A classical result of Erdős, Gyárfás 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.

中文翻译：

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单色循环分割的最小度条件

Erdős，Gyárfás和Pyber的经典结果表明，任何带有r边的完整图都有一个划分为$Ø（{\mathrm{[R}}^2\mathrm{日志}\mathrm{[R}）$单色循环。在这里，我们确定此属性的最小度阈值。更确切地说，我们证明存在一个常数c，使得n个顶点上的r边色图具有最小度至少$ñ/2+C\cdot \mathrm{[R}\mathrm{日志}ñ$ 有一个分区 $Ø（{\mathrm{[R}}^2）$单色循环。我们还提供了表明最小度条件和循环数基本紧密的构造。