Tournament Of Order Research Articles

A tabu-search for minimising the carry-over effects value of a round-robin tournament

A player b in a round-robin sports tournament receives a carry-over effect from another player a if some third player opposes a in round i and b in round i+1. Let γ(ab) denote the number of times player b receives a carry-over effect from player a during a tournament. Then the carry-over effects value of the entire tournament T on n players is given by Γ(T)=ΣΣγ(ij)^2. Furthermore, let Γ(n) denote the minimum carry-over effects value over all round-robin tournaments on n players. A strict lower bound on Γ(n) is n(n-1) (in which case there exists a round-robin tournament of order n such that each player receives a carry-over effect from each other player exactly once), and it is known that this bound is attained for n=2^r or n=20,22. It is also known that round-robin tournaments can be constructed from so-called starters; round-robin tournaments constructed in this way are called cyclic. It has previously been shown that cyclic round-robin tournaments have the potential of admitting small values for Γ(T), and in this paper a tabu-search is used to find starters which produce cyclic tournaments with small carry-over effects values. The best solutions in the literature are matched for n

Bounds and constructions for n-e.c. tournaments

Few families of tournaments satisfying the $n$-e.c. adjacency property are known. We supply a new random construction for generating infinite families of vertex-transitive $n$-e.c. tournaments by considering circulant tournaments. Switching is used to generate exponentially many $n$-e.c. tournaments of certain orders. With aid of a computer search, we demonstrate that there is a unique minimum order $3$-e.c. tournament of order $19,$ and there are no $3$-e.c. tournaments of orders $20,$ $21,$ and $22.$

Vertex-transitive tournaments of order a product of two distinct primes

In this paper, we begin a classification of the vertex-transitive tournaments of order pq where p and q are distinct odd primes. In particular we characterize the pq -circulant tournaments; see Theorem 4.3. Moreover, we determine 2-closed (in Wielandt's sense) odd-order transitive permutation groups of degree p and pq by using the classifications of vertex-transitive tournaments of order p and pq . We prove that each such 2-closed group containing a cyclic regular subgroup is the full automorphism group of some circulant tournament.

On the number of non-critical vertices in strong tournaments of order [formula omitted] with minimum out-degree [formula omitted] and in-degree [formula omitted

Let T be a strong tournament of order n ≥ 4 with given minimum out-degree δ + and in-degree δ − . By definition, a vertex w in T is non-critical if the subtournament T − w is also strong. In the present paper, we show that T contains at least min { n , 2 δ + + 2 δ − − 2 } non-critical vertices, and all tournaments for which this lower bound is attained are determined. For the case min { δ + , δ − } ≥ 2 , we also describe all strong tournaments of order n ≥ 2 δ + + 2 δ − that include exactly 2 δ + + 2 δ − − 1 non-critical vertices. From this description it follows that any strong tournament T of order n ≥ 2 δ + + 2 δ − + 2 with min { δ + , δ − } ≥ 2 contains at least 2 δ + + 2 δ − non-critical vertices. Finally, for the case min { δ + , δ − } ≥ 4 , we completely describe all strong tournaments of order n ≥ 2 δ + + 2 δ − + 2 that admit exactly 2 δ + + 2 δ − non-critical vertices. All of these results sharpen those obtained recently by K. Kotani in terms of δ = min { δ + , δ − } .

A tournament of order 14 with disjoint Banks and Slater sets

It is shown in this note that there exists a tournament of order 14 with disjoint Banks and Slater sets. Previously, the smallest such tournament was reported to be of order 16. In addition, it is shown that 11 is the minimum order of a tournament in which the Slater set is not a subset of the Banks set.

Scheduling spouse - avoiding mixed doubles round - robin tennis tournaments

The problem of scheduling a spouse-avoiding mixed doubles round-robin tennis tournament (SMDRTT) of order involves finding a playing schedule for married couples in such a way that no player teams up with his/her spouse, no player opposes his/her spouse, each player opposes every other player of the same sex exactly once, each player teams up with every player of the opposite sex (except his/her spouse) exactly once, and each player opposes every player of the opposite sex (except his/her spouse) exactly once.

