Cycle-pancyclism in tournaments III

Abstract

LetT be a hamiltonian tournament withn vertices and γ a hamiltonian cycle ofT. For a cycleC _k of lengthk inT we denoteI _γ(C _k, the number of arcs that γ andC _k have in common. Letf(n,k,T,γ)=max{I _γ(C _k)|C _k ⊂T} andf(n, k)=min{f(n, k, T, γ)|T is a hamiltonian tournament withn vertices, and γ a hamiltonian cycle ofT}. In a previous paper [3] we studied the case ofn≥2k−4 and proved that

• .f(n, 3)=1, f(n, 4)=1 andf(n, 5)=2 ifn≠2k−2;

• .f(n, k)=k−1 if and only ifn=2k−2;

• . fork>5,f(n, k)=k−2 if and only ifn≥2k−4,n≠2k−2 andn≡k (modk−2);

• . fork>5,f(n, k)=k−3 if and only ifn≥2k−4 andn≢k (modk−2).

In this paper we consider the case ofn≤2k−5 and complete the description off(n, k) by proving thatf(n, k)=k−4 if and only ifn≤2k−5.