On even rainbow or nontriangular directed cycles

Let G =(V,E) beann-vertex edge-colored graph. In 2013, H. Li proved that if every vertex v ∈ V is incident to at least (n +1)/2 distinctly colored edges, then G admits a rainbow triangle. We establish a corresponding result for fixed even rainbow ℓ-cycles C_ℓ: if every vertex v ∈ V is incident to at least (n +5)/3 distinctly colored edges, where n ≥ n₀(ℓ) is sufficiently large, then G admits an even rainbow ℓ-cycle C_ℓ. This result is best possible whenever ℓ ≢ 0 (mod 3). Correspondingly, we also show that for a fixed (even or odd) integer ℓ ≥ 4, every large n-vertex oriented graph ⃗G =(V,E)^⃗ with minimum outdegree at least (n +1)/3 admitsa (consistently) directed ℓ-cycle C^⃗_ℓ. Our latter result relates to one of Kelly, Kühn, and Osthus, who proved a similar statement for oriented graphs with large semi-degree.PAcwJQGWOHg5kjWrmX3lzXqkP56Our proofs are based on the stability method.