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

Let D be a semicomplete multipartite digraph, with partite sets (called color classes) V 1 ; V 2 ; : : :; V c , such that jV 1 j jV 2 j : : : jV c j. Deene f(D) = jV (D)j ? 3jV c j + 1 and g(D) = jV (D)j?jVc?1j?2jVcj+2 2. We deene the irregularity i(D) of D to be maxjd + (x) ? d ? (y)j over all vertices x and y of D (possibly x = y). We deene the local irregularity i l (D) of D to be maxjd + (x)?d ? (x)j over all vertices x of D and we deene the global irregularity of D to be i g (D) = maxfd + (x); d ? (x) : x 2 V (D)g ? minfd + (y); d ? (y) : y 2 V (D)g. In this paper we show that if i g (D) g(D) or if i l (D) minff(D); g(D)g 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 5]. Our result also gives support to the conjecture from 5] that all diregular c-partite tournaments (c 4) are pancyclic. Finally we show that our result in some sense is best possible, by giving an innnite class of non-Hamiltonian semicomplete multipartite digraphs, D, with i g (D) = i(D) = i l (D) = g(D) + 1 2 f(D) + 1.