On complementary cycles in locally semicomplete digraphs (Q1343256)

A digraph \(D\) is locally semicomplete if for each vertex \(x\) in \(D\) the subdigraphs induced by the positive and negative neighbor sets of \(x\) are both semicomplete. This paper gives the following result about 2- connected locally semicomplete digraphs: Such digraphs do not have their vertex set partitioned by two complementary dicycles if and only if they are 2-diregular and have odd order. From this theorem follow two conjectures of Bang-Jensen giving conditions for a 2-connected local tournament \(D\) to have a dicycle \(C\) such that \(D- V(C)\) is strong.