Long properly coloured cycles in edge‐coloured graphs

Let G be an edge‐coloured graph. The minimum colour degree δ c ( G ) of G is the largest integer k such that, for every vertex v , there are at least k distinct colours on edges incident to v . We say that G is properly coloured if no two adjacent edges have the same colour. In this paper, we show that, for any ε > 0 and n large, every edge‐coloured graph G with δ c ( G ) ≥ ( 1 ∕ 2 + ε ) n contains a properly coloured cycle of length at least min { n , ⌊ 2 δ c ( G ) ∕ 3 ⌋ } .