Circuits of Each Length in Tournaments

A tournament is a directed graph whose underlying graph is a complete graph. A circuit is an alternating sequence of vertices and arcs of the form v 1, a 1, v 2, a 2, v 3, . . . , v n-1, a n-1, v n in which vertex v n = v 1, arc a i = v i v i+1 for i = 1, 2, . . . , n?1, and $${a_i \neq a_j}$$ a i ? a j if $${i \neq j}$$ i ? j . In this paper, we shall show that every tournament T n in a subclass of tournaments has a circuit of each length k for $${3 \leqslant k \leqslant \theta(T_n)}$$ 3 ? k ? ? ( T n ) , where $${\theta(T_n) = \frac{n(n-1)}{2}-3}$$ ? ( T n ) = n ( n - 1 ) 2 - 3 if n is odd and $${\theta(T_n) = \frac{n(n-1)}{2}-\frac{n}{2}}$$ ? ( T n ) = n ( n - 1 ) 2 - n 2 otherwise. Note that a graph having ?(G) > n can be used as a host graph on embedding cycles with lengths larger than n to it if congestions are allowed only on vertices.