Problem
Research questions and friction points this paper is trying to address.
4-connected graph
girth
Charney-Davis curvature
legal system
A Four-Connected Graph without a Legal System
🤖 AI Summary
研究解决了是否存在一个具有特定属性的4-连通图的问题，通过从六角柱开始并附加三个基于K_{3,4}的帽来构造这样一个图。
📝 Abstract
In a 2021 paper, Jankiewicz, Norin, and Wise asked whether there exists a finite $4$-connected graph of girth at least four and nonnegative Charney--Davis curvature such that no $4$-connected ordinary subgraph admits a legal system. We construct such a graph by starting from the hexagonal prism and attaching three $K_{3,4}$-based caps along pairwise disjoint induced $4$-cycles. The key structural input is a restriction theorem showing that a legal system on an induced-$4$-cycle amalgam restricts to each side, so the obstruction carried by the negatively curved prism survives the attachments. The resulting $33$-vertex graph is $4$-regular and $4$-connected, has girth four and Charney--Davis curvature one, and, by $4$-regularity, is its own unique $4$-connected ordinary subgraph.
Innovation
Methods, ideas, or system contributions that make the work stand out.
4-connected graph
legal system
restriction theorem
induced-4-cycle amalgam
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