Complementary Cycles in Irregular Multipartite Tournaments

A tournament is a directed graph obtained by assigning a direction for each edge in an undirected complete graph. A digraphDis cycle complementary if there exist two vertex disjoint cyclesCandC′such thatV(D)=V(C)∪V(C′). LetDbe a locally almost regularc-partite tournament withc≥3and|γ(D)|≤3such that all partite sets have the same cardinalityr, and letC3be a3-cycle ofD. In this paper, we prove that ifD-V(C3)has no cycle factor, thenDcontains a pair of disjoint cycles of length3and|V(D)|-3, unlessDis isomorphic toT7,D4,2,D4,2⁎, orD3,2.