Vertices x and y dominate a tournament T if for all vertices z 6 = x; y; either x beats z or y beats z. Let dom(T) be the graph on the vertices of T with edges between pairs of vertices that dominate T. We show dom(T) is either an odd cycle with possible pendant vertices or a forest of caterpillars. Since dom(T) is the complement of the competition graph of the tournament formed by reversing th...

A tournament is a graph in which each pair of distinct vertices is connected by exactly one directed edge. Tournaments are an important graph class, for which isomorphism testing seems to be easier to compute than for the isomorphism problem of general graphs. We show that tournament isomorphism and tournament automorphism is hard under DLOGTIME uniform AC many-one reductions for the complexity...

The Brazilian soccer tournament is organized every year by the Brazilian Soccer Confederation. Its major sponsor is TV Globo, the largest media group and television network in Brazil, who imposes constraints on the games to be broadcast. Scheduling the games of this tournament is a very constrained problem, with two objectives: breaks minimization (fairness) and the maximization of the revenues...

We call the tournament T an m-coloured tournament if the arcs of T are coloured with m colours. In this paper we prove that for each n¿ 6, there exists a 4-coloured tournament Tn of order n satisfying the two following conditions: (1) Tn does not contain C3 (the directed cycle of length 3, whose arcs are coloured with three distinct colours), and (2) Tn does not contain any vertex v such that f...

Abstract We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and Fox Sudakov. First, t ≥ 3, we show that any two-colouring of the complete graph on n vertices is δ -far from being monochromatic contains an unavoidable t-colouring when ≫ −1/ , where -colouring a clique order 2 in which one colour forms either or disjoint cliques . Next, tournament transitive t-tournam...

This paper explores a novel way for analyzing the tournament structures to find a best suitable one for the tournament under consideration. It concerns about three aspects such as tournament conducting cost, competitiveness development and ranking precision. It then proposes a new method using progress tree to detect potential throwaway matches. The analysis performed using the proposed method ...

u. and u . is joined by one of the (oriented) arcs u .u . or u.u . . 1 J 13 J 1 The arcs in some set S are said to be consistent if it is possible to relabel the nodes of the tournament in such a way that if the arc u.u . is in S then i > j . (This is easily seen to be equiva1 3 lent to requiring that the tournament contains no oriented cycles composed entirely of arcs of S . ) Sets of consiste...

Let T be a 3-partite tournament. We say that a vertex v is −→ C3 -free if v does not lie on any directed triangle of T . Let F3(T ) be the set of the −→ C3 -free vertices in a 3-partite tournament and f3(T ) its cardinality. In this paper we prove that if T is a regular 3-partite tournament, then F3(T )must be contained in one of the partite sets of T . It is also shown that for every regular 3...

By a tournament we mean the outcome of a round-robin tournament in which there are no draws . Such a tournament may be represented by a graph in which the n players are represented by vertices labelled 1, 2, . . . , n, and the outcomes of the games are represented by directed edges so that every pair of vertices is joined by one directed edge . We call such a graph a complete directed graph . O...