Tournament Of Order Research Articles

Partition of regular balanced c-tournaments into strongly connected c-tournaments

A multipartite tournament will be called r-balanced if every partite set has r vertices. The study of the existence of strongly connected subtournaments in multipartite tournaments started in 1999 by Volkmann. In this paper we study a much more general problem: the existence of a partition of a c-partite tournament into strongly connected tournaments of order c, which will be called strong partition. Given a positive integer r, the strong partition number, ST(r), is the minimum integer c′ such that every regular r-balanced c-partite tournament, with c≥c′, has a strong partition. We prove that for every r≥2, ST(r) exists and that 5≤ST(2)≤7. Moreover, as a consequence of our technique, for every r, there exists a c such that for every c′>c, there exists a strong partition into maximal tournaments of minimum degree ⌊c′−24⌋+1.

Path decompositions of tournaments

In 1976, Alspach, Mason, and Pullman conjectured that any tournament $T$ of even order can be decomposed into exactly ${\rm ex}(T)$ paths, where ${\rm ex}(T):= \frac{1}{2}\sum_{v\in V(T)}|d_T^+(v)-d_T^-(v)|$. We prove this conjecture for all sufficiently large tournaments. We also prove an asymptotically optimal result for tournaments of odd order.

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 characterizations of tournaments

Characterizing graphs by the spectra of various matrices associated with the graphs has long been an important topic in spectral graph theory. However, it is generally very hard to show a given graph to be determined by its spectrum. This paper gives a simple criterion for a tournament to be determined by its adjacency spectrum. More precisely, let G be a tournament of order n with adjacency matrix A, and WA(G)=[e,Ae,...,An−1e] be its walk matrix, where e is the all-one vector. We show that, for any self-converse tournament G, if det⁡WA(G) is square-free, then G is uniquely determined by its adjacency spectrum among all tournaments. The result is somewhat unexpected, which was achieved by employing some recent tools developed by the second author for showing a graph to be determined by its generalized spectrum. The key observation is that for a tournament G, the complement of G coincides with the converse of G, and hence a certain equivalence between the ordinary adjacency spectrum and the generalized (skew)-adjacency spectrum for G can be established. Moreover, some equivalent formulations of the above result, as well as some related ones are also presented.

Rainbow triangles in arc-colored digraphs

Let D be an arc-colored digraph. The arc number a(D) of D is defined as the number of arcs of D. The color number c(D) of D is defined as the number of colors assigned to the arcs of D. A rainbow triangle in D is a directed triangle in which every pair of arcs has distinct colors. Let f(D) be the smallest integer such that if c(D)≥f(D), then D contains a rainbow triangle. In this paper we obtain f(K↔n) and f(Tn), where K↔n is a complete digraph of order n and Tn is a strongly connected tournament of order n. Moreover we characterize the arc-colored complete digraph K↔n with c(K↔n)=f(K↔n)−1 and containing no rainbow triangles. We also prove that an arc-colored digraph D on n vertices contains a rainbow triangle when a(D)+c(D)≥a(K↔n)+f(K↔n), which is a directed extension of the undirected case.

On the unavoidability of oriented trees

A digraph is n-unavoidable if it is contained in every tournament of order n. We first prove that every arborescence of order n with k leaves is (n+k−1)-unavoidable. We then prove that every oriented tree of order n (n≥2) with k leaves is (32n+32k−2)-unavoidable and (92n−52k−92)-unavoidable, and thus (218n−4716)-unavoidable. Finally, we prove that every oriented tree of order n with k leaves is (n+144k2−280k+124)-unavoidable.

Rainbow Triangles in Arc-Colored Tournaments

Let \(T_{n}\) be an arc-colored tournament of order n. The maximum monochromatic indegree \(\varDelta ^{-mon}(T_{n})\) (resp. outdegree \(\varDelta ^{+mon}(T_{n})\)) of \(T_{n}\) is the maximum number of in-arcs (resp. out-arcs) of a same color incident to a vertex of \(T_{n}\). The irregularity \(i(T_{n})\) of \(T_{n}\) is the maximum difference between the indegree and outdegree of a vertex of \(T_{n}\). A subdigraph H of an arc-colored digraph D is called rainbow if each pair of arcs in H have distinct colors. In this paper, we show that each vertex v in an arc-colored tournament \(T_{n}\) with \(\varDelta ^{-mon}(T_n)\le \varDelta ^{+mon}(T_n)\) is contained in at least \(\frac{\delta (v)(n-\delta (v)-i(T_n))}{2}-[\varDelta ^{-mon}(T_{n})(n-1)+\varDelta ^{+mon}(T_{n})d^+(v)]\) rainbow triangles, where \(\delta (v)=\min \{d^+(v), d^-(v)\}\). We also give some maximum monochromatic degree conditions for \(T_{n}\) to contain rainbow triangles, and to contain rainbow triangles passing through a given vertex. Finally, we present some examples showing that some of the conditions in our results are best possible.

