Complementary Cycles in Almost Regular Multipartite Tournaments

Abstract

A \(c\)-partite or multipartite tournament is an orientation of a complete \(c\)-partite graph. A digraph \(D\) is cycle complementary if there exist two vertex-disjoint directed cycles \(C\) and \(C’\) such that \(V(D) = V(C) \cup V(C’)\). The global irregularity of a digraph \(D\) is defined by
\[i_g(D) = \max\{\max(d^+(x), d^-(x)) – \min(d^+(y),d^-(y)) \mid x,y \in V(D)\}.\]
If \(i_g(D) = 0\), then \(D\) is regular, and if \(i_g(D) \leq 1\), then \(D\) is almost regular. We prove in this paper that every almost regular \(c\)-partite tournament with \(c \geq 3\) such that all partite sets have the same cardinality \(r \geq 4\) contains two complementary directed cycles of length \(3\) and \(|V(D)| – 3\).