Nesting directed cycle systems of even length

Nesting of cycle systems of odd length

On α-resolvable directed cycle systems for cycle length 4

A directedm-cycle system of order v with index λ, denotedm-DCS(v, λ), is a collection of directed cycles of length m whose directed edges partition the directed edges of λDKv. An m-DCS(v, λ) is α-resolvable if its directed cycles can be partitioned into classes such that each point of the design occurs in precisely α cycles in each class. The necessary conditions for the existence of such a des...

A small embedding for large directed even cycle systems

A directed m-cycle system is a collection of directed edges of the form {(XbX2), (X2' X3)(X3, X4), ... , (Xm-l, xm), (xm' xd}, where Xl, X2, X3, ... , Xm are distinct. A partial directed m-cycle system of order n is a pair (8, C), where C is a collection of edge disjoint m-cycles of the complete directed graph Dn with vertex set S. If the cycles in C partition the edge set of Dn we have the def...

The Even Cycle Problem for Directed Graphs

If each arc in a strongly connected directed graph of minimum in-degree and outdegree at least 3 is assigned a weight 0 or 1, then the resultingweighted directed graph has a directed cycle of even total weight. This provesa conjecture made by L. Lovasz in 1975 and has applications to colour-criticalhypergraphs, sign-nonsingular matrices, and permanents of matrices. MATHEMATICAL ...

Nesting kite and 4-cycle systems