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(r2logr)
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⋅rlogn has a partition into O(r2) monochromatic cycles. We also
provide constructions showing that the minimum degree condition and the number
of cycles are essentially tight.Comment: 22 pages (26 including appendix