On odd rainbow cycles in edge-colored graphs

Abstract

Let G=(V,E) be an n-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 prove that the same hypothesis ensures a rainbow ℓ-cycle Cℓ whenever n≥432ℓ. This result is sharp for all odd integers ℓ≥3, and extends earlier work of the authors for when ℓ is even.