Formula omitted]-Ary spanning trees contained in tournaments

A rooted tree is called a k-ary tree, if all non-leaf vertices have exactly k children, except possibly one non-leaf vertex has at most k−1 children. Denote by h(k) the minimum integer such that every tournament of order at least h(k) contains a k-ary spanning tree. It is well-known that every tournament contains a Hamiltonian path, which implies that h(1)=1. Lu et al. (1999) proved the existence of h(k), and showed that h(2)=4 and h(3)=8. The exact values of h(k) remain unknown for k≥4. A result of Erdős on the domination number of tournaments implies h(k)=Ω(klogk). In this paper, we prove that h(4)=10 and h(5)≥13.

Monetary Incentives in Italian Public Administration: A Stimulus for Employees? An Agent-Based Approach

The paper, focusing on the context of Public Administration (PA), addresses the effects of monetary incentives in employees’ performance. In the Italian PA, the monetary incentives are distributed according to the D.L.150/09 (i.e., the monetary incentives are divided among the employees according to the employees’ performance) which is based on the rank order tournament. The paper investigates if this mechanism has positive and sustainable impacts on the employees’ performance in the short, middle, and long term. The employees’ performance has been modeled as a function of ability and motivation. The results of the computational experiments show a positive impact of the monetary incentives, distributed according to merit criteria, on the employees’ performance in the short, middle, and long term.

On king–serf pair in tournaments

For a directed graph G, a vertex x is a king if every other vertex can be reached from x by a directed path of length at most 2 and is a serf if x can be reached from every other vertex by a directed path of length at most 2. A tournament T with vertex set V is called X-hamiltonian crossing if every hamiltonian path must have one end in X and the other in X̄=V−X. Our main results are the following: •A tournament T≠T4 is X-hamiltonian crossing if and only if T has a strong component decomposition V=V1∪⋯∪Vk such that V1⊆X and Vk⊆X̄, or V1⊆X̄ and Vk⊆X.•Let T be a tournament and (x,y) a king–serf pair in T−xy. Then there exists no hamiltonian (x,y)-path in T if and only if T is an x→y special tournament with U(S1;D1,D2)≠0̸ and U(S2;D1,D2)≠0̸.•Let T be a tournament of order n≥3. If x is a vertex of maximum out-degree and y a vertex of maximum in-degree then there exists a hamiltonian (x,y)-path if and only if xy is not exceptional.

Prize Structure and Performance: Evidence from NASCAR

The predictions that emerge from tournament theory have been tested in a number of sports-related settings. Since sporting events involving individuals (golf, tennis, running, auto racing) feature rank order tournaments with relatively large payoffs and easily observable outcomes, sports is a natural setting for such tests. In this paper, we test the predictions of tournament theory using a unique race-level data set from NASCAR. Most previous tests of tournament theory using NASCAR data used either season level data or race level data from a few seasons. Our empirical work uses race and driver level NASCAR data for 1114 races over the period 1975–2009. Our results support the predictions of tournament theory: the larger the spread in prizes paid in the race, measured by the standard deviation or interquartile range of prizes paid, the higher the average speed in the race. Our results account for the length of the track, number of entrants, number of caution flags, and unobservable year- and week-level heterogeneity.

On the Unavoidability of Oriented Trees

A digraph is n-unavoidable if it is contained in every tournament of order n. We first prove that every arborescence of order n with k leaves is (n + k − 1)-unavoidable. We then prove that every oriented tree of order n with k leaves is (32n+32k−2)-unavoidable and (92n−52k−92)-unavoidable, and thus (218(n−1))-unavoidable. Finally, we prove that every oriented tree of order n with k leaves is (n+144k2−280k+124)-unavoidable.

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.

Subdivisions in Digraphs of Large Out-Degree or Large Dichromatic Number

In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the existence of $f(5)$ remains unknown. In this paper, we show that if $D$ is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from $x$ to $y$ and a directed path from $y$ to $x$, then every digraph with minimum out-degree large enough contains a subdivision of $D$. Additionally, we study Mader's conjecture considering another graph parameter. The dichromatic number of a digraph $D$ is the smallest integer $k$ such that $D$ can be partitioned into $k$ acyclic subdigraphs. We show that any digraph with dichromatic number greater than $4^m (n-1)$ contains every digraph with $n$ vertices and $m$ arcs as a subdivision. We show that any digraph with dichromatic number greater than $4^m (n-1)$ contains every digraph with $n$ vertices and $m$ arcs as a subdivision.

