arXiv 2302.021902023-02math.CO

一个用于加性着色的Alon-Tarsi型定理

An Alon-Tarsi Style Theorem for Additive Colorings

Ian Gossett

备注：: 12 pages. Preprint version. Peer-reviewed version to appear in Graphs and Combinatorics
发表信息：: Graphs and Combinatorics 40, 67 (2024)
DOI：: 10.1007/s00373-024-02797-2

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AI总结：

该研究给出Alon-Tarsi列表着色定理的新证明，借助其思路提出加性着色的Alon-Tarsi型定理，引入新有向图$\boldsymbol{\text{W}}(D)$并给出加性列表着色存在的条件，还将结果应用于一类三部图。

摘要：

我们首先给出Alon-Tarsi列表着色定理的一种替代证明。我们利用该证明的思路得到如下结果，它是Alon-Tarsi定理的加性着色类比：设$G$是一个图，$D$是$G$的一个定向。我们引入一个新的有向图$\boldsymbol{\text{W}}(D)$，使得若每个顶点$v$在$D$中的出度为$d_v$，且$\boldsymbol{\text{W}}(D)$中具有偶数条边的欧拉子有向图的数量与具有奇数条边的欧拉子有向图的数量不相等，则对于将大小为$d_v+1$的正整数列表$L(v)$分配给$G$的顶点的任意指派，都存在$G$的一个加性着色，为每个顶点$v$指派$L(v)$中的一个元素。作为应用，我们证明了一类三部图$G$的加性列表着色结果，这类图的其中一个颜色类仅包含邻域为完全图的顶点。

英文摘要：

We first give an alternative proof of the Alon-Tarsi list coloring theorem. We use the ideas from this proof to obtain the following result, which is an additive coloring analog of the Alon-Tarsi Theorem: Let $G$ be a graph and let $D$ be an orientation of $G$. We introduce a new digraph $\mathcal{W}(D)$, such that if the out-degree in $D$ of each vertex $v$ is $d_v$, and if the number of Eulerian subdigraphs of $\mathcal{W}(D)$ with an even number of edges differs from the number of Eulerian subdigraphs of $\mathcal{W}(D)$ with an odd number of edges, then for any assignment of lists $L(v)$ of $d_v+1$ positive integers to the vertices of $G$, there is an additive coloring of $G$ assigning to each vertex $v$ an element from $L(v)$. As an application, we prove an additive list coloring result for tripartite graphs $G$ such that one of the color classes of $G$ contains only vertices whose neighborhoods are complete.

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摘要：

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