How Close to Regular Must a Semicomplete Multipartite Digraph Be to Secure Hamiltonicity?

Abstract.

 Let D be a semicomplete multipartite digraph, with partite sets V ₁, V ₂,…, V _c, such that |V ₁|≤|V ₂|≤…≤|V _c|. Define f(D)=|V(D)|−3|V _c|+1 and . We define the irregularity i(D) of D to be max|d ⁺(x)−d ⁻(y)| over all vertices x and y of D (possibly x=y). We define the local irregularity i _l(D) of D to be max|d ⁺(x)−d ⁻(x)| over all vertices x of D and we define the global irregularity of D to be i _g(D)=max{d ⁺(x),d ⁻(x) : x∈V(D)}−min{d ⁺(y),d ⁻(y) : y∈V(D)}. In this paper we show that if i _g(D)≤g(D) or if i _l(D)≤min{f(D), g(D)} then D is Hamiltonian. We furthermore show how this implies a theorem which generalizes two results by Volkmann and solves a stated problem and a conjecture from [6]. Our result also gives support to the conjecture from [6] that all diregular c-partite tournaments (c≥4) are pancyclic, and it is used in [9], which proves this conjecture for all c≥5. Finally we show that our result in some sense is best possible, by giving an infinite class of non-Hamiltonian semicomplete multipartite digraphs, D, with i _g(D)=i(D)=i _l(D)=g(D)+½≤f(D)+1.