Regular Tournaments Research Articles

The Terwilliger algebras of doubly regular tournaments

The Terwilliger algebras of doubly regular tournaments

Self-orthogonal codes over a non-unital ring from two class association schemes

There is a non-unital ring [Formula: see text] of order 4 defined by generators and relations as [Formula: see text] In this paper, we present special constructions of linear codes over [Formula: see text] from the adjacency matrices of two class association schemes. These consist of either Strongly Regular Graphs (SRGs) or Doubly Regular Tournaments (DRTs). We investigate the conditions under which these codes are self-orthogonal, quasi self-dual, or Type IV. As a byproduct of this study, some diophantine relations on the weight of sum of vectors with coefficients in [Formula: see text] are derived. Some examples of codes with minimum distance better than that of Type IV codes over unital rings of the same order in modest lengths are given.

On Seymour's and Sullivan's second neighbourhood conjectures

AbstractFor a vertex of a digraph, (, respectively) is the number of vertices at distance 1 from (to, respectively) and is the number of vertices at distance 2 from . In 1995, Seymour conjectured that for any oriented graph there exists a vertex such that . In 2006, Sullivan conjectured that there exists a vertex in such that . We give a sufficient condition in terms of the number of transitive triangles for an oriented graph to satisfy Sullivan's conjecture. In particular, this implies that Sullivan's conjecture holds for all orientations of planar graphs and triangle‐free graphs. An oriented graph is an oriented split graph if the vertices of can be partitioned into vertex sets and such that is an independent set and induces a tournament. We also show that the two conjectures hold for some families of oriented split graphs, in particular, when induces a regular or an almost regular tournament.

Arc-pancyclicity of hypertournaments with irregularity at most two

A k-hypertournament H consists of a pair (V(H),A(H)), where V(H) is a non-empty finite set of vertices and A(H) contains exactly one permutation of S as an arc for all k-subsets S of V(H). The irregularity of hypertournament H=(V(H),A(H)) is defined as maxv∈V(H)⁡{|deg+(v)−deg−(v)|}. A hypertournament with irregularity 0 is called as a ∑-regular or degree regular hypertournament which was introduced by Surmacs and is a generalization of regular tournament. A hypertournament with irregularity 1 is called as an almost ∑-regular or almost degree regular hypertournament which is a generalization of almost regular tournament.It is well known that every regular tournament is arc-pancyclic and every almost regular tournament with at least 8 vertices is arc-4-pancyclic. Surmacs (2017) [7] proved that every ∑-regular k-hypertournament on n-vertices, where 2≤k≤n−2, is arc-pancyclic. In this paper, we prove that every k-hypertournament on n vertices with irregularity at most two, where 3≤k≤n−2, is arc-pancyclic, which is a generalization of above results.

On d-panconnected tournaments with large semidegrees

We prove the following new results.(a)Let T be a regular tournament of order 2n+1≥11 and S a subset of V(T). Suppose that |S|≤12(n−2) and x, y are distinct vertices in V(T)∖S. If the subtournament T−S contains an (x,y)-path of length r, where 3≤r≤|V(T)∖S|−2, then T−S also contains an (x,y)-path of length r+1.(b)Let T be an m-irregular tournament of order p, i.e., |d+(x)−d−(x)|≤m for every vertex x of T. If m≤13(p−5) (respectively, m≤15(p−3)), then for every pair of vertices x and y, T has an (x,y)-path of any length k, 4≤k≤p−1 (respectively, 3≤k≤p−1 or T belongs to a family G of tournaments, which is defined in the paper). In other words, (b) means that if the semidegrees of every vertex of a tournament T of order p are between 13(p+1) and 23(p−2) (respectively, between 15(2p−1) and 15(3p−4)), then the claims in (b) hold.Our results improve in a sense related results of Alspach (1967), Jacobsen (1972), Alspach et al. (1974), Thomassen (1978) and Darbinyan (1977, 1978, 1979), and are sharp in a sense.

Spectral Slater index of tournaments

The Slater index $i(T)$ of a tournament $T$ is the minimum number of arcs that must be reversed to make $T$ transitive. In this paper, we define a parameter $\Lambda(T)$ from the spectrum of the skew-adjacency matrix of $T$, called the spectral Slater index. This parameter is a measure of remoteness between the spectrum of $T$ and that of a transitive tournament. We show that $\Lambda(T)\leq8\, i(T)$ and we characterize the tournaments with maximal spectral Slater index. As an application, an improved lower bound on the Slater index of doubly regular tournaments is given.

K-Spectrally monomorphic tournaments

A tournament is k-spectrally monomorphic if all the k×k principal submatrices of its adjacency matrix have the same characteristic polynomial. Transitive n-tournaments are trivially k-spectrally monomorphic. We show that there are no others for k∈{3,…,n−3}. Furthermore, we prove that for n≥5, a non-transitive n-tournament is (n−2)-spectrally monomorphic if and only if it is doubly regular. Finally, we give some results on (n−1)-spectrally monomorphic regular tournaments.

