First, some definitions. A tournament is regular of degree k if each point has indegree k and outdegree k: clearly such a tournament has 2k +1 points. The trivial tournament has just one point. A tournament T is doubly regular with subdegree t if it is non-trivial and any two points of T jointly dominate precisely t points; equivalently if T is non-trivial and for each point v of T, the subtour...

In this thesis, we investigate several properties of k-tournaments, where k 2 3. These properties fall into three broad areas. The first contains properties related to the ranking of the participants in a k-tournament, including a representation theorem for posets. The second contains properties related to the representation of a finite group as the automorphism group of a k-tournament, with va...

‎in this paper‎, ‎the type-{rm ii} matrices on (negative) latin square graphs are considered and it is proved that‎, ‎under‎ ‎certain conditions‎, ‎the nomura algebras of such type-{rm ii} matrices are trivial‎. ‎in addition‎, ‎we construct type-{rm ii} matrices‎ ‎on doubly regular tournaments and show that the nomura algebras of such matrices are also trivial‎.

We show that every sufficiently large regular tournament can almost completely be decomposed into edge-disjoint Hamilton cycles. More precisely, for each η > 0 every regular tournament G of sufficiently large order n contains at least (1/2 − η)n edge-disjoint Hamilton cycles. This gives an approximate solution to a conjecture of Kelly from 1968. Our result also extends to almost regular tournam...

A tournament is an orientation of a complete graph and a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If D is a digraph, then let d + (x) be the outdgree and d ? (x) the indegree of the vertex x in D. The minimum (maximum) out-degree and the minimum (maximum) indegree of D are denoted by + ((+) and ? ((?), respectively. In addition, we deene = minf + ; ?...

We show that every sufficiently large regular tournament can almost completely be decomposed into edge-disjoint Hamilton cycles. More precisely, for each η > 0 every regular tournament G of sufficiently large order n contains at least (1/2− η)n edge-disjoint Hamilton cycles. This gives an approximate solution to a conjecture of Kelly from 1968. Our result also extends to almost regular tourname...

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 a recent paper, it was proved that if T is a regular 3-partite tournament, then f3(T ) < n 9 , where n is the order of T . In this paper, we prove that f3(T ) ≤ n 12 . We ...

If x is a vertex of a digraph D, then we denote by d(x) and d−(x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined by ig(D) = max{d+(x), d−(x)}−min{d+(y), d−(y)} over all vertices x and y of D (including x = y). If ig(D) = 0, then D is regular and if ig(D) ≤ 1, then D is almost regular. A c-partite tournament is an orientation of a complete c-...