Hamiltonian Cycles Avoiding Prescribed Arcs in Tournaments

In [6], Thomassen 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, . . . , Xl be a partition of the vertex set V of T such that |X1| ≤ |X2| ≤ . . . ≤ |Xl|. If k ≥ ∑l−1 i=1b|Xi|/2c + |Xl|, then T − ∪i=1{xy ∈ A : x, y ∈ Xi} has a Hamiltonian cycle. The bound on k is sharp.