Abstract
A classical result of Erdős, Gyárfás 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.
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