论文标题
Aharoni的彩虹循环猜想达到了加性常数
Aharoni's rainbow cycle conjecture holds up to an additive constant
论文作者
论文摘要
在2017年，Aharoni提出了Caccetta-Häggkvist的以下概括：如果$ G $是一个简单的$ n $ n $ n $ vertex edgex edge-edge-edge-edge-edge-enge-edge the Gragh，the $ n $ color类的大小至少$ r $，那么$ g $，那么$ g $最多包含$ \ lceil n/r \ lceil n/r \ rceil $。 在本文中，我们证明，对于修复了$ r $，Aharoni的猜想达到了增材常数。具体来说，我们表明，对于每个固定$ r \ geq 1 $，存在一个常数$α_r\ in O（r^5 \ log^2 r）$，这样，如果$ g $是一个简单的$ n $ n $ n $ vertex edge-vertex edge-endex edge-vertex edge-vertex edge-vertex edge-vertex edge the $ n $ n $ color of a $ color of color of color of $ r $，那么$ g $，则$ g $最多包含$ n/r n/r r + a的彩虹周期。
In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture: if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $\lceil n/r \rceil$. In this paper, we prove that, for fixed $r$, Aharoni's conjecture holds up to an additive constant. Specifically, we show that for each fixed $r \geq 1$, there exists a constant $α_r \in O(r^5 \log^2 r)$ such that if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $n/r + α_r$.
