Almost all almost regular c-partite tournaments with c ≥ 5 are vertex pancyclic

A tournament is an orientation of a complete graph and a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If D is a digraph, then let d⁺(x) be the outdegree and d^-(x) the indegree of the vertex x in D. The minimum (maximum) outdegree and the minimum (maximum) indegree of D are denoted by δ⁺ (Δ⁺) and δ^- (Δ^-), respectively. In addition, we define δ = min(δ⁺,δ^-) and Δ =max(Δ⁺,Δ^-). A digraph is regular when δ = Δ and almost regular when Δ - δ 1. Recently, the third author proved that all regular c-partite tournaments are vertex pancyclic when c ≥ 5, and that all, except possibly a finite number, regular 4-partite tournaments are vertex pancyclic. Clearly, in a regular multipartite tournament, each partite set has the same cardinality. As a supplement of Yeo's result we prove first that an almost regular c-partite tournament with c ≥ 5 is vertex pancyclic, if all partite sets have the same cardinality. Second, we show that all almost regular c-partite tournaments are vertex pancyclic when c ≥ 8, and third that all, except possibly a finite number, almost regular c-partite tournaments are vertex pancyclic when c ≥ 5.