Participation incentives in rank order tournaments with endogenous entry

Rank order tournaments, in which the payment to an agent is based upon relative observed performance, are a commonly used compensation scheme. In practice, agents often compete in some (but not all) events in a set of tournaments. The present study considers two mutually exclusive tournaments, in which agents themselves decide which event to enter. An agent bases this decision upon the combination of three distinct effects: a prize effect, a winning probability effect, and an effort cost effect. The precise impact of each of these effects is analyzed. Of particular interest is the possibility that a field of higher quality may be attracted to the event with smaller prizes.

Rank Order Tournaments and Incentive Alignment: The Effect on Firm Performance

In this article, we study the effectiveness of promotion-based tournament incentives. We simultaneously investigate tournament incentives for the VP and performance- or equity- based (alignment) incentives for the VP and the CEO. We find that tournament incentives, as measured by the pay differential between the CEO and VPs, relate positively to firm performance. We show that the effect of tournament incentives on firm performance is weaker when the firm has a new CEO and more so when the new CEO is an outsider, and when the firm belongs to a homogeneous industry. On the other hand, tournament effects are stronger when the CEO is close to retirement. Our analysis is robust to corrections for endogeneity of all our incentive measures as well for several alternate measures of tournament incentives and firm performance.

Circulant tournaments of prime order are tight

We say that a tournament is tight if for every proper 3-coloring of its vertex set there is a directed cyclic triangle whose vertices have different colors. In this paper, we prove that all circulant tournaments with a prime number p ≥ 3 of vertices are tight using results relating to the acyclic disconnection of a digraph and theorems of additive number theory.

A note on complete subdivisions in digraphs of large outdegree

AbstractMader conjectured that for all $\ell$ there is an integer $\delta^+(\ell)$ such that every digraph of minimum outdegree at least $\delta^+(\ell)$ contains a subdivision of a transitive tournament of order $\ell$. In this note, we observe that if the minimum outdegree of a digraph is sufficiently large compared to its order then one can even guarantee a subdivision of a large complete digraph. More precisely, let $\vec G$ be a digraph of order n whose minimum outdegree is at least d. Then $\vec G$ contains a subdivision of a complete digraph of order $\lfloor d^{2}/(8n^{3/2}) \rfloor$. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 1–6, 2008

Cycles in a tournament with pairwise zero, one or two given vertices in common

Chen et al. [Partitioning vertices of a tournament into independent cycles, J. Combin. Theory Ser. B 83 (2001) 213–220] proved that every k -connected tournament with at least 8 k vertices admits k vertex-disjoint cycles spanning the vertex set, which answered a question posed by Bollobas. In this paper, we prove, as a consequence of a more general result, that every k -connected tournament of diameter at least 4 contains k vertex-disjoint cycles spanning the vertex set. Then, for a connected tournament of diameter at most 3, we determine a relation between the maximum number of vertex-disjoint cycles and the maximum number of vertex-disjoint cycles spanning the vertex set of T. Also, by using a lemma of Chen et al. [Partitioning vertices of a tournament into independent cycles, J. Combin. Theory Ser. B 83 (2001) 213–220], we prove that a k -connected tournament of order at least 5 k - 3 , of diameter distinct from 3 (resp. 3) admits k (resp. k - 1 ) vertex-disjoint cycles spanning the vertex set of T, with only one exception. Finally, we give results on cycles with pairwise one or two vertices in common. A few open problems are raised.

Interval Tournaments

AbstractA tournament is an orientation of a complete graph. A directed graph is an interval digraph if for each vertex v there corresponds an ordered pair of intervals (Sv, Tv) such that u → v if and only if Su ∩ Tv ≠ ∅︁. A bipartite graph is an interval bigraph if to each vertex there corresponds an interval such that vertices are adjacent if and only if their corresponding intervals intersect and each vertex belongs to a different partite set. We use the equivalence of the models for interval digraphs and interval bigraphs to characterize tournaments that are interval digraphs via forbidden subtournaments and prove that a tournament on n vertices is an interval digraph if and only if it has a transitive (n − 1)‐subtournament. We also characterize the obstructions to the existence of a transitive subtournament of order n − 1 in a tournament of order n. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 72–81, 2007

A note on decompositions of transitive tournaments

For any positive integer n, we determine all connected digraphs G of size at most four, such that a transitive tournament of order n is G-decomposable. Among others, these results disprove a generalization of a theorem of Sali and Simonyi [Orientations of self-complementary graphs and the relation of Sperner and Shannon capacities, European J. Combin. 20 (1999), 93–99].

