Properties Of Tournaments Research Articles

Cycles in [formula omitted]-traceable oriented graphs

A digraph of order at least k is termed k - traceable if each of its subdigraphs of order k is traceable. It turns out that several properties of tournaments—i.e., the 2-traceable oriented graphs—extend to k -traceable oriented graphs for small values of k . For instance, the authors together with O. Oellermann have recently shown that for k = 2 , 3 , 4 , 5 , 6 , all k -traceable oriented graphs are traceable. Moon [J.W. Moon, On subtournaments of a tournament, Canad. Math. Bull. 9(3) (1966) 297–301] observed that every nontrivial strong tournament T is vertex-pancyclic—i.e., through each vertex there is a cycle of every length from 3 up to the order of T . The present paper reports results pertaining to various cycle properties of strong k -traceable oriented graphs and explores the extent to which pancyclicity is retained by strong k -traceable oriented graphs. For each k ≥ 2 there are infinitely many k -traceable oriented graphs—e.g. tournaments. However, we establish an upper bound (linear in k ) on the order of k -traceable oriented graphs having a strong component with girth greater than 3. As an application of our findings, we show that the Path Partition Conjecture holds for 1-deficient oriented graphs having a strong component with girth at least 6. (A digraph is 1-deficient if its order is exactly one more than the order of its longest paths.)

Properties of Tournaments Among Well-Matched Players

1. TOURNAMENTS. In an n-player round robin tournament, each player plays one match against each of the other n - 1 players. The win-loss outcomes of these matches can be conveniently recorded in a tournament matrix A = [aij] as follows: First label the players in any order as 1, 2,.. ., n. For each pair i and j, set ai1 = 1 if player i defeats player j, and set aij = 0 otherwise. If i j j, then exactly one of aij and aji is nonzero; when i = j, aii = 0. What properties of the matrix A are related to the strengths of the players? The simplest measure of strength is the number of matches that the player wins, and the row sums of A count the number of matches won by each player. We are interested in understanding tournaments among players who are well matched in the sense that each player wins about half of the matches played. If the number of players is odd, many properties of A are very well understood. If the number of players is even, however, the properties of A are far less well understood. Indeed, there are many easily stated questions that lead to hard, open problems. Some of these problems are the focus of this paper. We let I denote the identity matrix, we let J denote the square matrix all of whose entries are ones, and we let e denote the column vector whose entries are all ones. A matrix A whose entries are zeros and ones is a tournament matrix exactly when

Properties of Tournaments among Well-Matched Players

Properties of Tournaments among Well-Matched Players