THE NUMBER OF PANCYCLIC ARCS CONTAINED IN A HAMILTONIAN CYCLE OF A TOURNAMENT

A tournament T is an orientation of a complete graph and an arc in T is called pancyclic if it is contained in a cycle of length l for all $3{\leq}l{\leq}n$, where n is the cardinality of the vertex set of T. In 1994, Moon [5] introduced the graph parameter h(T) as the maximum number of pancyclic arcs contained in the same Hamiltonian cycle of T and showed that $h(T){\geq}3$ for all strong tournaments with $n{\geq}3$. Havet [4] later conjectured that $h(T){\geq}2k+1$ for all k-strong tournaments and proved the case k = 2. In 2005, Yeo [7] gave the lower bound $h(T){\geq}\frac{k+5}{2}$ for all k-strong tournaments T. In this note, we will improve his bound to $h(T){\geq}\frac{2k+7}{3}$.