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 v1, a1, v2, a2, v3, . . . , vn-1, an-1, vn in which vertex vn = v1, arc ai = vivi+1 for i = 1, 2, . . . , n−1, and ai≠aj\documentclass[12pt]{minimal}
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\begin{document}$${a_i \neq a_j}$$\end{document} if i≠j\documentclass[12pt]{minimal}
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\begin{document}$${i \neq j}$$\end{document}. In this paper, we shall show that every tournament Tn in a subclass of tournaments has a circuit of each length k for 3⩽k⩽θ(Tn)\documentclass[12pt]{minimal}
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\begin{document}$${3 \leqslant k \leqslant \theta(T_n)}$$\end{document}, where θ(Tn)=n(n-1)2-3\documentclass[12pt]{minimal}
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\begin{document}$${\theta(T_n) = \frac{n(n-1)}{2}-3}$$\end{document} if n is odd and θ(Tn)=n(n-1)2-n2\documentclass[12pt]{minimal}
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\begin{document}$${\theta(T_n) = \frac{n(n-1)}{2}-\frac{n}{2}}$$\end{document} 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.