CEO power and labor productivity

PurposeThis paper aims to examine how Chief Executive Officer (CEO) power affects firm-level labor productivity.Design/methodology/approachThe authors rely on regression analysis to examine the relation between CEO power and labor productivity.FindingsFollowing prior research (i.e. the sequential rank order tournament theory), the authors predict that powerful CEOs lead to high labor productivity. They find a significant and positive relationship between CEO power and labor productivity. They further decompose labor productivity into labor efficiency and labor cost components and find a positive (negative) relationship between CEO power and labor efficiency (cost) component, suggesting that more powerful CEOs better manage labor efficiency and control labor cost. The results are also robust to various additional tests.Originality/valueThis study contributes to two streams of research: the CEO power literature in finance and the labor productivity and cost literature in accounting. To the best of the authors’ knowledge, it is the first study that performs a direct empirical test on the relation between CEO power and labor productivity.

On Min–Max Pair in Tournaments

Let T be a tournament of order \(n\ge 3\). A pair of distinct vertices x, y of T is called a min–max pair if one of x and y is of minimum out-degree, while the other is of maximum out-degree. Let xy be an arc such that x, y is a min–max pair. We call xy a min–max arc if x has minimum out-degree, and max–min arc otherwise. We prove that if yx is a min–max arc, then there exists a hamiltonian path from x to y; if xy is a max–min arc, then there exists a hamiltonian path from x to y with the exception of a few cases.

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.

χ‐bounded families of oriented graphs

AbstractA famous conjecture of Gyárfás and Sumner states for any tree T and integer k, if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star S and integer k, if the chromatic number of a digraph is large enough, either the digraph contains a clique of size k or it contains S as an induced subgraph. As an evidence, we prove that for any oriented star S, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order 3 or S as an induced subdigraph. We then study for which sets of orientations of P4 (the path on four vertices) similar statements hold. We establish some positive and negative results.

Arc-disjoint hamiltonian paths in non-round decomposable local tournaments

Thomassen proved that a strong tournament T has a pair of arc-disjoint Hamiltonian paths with distinct initial vertices and distinct terminal vertices if and only if T is not an almost transitive tournament of odd order, where an almost transitive tournament is obtained from a transitive tournament with acyclic ordering u1,u2,…,un (i.e., ui→uj for all 1≤i<j≤n) by reversing the arc u1un. A digraph D is a local tournament if for every vertex x of D, both the out-neighbors and the in-neighbors of x induce tournaments. Bang-Jensen, Guo, Gutin and Volkmann split local tournaments into three subclasses: the round decomposable; the non-round decomposable which are not tournaments; the non-round decomposable which are tournaments. In 2015, we proved that every 2-strong round decomposable local tournament has a Hamiltonian path and a Hamiltonian cycle which are arc-disjoint if and only if it is not the second power of an even cycle. In this paper, we discuss the arc-disjoint Hamiltonian paths in non-round decomposable local tournaments, and prove that every 2-strong non-round decomposable local tournament contains a pair of arc-disjoint Hamiltonian paths with distinct initial vertices and distinct terminal vertices. This result combining with the one on round decomposable local tournaments extends the above-mentioned result of Thomassen to 2-strong local tournaments.

On a generalisation of Mantel’s Theorem to Uniformly Dense Hypergraphs

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem in extremal combinatorics. While for $k=2$ this problem is well understood, due to the work of Tur\'an and of Erd\H{o}s and Stone, only very little is known for $k$-uniform hypergraphs for $k>2$. We focus on the case when $F$ is a $k$-uniform hypergraph with three edges on $k+1$ vertices. Already this very innocent (and maybe somewhat particular looking) problem is still wide open even for $k=3$. We consider a variant of the problem where the large hypergraph $H$ enjoys additional hereditary density conditions. Questions of this type were suggested by Erd\H os and S\'os about 30 years ago. We show that every $k$-uniform hypergraph $H$ with density $>2^{1-k}$ with respect to every large collections of $k$-cliques induced by sets of $(k-2)$-tuples contains a copy of $F$. The required density $2^{1-k}$ is best possible as higher order tournament constructions show. Our result can be viewed as a common generalisation of the first extremal result in graph theory due to Mantel (when $k=2$ and the hereditary density condition reduces to a normal density condition) and a recent result of Glebov, Kr\'al', and Volec (when $k=3$ and large subsets of vertices of $H$ induce a subhypergraph of density $>1/4$). Our proof for arbitrary $k\geq 2$ utilises the regularity method for hypergraphs.