有向图的无环双着色：路径、随机锦标赛与临界阶
Acyclic Dicolourings of Oriented Graphs: Paths, Random Tournaments, and Critical Orders
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中文总结 AI 辅助
本文研究有向图的无环双着色，证明Gallai-Roy型上界，将随机锦标赛的无环双色数下界改进两倍，并确定最小阶$m(3)=5$、$m(4)=7$及其锦标赛见证者。
中文摘要 AI 辅助
有向图的无环双着色是一种顶点划分，其中每个颜色类以及由两个类诱导的每个二分有向子图都是无环的。我们首先证明Gallai-Roy型界$\vec{\chi}_a(D)\leq L(D)$，其中$L(D)$是有向路径的最大阶。设$r=\log_2(8/7)$。Bang-Jensen、Picasarri-Arrieta和Yeo先前构造了阶为$n$的锦标赛，其无环双色数至少为$n-(\frac8r)\log_2 n-\log_2\log_2 n$。我们将主导对数系数改进两倍：对于均匀随机锦标赛$\mathcal{T}_n$，渐近几乎必然地，$\vec{\chi}_a(\mathcal{T}_n)\geq n-\frac4r\log_2 n+\frac2r\log_2\log_2 n-O(1)$。最后，若$m(k)$表示无环双色数至少为$k$的有向图的最小阶，锦标赛完备化表明相同的最小值可在锦标赛上取得。我们证明$m(3)=5$和$m(4)=7$，并分类这些最小阶下的锦标赛见证者：在阶五时有一个，在阶七时有两个。
英文摘要
An acyclic dicolouring of an oriented graph is a vertex partition in which every colour class and every bipartite subdigraph induced by two classes is acyclic. We first prove the Gallai--Roy-type bound $\vecχ_a(D)\leq L(D)$, where $L(D)$ is the maximum order of a directed path. Let $r=\log_2(8/7)$. Bang-Jensen, Picasarri-Arrieta, and Yeo previously constructed tournaments of order $n$ whose acyclic dichromatic number is at least $n-(\frac8r)\log_2 n-\log_2\log_2 n$. We improve the leading logarithmic coefficient by a factor of two: for the uniform random tournament $\mathcal{T}_n$, asymptotically almost surely, $\vecχ_a(\mathcal{T}_n)\geq n-\frac4r\log_2 n+\frac2r\log_2\log_2 n-O(1)$. Finally, if $m(k)$ denotes the minimum order of an oriented graph with acyclic dichromatic number at least $k$, tournament completion shows that the same minimum is obtained over tournaments. We prove $m(3)=5$ and $m(4)=7$ and classify the tournament witnesses at these minimum orders: there is one at order five and two at order seven.
发表机构
- Nankai University（南开大学）
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