ВОЗМОЖНОСТИ ДЛЯ СПОНСОРОВ И ПАРТНЕРОВ В ИНДУСТРИИ СМЕШАННЫХ ЕДИНОБОРСТВ

The research purpose is to evaluate and analyze the possibilities of using sponsorship in mixed martial arts in Russia. Methods and organization of the research. The following methods were used during the research: the analysis of scientific and methodological literature, documentary and Internet sources, study of the organization of promotions and mixed martial arts tournaments. Scientific articles, official documents and Internet sites of Russian and foreign mixed martial arts promotions were analyzed to achieve the goal. The analysis of materials on the possibility of using sponsorship in Russia, which were highlighted during the research, was carried out. Research results. The article presents the results of the research on the specific use of sponsorship in mixed martial arts in Russia and in the world. The sponsorship of promotions, regular tournaments and participating mixed martial arts fighters is described. The interest in mixed martial arts among different audiences is determined. The opportunities for potential sponsors and partners who will want to get their promotion through mixed martial arts are analyzed. Conclusion. According to the results of the research, we have highlighted promising opportunities for additional funding of mixed martial arts through organizing work with sponsors and partners. The data obtained allow experts to form an idea of the available opportunities in mixed martial arts, as a new channel of promotion and communication with the sports audience.

Self-orthogonal codes over a non-unital ring and combinatorial matrices

There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as \(E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b^2=b,\,ab=a,\, ba=b\rangle .\) We study a special construction of self-orthogonal codes over E, based on combinatorial matrices related to two-class association schemes, Strongly Regular Graphs (SRG), and Doubly Regular Tournaments (DRT). We construct quasi self-dual codes over E, and Type IV codes, that is, quasi self-dual codes whose all codewords have even Hamming weight. All these codes can be represented as formally self-dual additive codes over \(\mathbb {F}_4.\) The classical invariant theory bound for the weight enumerators of this class of codes improves the known bound on the minimum distance of Type IV codes over E.

On Explicit Random-Like Tournaments

We give a new theorem describing a relation between the quasi-random property of regular tournaments and their spectra. This provides many solutions to a constructing problem mentioned by Erdős and Moon (Can Math Bull 8(3):269–271, 1965) and Spencer (Graphs Comb 1(4):357–382, 1985).

Formally self-dual LCD codes from two-class association schemes

Linear codes with complementary duals, shortly named LCD codes, are linear codes whose intersection with their duals is trivial. In this paper, we outline a construction for LCD codes over finite fields from the adjacency matrices of two-class association schemes. These schemes consist of either strongly regular graphs (SRGs) or doubly regular tournaments (DRTs). Under certain conditions, the method yields formally self-dual codes. Further, we propose a decoding algorithm that can be feasible for the LCD codes obtained using one of the given methods.

Decomposing tournaments into paths

We consider a generalisation of Kelly's conjecture which is due to Alspach, Mason, and Pullman from 1976. Kelly's conjecture states that every regular tournament has an edge decomposition into Hamilton cycles, and this was proved by K\"uhn and Osthus for large tournaments. The conjecture of Alspach, Mason, and Pullman asks for the minimum number of paths needed in a path decomposition of a general tournament $T$. There is a natural lower bound for this number in terms of the degree sequence of $T$ and it is conjectured that this bound is correct for tournaments of even order. Almost all cases of the conjecture are open and we prove many of them.

Cyclic Triangle Factors in Regular Tournaments

Both Cuckler and Yuster independently conjectured that when $n$ is an odd positive multiple of $3$ every regular tournament on $n$ vertices contains a collection of $n/3$ vertex-disjoint copies of the cyclic triangle. Soon after, Keevash \& Sudakov proved that if $G$ is an orientation of a graph on $n$ vertices in which every vertex has both indegree and outdegree at least $(1/2 - o(1))n$, then there exists a collection of vertex-disjoint cyclic triangles that covers all but at most $3$ vertices. In this paper, we resolve the conjecture of Cuckler and Yuster for sufficiently large $n$.

Difference Families, Skew Hadamard Matrices, and Critical Groups of Doubly Regular Tournaments

In this paper we investigate the structure of the critical groups of doubly-regular tournaments (DRTs) associated with skew Hadamard difference families (SDFs) with one, two, or four blocks. Brown and Ried found that the existence of a skew Hadamard matrix of order $n+1$ is equivalent to the existence of a DRT on $n$ vertices. A well known construction of a skew Hadamard matrix order $n$ is by constructing skew Hadamard difference sets in abelian groups of order $n-1$. The Paley skew Hadamard matrix is an example of one such construction. Szekeres and Whiteman constructed skew Hadamard matrices from skew Hadamard difference families with two blocks. Wallis and Whiteman constructed skew Hadamard matrices from skew Hadamard difference families with four blocks. In this paper we consider the critical groups of DRTs associated with skew Hadamard matrices constructed from skew Hadamard difference families with one, two or four blocks. We compute the critical groups of DRTs associated with skew Hadamard difference families with two or four blocks. We also compute the critical group of the Paley tournament and show that this tournament is inequivalent to the other DRTs we considered. Consequently we prove that the associated skew Hadamard matrices are not equivalent.

