On 5‐Cycles and 6‐Cycles in Regular n‐Tournaments

Abstract

Let T be a tournament of order n and c m (T ) be the number of cycles of length m in T. For m = 3 and odd n, the maximum of c m (T ) is achieved for any regular tournament of order n (M. G. Kendall and B. Babington Smith, 1940), and in the case m = 4, it is attained only for the unique regular locally transitive tournament RLT n of order n (U. Colombo, 1964). A lower bound was also obtained for c 4 (T ) in the class R n of regular tournaments of order n (A. Kotzig, 1968). This bound is attained if and only if T is doubly regular (when n ≡ 3 mod 4) or nearly doubly regular (when n ≡ 1 mod 4) (B. Alspach and C. Tabib, 1982). In the present article, we show that for any regular tournament T of order n, the equality 2c 4 (T ) + c 5 (T ) = n(n − 1)(n + 1)(n − 3)(n + 3)/160 holds. This allows us to reduce the case m = 5 to the case m = 4. In turn, the pure spectral expression for c 6 (T ) obtained in the class R n implies that for a regular tournament T of order n ≥ 7, the inequality c 6 (T ) ≤ n(n − 1)(n + 1)(n − 3)(n 2 − 6n + 3)/384 holds, with equality if and only if T is doubly regular or T is the unique regular tournament of order 7 that is neither doubly regular nor locally transitive. We also determine the value of c 6 (RLT n ) and conjecture that this value coincides with the minimum number of 6-cycles in the class R n for each odd n ≥ 7.

The Shifted Turán Sieve Method on Tournaments

“…For example, in [17], the number of 6-cycles in a regular tournament of order n is bounded by (n + 1)n(n − 1)(n − 3)(n 2 − 6n + 3)/384, yet the average number of 6-cycles in Corollary 2.2 is (n + 1)n(n − 1)(n − 3)(n 2 − 6n + 8)/384. Thus, the bound in [17] is very close to the average. The upper bounds given by Theorems 1.4 and 1.5 are not always tight.…”

On the Number of 5‐Cycles in a Tournament

“…David Berman [3,4], maximized the number of 5-cycles in a narrow family of tournaments (specifically, those that are "semi-transitive"). More recently, Savchenko [16] has established bounds on 5-cycles and 6-cycles in regular tournaments. Computing the number of 5-cycles in a general n-tournament has remained elusive, however.…”