A Note on the Number of Hamiltonian Paths in Strong Tournaments

We prove that the minimum number of distinct hamiltonian paths in a strong tournament of order $n$ is $5^{{n-1}\over{3}}$. A known construction shows this number is best possible when $n \equiv 1 \hbox{ mod } 3$ and gives similar minimal values for $n$ congruent to $0$ and $2$ modulo $3$.

Infinite families of non-Cayley vertex-transitive tournaments

In this work is proven the existence of non-Cayley vertex-transitive tournaments of order p k , for each prime p ⩾ 3 and k ⩾ 4 , and an explicit construction is given. These tournaments are special cases of ( p k - 1 , p ) -metacirculant digraphs, and have the same automorphism group as the first non-Cayley vertex-transitive digraphs of order p k , given by Marušič in 1985. Moreover, from these tournaments, new non-Cayley vertex-transitive tournaments that realise many more degrees are constructed.

Pancyclic arcs and connectivity in tournaments

AbstractA tournament is an orientation of the edges of a complete graph. An arc is pancyclic in a tournament T if it is contained in a cycle of length l, for every 3 ≤ l ≤ |T|. Let p(T) denote the number of pancyclic arcs in a tournament T. In 4, Moon showed that for every non‐trivial strong tournament T, p(T) ≥ 3. Actually, he proved a somewhat stronger result: for any non‐trivial strong tournament h(T) ≥ 3 where h(T) is the maximum number of pancyclic arcs contained in the same hamiltonian cycle of T. Moreover, Moon characterized the tournaments with h(T) = 3. All these tournaments are not 2‐strong. In this paper, we investigate relationship between the functions p(T) and h(T) and the connectivity of the tournament T. Let pk(n) := min {p(T), T k‐strong tournament of order n} and hk(n) := min{h(T), T k‐strong tournament of order n}. We conjecture that (for k ≥ 2) there exists a constant αk> 0 such that pk(n) ≥ αkn and hk(n) ≥ 2k+1. In this paper, we establish the later conjecture when k = 2. We then characterized the tournaments with h(T) = 4 and those with p(T) = 4. We also prove that for k ≥ 2, pk(n) ≥ 2k+3. At last, we characterize the tournaments having exactly five pancyclic arcs. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 87–110, 2004

Trees in tournaments

In 1971, Sumner conjectured that any tournament of order 2(n−1) contains any oriented tree of order n. Since then several bounds have been established that get closer and closer to the suggested bound 2(n−1). In this paper we prove that any tournament of order 3(n−1) contains any oriented tree of order n.

All regular multipartite tournaments that are cycle complementary

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. A digraph D is cycle complementary if there exist two vertex-disjoint directed cycles spanning the vertex set V( D) of D. In this paper we prove that each regular c-partite tournament D of order | V( D)|⩾6 with c⩾3 is cycle complementary, unless D is isomorphic to T 7 or to D 3,2, where T 7 is a 3-regular tournament of order 7, and D 3,2 is a 2-regular 3-partite tournament such that there are exactly two vertices in each partite set.

Links between the Slater Index and the Ryser Index of Tournaments

Given a tournament T, the Slater index i(T) of T is the minimum number of arcs that must be reversed to make T transitive. The Ryser index τ˜(T) of T, defined from the out-degrees of T, measures a remoteness between T and the transitive tournaments of same order. In this paper, we study some links between i(T) and τ˜(T). More precisely, calling I(n, τ) the maximum value of i(T) over the set of tournaments on n vertices and such that τ˜(T)=τ, we compute an upper bound of I(n,τ) for every value of τ. Then we use this upper bound to study a conjecture stated by J.-C. Bermond on the regularity (i.e., the fact that all the out-degrees are equal or almost equal) of the tournaments with a maximum Slater index by showing that the out-degrees of such tournaments cannot be “too far” from the ones of the regular tournaments.

An experimental study on tournament design

Since recently, rank order tournaments have become quite popular for providing incentives in employment relationships. However, the consequences of different tournament designs are widely unexplored. This paper experimentally investigates different tournament design alternatives along two dimensions: tournament size and prize structure. We find that average effort tends to increase with a higher proportion of winner prizes. Additionally, variability of effort is lower if the number of winner prizes is high. Especially two-person tournaments are prone to collusion. Furthermore, we observe a restart effect.