Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper, the following question is studied: What is the maximum intersection with γ of a directed cycle of length k? It is proved that for an even k in the range 4 ≤ k ≤ [(n+4)/2], there exists a directed cycle $C_{h\left(k\right)}$ of length h(k), h(k) ∈ k,k-2 with $|A\left(C_{h\left(k\right)}\right)\cap A\left(\gamma \right)|\ge h\left(k\right)-3$ and the result is best possible. In a forthcoming paper the case of directed cycles of length k, k even and k < [(n+4)/2]...

Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper the following question is studied: What is the maximum intersection with γ of a directed cycle of length k contained in T[V(γ)]? It is proved that for an even k in the range (n+6)/2 ≤ k ≤ n-2, there exists a directed cycle $C_{h\left(k\right)}$ of length h(k), h(k) ∈ k,k-2 with $|A\left(C_{h\left(k\right)}\right)\cap A\left(\gamma \right)|\ge h\left(k\right)-4$ and the result is best possible. In a previous paper a similar result for 4 ≤ k ≤ (n+4)/2 was proved.

If $x$ is a vertex of a digraph $D$, then we denote by $d^{+}\left(x\right)$ and $d^{-}\left(x\right)$ the outdegree and the indegree of $x$, respectively. A digraph $D$ is called regular, if there is a number $p\in \mathbb{N}$ such that $d^{+}\left(x\right)=d^{-}\left(x\right)=p$ for all vertices $x$ of $D$. A $c$-partite tournament is an orientation of a complete $c$-partite graph. There are many results about directed cycles of a given length or of directed cycles with vertices from a given number of partite sets. The idea is now to combine the two properties. In this article, we examine in particular, whether...

Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k,l)-kernel N of D is a k-independent set of vertices (if u,v ∈ N then d(u,v) ≥ k) and l-absorbent (if u ∈ V(D)-N then there exists v ∈ N such that d(u,v) ≤ l). A k-kernel is a (k,k-1)-kernel. A digraph D is cyclically k-partite if there exists a partition ${V_i}_{i=0}^{k-1}$ of V(D) such that every arc in D is a $V_i{V}_{i+1}-arc$ (mod k). We give a characterization for an unilateral digraph to be cyclically k-partite through the lengths...