AI 中文总结
本文以超线性形式否定Gyárfás的单色路径划分猜想，证明足够大r时r边染色完全图需至少(1-o(1))rloglogr个单色路径，其构造还扩展到平衡二分图场景，否定了Pokrovskiy的相关猜想。
AI 中文摘要
1989年，Gyárfás提出猜想：每一个r边染色完全图的顶点集可划分为至多r个顶点不相交的单色路径。随后Erdős、Gyárfás和Pyber提出了关于单色环的类似猜想。Pokrovskiy证明了r=3时Gyárfás的猜想成立，但通过构造需要至少r+1个单色环的染色，否定了r≥3时Erdős、Gyárfás和Pyber的猜想。本文以定量性强的超线性形式否定了Gyárfás的猜想：对每个足够大的r，存在一个r边染色完全图，其需要至少(1-o(1))rloglogr个顶点不相交的单色路径。因此，单色环划分数也随r呈超线性增长。本文的构造还可扩展到平衡二分图场景，否定了Pokrovskiy的一个猜想。
英文摘要
In 1989, Gyárfás conjectured that the vertex set of every $r$-edge-coloured complete graph can be partitioned into at most $r$ vertex-disjoint monochromatic paths. Erdős, Gyárfás, and Pyber subsequently proposed the analogous conjecture for monochromatic cycles. Pokrovskiy proved Gyárfás's conjecture for $r=3$, while disproving the conjecture of Erdős, Gyárfás, and Pyber for every $r\ge3$ by constructing colourings that require at least $r+1$ monochromatic cycles. In this paper, we disprove Gyárfás's conjecture in a quantitatively strong superlinear form: for every sufficiently large $r$, there exists an $r$-edge-coloured complete graph that requires at least $(1-o(1))r\log\log r$ vertex-disjoint monochromatic paths. Consequently, the monochromatic cycle-partition number is also superlinear in $r$. Our construction also disproves two conjectures of Pokrovskiy: one on monochromatic cycle coverings and the other on path coverings in the balanced bipartite setting.
Comments17 pages; added new results
