Abstract
We discuss the complexity of finding a cycle of even length in a digraph. In particular, we observe that finding a cycle of prescribed parity through a prescribed edge is NP-complete. Also, we settle a problem of Lovász [11] and disprove a conjecture of Seymour [15] by describing, for each natural number k, a digraph of minimum outdegree k and with no even cycle. We prove that a digraph of order n and minimum outdegree [log2n] + 1 contains, for each edge set E, a cycle containing an even number of edges of E and we show that this is best possible. A modification of the construction yields counterexamples to Hamidoune's conjecture [5] on local connectivity in digraphs of large indegrees and outdegrees.
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