Hamilton cycles, avoiding prescribed arcs, in close-to-regular tournaments

Abstract

The local irregularity of a digraph D is defined as il(D) = max {|d+ (x) − d− (x)| : x ϵ V(D)}. Let T be a tournament, let Γ = {V1, V2, …, Vc} be a partition of V(T) such that |V1| ≥ |V2| ≥ … ≥ |Vc|, and let D be the multipartite tournament obtained by deleting all the arcs with both end points in the same set in Γ. We prove that, if |V(T)| ≥ max{2il(T) + 2|V1| + 2|V2| − 2, il(T) + 3|V1| − 1}, then D is Hamiltonian. Furthermore, if T is regular (i.e., il(T) = 0), then we state slightly better lower bounds for |V(T)| such that we still can guarantee that D is Hamiltonian. Finally, we show that our results are best possible. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 123–136, 1999