On 4-cycles and 5-cycles in regular tournaments

Citation

Rowlinson P (1986) On 4-cycles and 5-cycles in regular tournaments. Bulletin of the London Mathematical Society, 18 (2), pp. 135-139. https://doi.org/10.1112/blms/18.2.135

Abstract
First paragraph: 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 subtournament Tv on the points dominated by v is regular of degree t. By counting arcs in Tv we see that v has outdegree 2t +1, and it follows that T is regular of degree 2t+1. Reid and Brown [4] have shown that the existence of a doubly regular tournament with subdegree t is equivalent to the existence of a skew-Hadamard matrix of order 4t+4. The simplest examples of doubly regular tournaments are provided by the quadratic residue tournaments QRp, where p is a prime congruent to 3 modulo 4: the points of QRp are the p elements of the field Zp, and u dominates v if and only if u - v is a square in Zp.

Journal
Bulletin of the London Mathematical Society: Volume 18, Issue 2