On 4‐Cycles and 5‐Cycles in Regular Tournaments

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 /. By counting arcs in Tv we see that v has outdegree 2t +1 , and it follows that Tis regular of degree 2t+1. Reid and Brown [4] have shown that the existence of a doubly regular tournament with subdegree / is equivalent to the existence of a skew-Hadamard matrix of order At+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 inZp. Let m be an integer ^ 3. We say that a tournament T has property &m if T is non-trivial and each arc of T lies in the same non-zero number of w-cycles. It is well-known (see [4]) that a tournament has property ^ 3 if and only if it is doubly regular. Moreover, as we shall see, if such a tournament has more than three points then it also has properties ^4 and &h. Although a tournament with property ^ or 8Ph is necessarily regular, neither property alone is strong enough to ensure double regularity. To see this consider first the 9-point tournament Z3[Z3], defined as follows: it has three point-disjoint 3-cycles Cv C2, C3 and each point of Q dominates each point of Q+ 1 (where subscripts are reduced modulo 3). It is straightforward to check that this tournament has property ^4 but not ^3. Secondly, the 5-point tournament which is the union of two arc-disjoint 5-cycles has property ^ 5 but not ^3. We prove that a tournament which has both property ^4 and property &h is doubly regular. Our methods are algebraic and serve also to show that a regular tournament is doubly regular if and only if the minimal polynomial of its adjacency matrix has degree 3. 2. Some preliminary results