Circuits of Each Length in Tournaments

Abstract

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 ${a_i \neq a_j}$ if ${i \neq j}$. In this paper, we shall show that every tournament T _n in a subclass of tournaments has a circuit of each length k for ${3 \leqslant k \leqslant \theta(T_n)}$, where ${\theta(T_n) = \frac{n(n-1)}{2}-3}$ if n is odd and ${\theta(T_n) = \frac{n(n-1)}{2}-\frac{n}{2}}$ otherwise. Note that a graph having θ(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.