Cycles in 3-anti-circulant digraphs

A digraph D is a 3-anti-circulant digraph, if for any four distinct vertices x1, x2, x3, x4 ∈ V (D), x1 → x2 ← x3 → x4 implies x4 → x1. In this paper, we characterize the structure of 3-anti-circulant digraphs containing a cycle factor and show that the structure is very close to semicomplete and semicomplete bipartite digraphs. Laborde et al. conjectured that every digraph has an independent set intersecting every longest path. It has been shown that the conjecture is true for 3-anti-circulant digraphs. In this paper, we generalize the result to the longest cycle and prove that there exists an independent set intersecting every longest cycle for 3-anti-circulant digraphs.