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 $C^{\prime }$ such that $V(D)=V(C)\cup V(C^{\prime })$. 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 $i_g(D)\le 1$, then $D$ is almost regular. We prove in this paper that every almost regular $c$-partite tournament with $c\ge 3$ such that all partite sets have the same cardinality $r\ge 4$ contains two complementary directed cycles of length $3$ and $|V(D)|–3$.