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 .