| The cyclic chromatic number is the smallest number of colours needed to colour the vertices of a tournament so that no cyclic triple is monochromatic. Bagga, Beineke, and Harary (2) conjectured that every tournament score vector belongs to a tournament with cyclic chromatic number 1 or 2. We prove this conjecture and derive other related results. We also develop the upper and lower bounds of the largest cyclic chromatic number among all n-tournaments, improving those in (2).; A digraph D is primitive if for some positive integer k, there is a walk from any vertex to any vertex of length exactly k. The minimum such k is called the exponent of D. Brualdi and Liu (7) introduced the generalized exponents for primitive matrices, and Liu discussed in (23) the generalized exponents of tournament matrices. We discuss the generalized exponents of primitive matrices, in particular, the generalized exponents of primitive tournaments, and thereby derive a scheme for ranking players in a tournament by using the generalized exponents.; Perron values of primitive tournaments are another aspect of our studies. We try to find some relationships between the exponents and Perron values. We discuss the exponents and Perron values of a special family of tournaments, those tournaments with score vector (1, 1, 2, ..., {dollar}n - 3, n - 2, n - 2),{dollar} and conjecture that there is an inverse relationship between the exponents and Perron values of tournaments. We also study those upper bounds and limits of Perron values of some special tournament sequences. |
