Paths and cycles containing given arcs, in close to regular multipartite tournaments

The global irregularity of a digraph D is defined by ig(D)=max{d+(x),d−(x)}−min{d+(y),d−(y)} over all vertices x and y of D (including x=y). In this paper we prove that if D is a c-partite tournament such that c⩾4 and then there exists a path of length l between any two given vertices for all 42⩽l⩽|V(D)|−1. There are many consequences of this result. For example we show that all sufficiently large regular c-partite tournaments with c⩾4 have a Hamilton cycle through any given arc, and the condition c⩾4 is best possible. Sufficient conditions are furthermore given for when a c-partite tournament with c⩾4 has a Hamilton cycle containing a given path or a set of given arcs. We show that all sufficiently large c-partite tournaments with c⩾5 and bounded ig are vertex-pancyclic and all sufficiently large regular 4-partite tournaments are vertex-pancyclic. Finally we give a lower bound on the number of Hamilton cycles in a c-partite tournament with c⩾4.