Entropy of tournament digraphs

The Rényi α-entropy Hα of complete antisymmetric directed graphs (i.e., tournaments) is explored. We optimize Hα when α=2 and 3, and find that as α increases Hα's sensitivity to what we refer to as ‘regularity’ increases as well. A regular tournament on n vertices is one with each vertex having out-degree n−12, but there is a lot of diversity in terms of structure among the regular tournaments; for example, a regular tournament may be such that each vertex's out-set induces a regular tournament (a doubly regular tournament) or a transitive tournament (a rotational tournament). As α increases, on the set of regular tournaments, Hα has maximum value on doubly regular tournaments and minimum value on rotational tournaments. The more ‘regular’, the higher the entropy. We show, however, that among all tournaments on a fixed number of vertices H2 and H3 are maximized by any regular tournament on that number of vertices. We also provide a calculation that is equivalent to the von Neumann entropy, but may be applied to any directed or undirected graph and shows that the von Neumann entropy is a measure of how quickly a random walk on the graph or directed graph settles.

A Note on Min–Max Pair in Tournaments

We prove the following results in this paper. Let T be a tournament of order $$n\ge 3$$ with vertex set V and arc set E, x a vertex of maximum out-degree and y a vertex of minimum out-degree of T. If $$yx\in E$$ then there exists a path of length i from x to y for any i with $$2\le i \le n-1$$ ; and if $$xy\in E$$ , then there exists a path of length i from x to y for any i with $$3\le i \le n-1$$ unless xy is exceptional. We also give a very short discussion to almost regular tournaments and prove a result of Jakobsen.

Ranking in young tennis players—a study to determine possible correlates

The purpose of this study was to examine the associations of physical, functional, experiential, and training-related characteristics with ranking in a cohort of competitive U12 tennis players. A total of 119 (boys = 68, girls = 51) nationally ranked Turkish players aged 9.6–12.3 years (10.9 ± 0.7) were measured on stature, sitting height, body mass, skinfolds, grip strength, and agility. Age at peak height velocity (APHV), percentage of predicted adult stature (PAS%), body mass index (BMI), body fat percentages (BF%) and growth status were calculated. Weekly training hours and experiences in regular tennis training and tournament play were recorded. Compared to boys, girls were found to be significantly taller and more advanced in maturation. There were no significant relationships between growth, APHV, BMI, BF% and rankings. In girls, PAS% was significantly correlated with ranking. Results revealed that variables regarding experience, training volume, and motor performance were significantly associated with ranking in both boys and girls. Age to start regular training (r = −0.540) for the girls and weekly training volume (r = −0.489) for the boys were the most correlated variables. These results were confirmed by logistic regression models. The findings highlight the possible positive consequences of early participation in regular tennis training and tournaments, rather than growth and body composition, on the ranking of U12 tennis players in both genders.

On Orthogonal Matrices with Zero Diagonal

This paper considers real orthogonal $n\times n$ matrices whose diagonal entries are zero and off-diagonal entries nonzero, which are referred to as $\OMZD(n)$. It is shown that there exists an $\OMZD(n)$ if and only if $n\neq 1,\ 3$, and that a symmetric $\OMZD(n)$ exists if and only if $n$ is even and $n\neq 4$. Also, a construction of $\OMZD(n)$ obtained from doubly regular tournaments is given. Finally, the results are applied to determine the minimum number of distinct eigenvalues of matrices associated with some families of graphs, and the related notion of orthogonal matrices with partially-zero diagonal is considered.

Automorphism groups of Walecki tournaments with zero and odd signatures

Walecki tournaments were defined by Alspach in 1966. They form a class of regular tournaments that posses a natural Hamilton directed cycle decomposition. It has been conjectured by Kelly in 1964 that every regular tournament possesses such a decomposition. Therefore Walecki tournaments speak in favor of the conjecture. A second interest in Walecki tournaments arises from the mapping between cycles of the complementing circular shift register and isomorphism classes of Walecki tournaments. The problem of enumerating non-isomorphic Walecki tournaments has not been solved to date. We characterize the arc structure of Walecki tournaments whose corresponding binary sequences have zero and odd signature. Automorphism groups are determined for zero signature Walecki tournaments and for odd signature Walecki tournaments with the zero signature Walecki subtournaments. Walecki tournaments possess a broad range of subtournaments isomorphic to some Walecki tournament. Subtournaments of odd signature Walecki tournaments induced by the outsets of the central vertex are proven to be either regular or almost regular.

Enumerating Unlabelled Embeddings of Digraphs

AbstractA 2-cell embedding of an Eulerian digraph D into a closed surface is said to be directed if the boundary of each face is a directed closed walk in D. In this paper, a method is developed with the purpose of enumerating unlabelled embeddings for an Eulerian digraph. As an application, we obtain explicit formulas for the number of unlabelled embeddings of directed bouquets of cycles Bn, directed dipoles OD2n and for a class of regular tournaments T2n+1.