Hamiltonian Cycles Avoiding Prescribed Arcs in Tournaments

Abstract

Thomassen [6] conjectured that if I is a set of k−1 arcs in a k-strong tournament T, then T−I has a Hamiltonian cycle. This conjecture was proved by Fraisse and Thomassen [3]. We prove the following stronger result. Let T=(V, A) be a k-strong tournament on n vertices and let X1, X2, [ctdot ], Xl be a partition of the vertex set V of T such that [mid ]X1[mid ][les ][mid ]X2[mid ] [les ][ctdot ][les ][mid ]Xl[mid ]. If k[ges ][sum ] l−1i=1[lfloor ] [mid ]Xi[mid ]/2[rfloor ]+[mid ]Xl[mid ], then T−∪li=1 {xy∈A[ratio ]x, y∈Xi} has a Hamiltonian cycle. The bound on k is sharp.