some properties of primitive permutation groups of odd order are proved and a counting formula for point-primitive tournaments of order p 2 k ( p prime number, k integer ⪖0) is derived from them.