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arXiv 2206.13635math.CO
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排除子式的超图着色

Coloring hypergraphs with excluded minors

Raphael Steiner

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15 pages, Final version, revised according to the reviewer's comments
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AI总结：

本文证明Hadwiger猜想可自然扩展到超图着色，建立不含$K_t$-子式的超图色数界$h(t)=O(t\log\log t)$，并猜想精确值为$\lceil\frac{3}{2}(t-1)\rceil$，对独立数至多2的超图给出证明，同时应用于普通图着色。

摘要：

Hadwiger猜想是图论中最著名的开放问题之一，它断言每个不含$K_t$作为子式的图都是$(t-1)$-可着色的。本文旨在证明Hadwiger问题向超图着色存在自然扩展，并推导出一些初步结果和应用。将普通图子式推广到超图，我们称超图$H_1$是超图$H_2$的子式，如果可以通过有限次顶点删除、超边删除和超边收缩从$H_2$得到与$H_1$同构的超图。我们首先证明Hadwiger猜想对超图的弱扩展成立：对于每个$t \ge 1$，存在有限（最小）整数$h(t)$使得每个不含$K_t$-子式的超图都是$h(t)$-可着色的，并且我们证明$$\left\lceil\frac{3}{2}(t-1)\right\rceil \le h(t) \le 2g(t)$$其中$g(t)$表示不含$K_t$-子式的图的最大色数。利用Delcourt和Postle最近的结果$g(t)=O(t \log \log t)$，这得出$h(t)=O(t \log \log t)$。我们进一步猜想$h(t)=\left\lceil\frac{3}{2}(t-1)\right\rceil$，即对于所有$t \ge 1$，每个不含$K_t$-子式的超图都是$\left\lceil\frac{3}{2}(t-1)\right\rceil$-可着色的，并证明该猜想对所有独立数至多为2的超图成立。通过考虑特殊类别的超图，上述结果对普通图着色也有一些有趣的应用，例如：色数为$C k t \log \log t$的图包含具有$k$-边连通分支集的$K_t$-子式；色数为$C q t \log \log t$的图包含具有模$q$连通分支集的$K_t$-子式；通过考虑有向图的圈超图，我们恢复了关于大二色数有向图中强子式的已知结果作为特例。

英文摘要：

Hadwiger's conjecture, among the most famous open problems in graph theory, states that every graph that does not contain $K_t$ as a minor is properly $(t-1)$-colorable. The purpose of this work is to demonstrate that a natural extension of Hadwiger's problem to hypergraph coloring exists, and to derive some first partial results and applications. Generalizing ordinary graph minors to hypergraphs, we say that a hypergraph $H_1$ is a minor of a hypergraph $H_2$, if a hypergraph isomorphic to $H_1$ can be obtained from $H_2$ via a finite sequence of vertex- and hyperedge-deletions, and hyperedge contractions. We first show that a weak extension of Hadwiger's conjecture to hypergraphs holds true: For every $t \ge 1$, there exists a finite (smallest) integer $h(t)$ such that every hypergraph with no $K_t$-minor is $h(t)$-colorable, and we prove $$\left\lceil\frac{3}{2}(t-1)\right\rceil \le h(t) \le 2g(t)$$ where $g(t)$ denotes the maximum chromatic number of graphs with no $K_t$-minor. Using the recent result by Delcourt and Postle that $g(t)=O(t \log \log t)$, this yields $h(t)=O(t \log \log t)$. We further conjecture that $h(t)=\left\lceil\frac{3}{2}(t-1)\right\rceil$, i.e., that every hypergraph with no $K_t$-minor is $\left\lceil\frac{3}{2}(t-1)\right\rceil$-colorable for all $t \ge 1$, and prove this conjecture for all hypergraphs with independence number at most $2$. By considering special classes of hypergraphs, the above additionally has some interesting applications for ordinary graph coloring, such as: -graphs of chromatic number $C k t \log \log t$ contain $K_t$-minors with $k$-edge-connected branch-sets, -graphs of chromatic number $C q t \log \log t$ contain $K_t$-minors with modulo-$q$-connected branch sets, -by considering cycle hypergraphs of digraphs we recover known results on strong minors in digraphs of large dichromatic number as special cases.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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