Complementary cycles in irregular multipartite tournaments (Q1793296)

Summary: A tournament is a directed graph obtained by assigning a direction for each edge in an undirected complete graph. A digraph \(D\) is cycle complementary if there exist two vertex disjoint cycles \(C\) and \(C'\) such that \(V(D) = V(C) \cup V(C')\). Let \(D\) be a locally almost regular \(c\)-partite tournament with \(c \geq 3\) and \(| \gamma(D) | \leq 3\) such that all partite sets have the same cardinality \(r\), and let \(C_3\) be a \(3\)-cycle of \(D\). In this paper, we prove that if \(D - V(C_3)\) has no cycle factor, then \(D\) contains a pair of disjoint cycles of length \(3\) and \(| V(D) | - 3\), unless \(D\) is isomorphic to \(T_7\), \(D_{4,2}\), \(D_{4,2}^\ast\), or \(D_{3,2}\).