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

The global irregularity of a digraph D is defined by i_g (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 | V (D) | > 476 i_g (D) + 13 917 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 i_g 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.