3-平均边着色中的单色圈划分
Monochromatic cycle partitions in 3-mean edge-colourings
AI总结：
针对Conlon与Stein提出的边着色完全图划分为有限个单色圈的问题，证明顶点关联颜色平均数至多为3时该命题成立，解决了r=3的首个公开情况。
AI中文摘要：
给定自然数r及边着色完全图，若顶点关联颜色的平均数至多为r，Conlon与Stein曾提出是否存在划分为有限个单色圈的顶点划分，并证明r=2时成立；我们解决首个公开问题，证明r=3时对应命题成立。
英文摘要：
Given $r \in \mathbb{N}$ and an edge-coloured complete graph such that the average number of colours incident with a vertex is at most $r$, Conlon and Stein asked whether there is a vertex partition into a bounded number of monochromatic cycles, and proved this for $r=2$. We settle the first open case by proving the corresponding statement for $r=3$.
