发表机构
University College London; ETH Zürich（伦敦大学学院; 苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文反驳了支配性Hadwiger猜想，该猜想断言色数至少为t的图必含支配性K_t-模型。作者基于ChatGPT 6 Astra Ultra发现的构造，利用Suzuki-Tits卵形随机子采样块几何图的补图作为反例。
AI 中文摘要
Hadwiger猜想（1943年）指出，每个色数至少为t的图G都包含一个K_t-模型：即t个顶点不相交的连通子图T_1,…,T_t的集合，使得对于所有1≤i<j≤t，T_j中某个顶点在T_i中有邻居。将该定义中的“某个”替换为“每个”，便产生了显著更强的支配性K_t-模型的概念，该概念由Illingworth和Wood（2024）提出。他们提出了一个问题：每个色数至少为t的图是否都包含一个支配性K_t-模型。这一陈述是对Hadwiger猜想的显著加强，并被称为支配性Hadwiger猜想。我们提供了对由ChatGPT 6 Astra Ultra发现的该猜想的一个反例的自行阐述。该构造是补图，即一个伪随机三角形自由图，该图通过对基于Suzuki-Tits卵形的块几何图进行随机子采样而获得。
英文摘要
Hadwiger's conjecture (1943) states that every graph $G$ with chromatic number at least $t$ contains a $K_t$-model: a collection of $t$ vertex-disjoint connected subgraphs $T_1,\dots,T_t$ such that for all $1\le i<j\le t$ some vertex in $T_j$ has a neighbour in $T_i$. Replacing "some" in this definition by "every" gives rise to the significantly stronger notion of a dominating $K_t$-model introduced by Illingworth and Wood (2024). They raised the question whether every graph of chromatic number at least $t$ contains a dominating $K_t$-model. This statement is a significant strengthening of Hadwiger's conjecture and has come to be known as the dominating Hadwiger conjecture. We provide our own exposition of a disproof of this conjecture found by ChatGPT 6 Astra Ultra. The construction is the complement of a pseudorandom triangle-free graph that is obtained by randomly subsampling a block geometric graph based on the Suzuki-Tits ovoid.
Comments12.5 pages main part including references, 5.5 pages appendix, no figures
