An edge-coloured version of Dirac's theorem

Let $G$ be an edge-coloured graph. The minimum colour degree ${\delta }^c(G)$ 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 every edge-coloured graph $G$ with ${\delta }^c(G)\ge 2\mathrm{\mid }G\mathrm{\mid }\mathrm{/}3$ contains a properly coloured $2$-factor. Furthermore, we show that for any $\epsilon >0$ there exists an integer $n_0$ such that every edge-coloured graph $G$ with $\mathrm{\mid }G\mathrm{\mid }=n\ge n_0$ and ${\delta }^c(G)\ge (2\mathrm{/}3+\epsilon )n$ contains a properly coloured cycle of length $\mathrm{\ell }$ for every $3 \le \ell \le n$\mathrm{.}Thisresultisbestpossibleinthesensethatthestatementisfalsefor$ \delta^c(G) < 2n / 3 $.