5 We prove that every tournament with minimum out-degree at least 2k− 1 contains k disjoint 6 3-cycles. This provides additional support for the conjecture by Bermond and Thomassen that 7 every digraph D of minimum out-degree 2k − 1 contains k vertex disjoint cycles. We also prove 8 that for every > 0, when k is large enough, every tournament with minimum out-degree at least 9 (1.5+ )k contains...

In this paper, we construct infinite families of tight regular tournaments. In particular, we prove that two classes of regular tournaments, tame molds and ample tournaments are tight. We exhibit an infinite family of 3-dichromatic tight tournaments. With this family we positively answer to one case of a conjecture posed by V. Neumann-Lara. Finally, we show that any tournament with a tight mold...

The well known state-space formulae for doubly coprime factorizations of transfer functions of stabilizable and detectable linear systems are extended to a large class of infinite-dimensional systems known as regular linear systems. Abstract-Mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system, as well as fo...

In the world of discrete mathematics, we encounter a bewildering variety of topics with no apparent connection between them. But appearances are deceptive. For example, combinatorics tells us about difference sets, block designs, and triple systems. Geometry leads us to finite projective planes and Latin squares. Graph theory introduces us to round–robin tournaments and map colorings, linear al...

This design, which we call the (7, 3, 1) design, makes appearances in many areas of mathematics. It seems to turn up again and again in unexpected places. An earlier paper in this MAGAZINE [4] described (7, 3, 1)’s appearance in a number of different areas, including finite projective planes, as the Fano plane (FIGURE 1); graph theory, as the Heawood graph and the doubly regular round-robin tou...

Thomassen (J. Combin. Theory Ser. B 28, 1980, 142–163) proved that every strong tournament contains a vertex x such that each arc going out from x is contained in a Hamiltonian cycle. In this paper, we extend the result of Thomassen and prove that a strong tournament contains a vertex x such that every arc going out from x is pancyclic, and our proof yields a polynomial algorithm to nd such a v...

In this paper we prove that if T is a regular n-partite tournament with n ≥ 4, then each arc of T lies on a cycle whose vertices are from exactly k partite sets for k = 4, 5, . . . , n. Our result, in a sense, generalizes a theorem due to Alspach.