  A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any
$r$-edge-coloured complete graph has a partition into $O(r^2 \log r)$
monochromatic cycles. Here we determine the minimum degree threshold for this
property. More precisely, we show that there exists a constant $c$ such that
any $r$-edge-coloured graph on $n$ vertices with minimum degree at least $n/2 +
c \cdot r \log n$ has a partition into $O(r^2)$ monochromatic cycles. We also
provide constructions showing that the minimum degree condition and the number
of cycles are essentially tight.
