发表机构
Kyungpook National University; California State University, Chico; Korea University（庆北国立大学; 加州州立大学奇科分校; 高丽大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于删除顶点对的诱导子图检测方法，分类了22条边、八个4度顶点和四个3度顶点的无三角形内蕴纽结图，证明恰好有四个。
AI 中文摘要
内蕴纽结图是指其每个空间嵌入都包含一个非平凡纽结的圈。对此类图进行分类是空间图理论中的一个核心问题。已知每个内蕴纽结图至少有21条边，且21条边的情形已完全解决。然而，对于22条边的情形，分类仍不完整。特别地，已知恰好有八个具有至少5度顶点的无三角形例子，仅剩下所有顶点度数均为3或4的情形。在本文中，我们引入了一种基于删除顶点对所得诱导子图来检测内蕴纽结性的方法。利用该方法，我们对所有具有22条边、八个4度顶点和四个3度顶点的无三角形内蕴纽结图进行了分类。我们证明恰好有四个：$E_9\!+\!e$族中的Cousins 43、105和109，以及$H_9\!+\!e$族中的图$H_{12}\! +\! e$。
英文摘要
An intrinsically knotted graph is one for which every spatial embedding contains a nontrivially knotted cycle. Classifying such graphs is a central problem in spatial graph theory. It is known that every intrinsically knotted graph has at least 21 edges, and the case of 21 edges has been completely resolved. For 22 edges, however, the classification remains incomplete. In particular, exactly eight triangle-free examples with a vertex of degree at least 5 are known, leaving only the case in which all vertices have degree 3 or 4. In this paper, we introduce a method for detecting intrinsic knottedness based on induced subgraphs obtained by deleting pairs of vertices. Using this method, we classify all triangle-free intrinsically knotted graphs with 22 edges having eight vertices of degree~4 and four of degree~3. We prove that there are exactly four: Cousins 43, 105, and 109 in the $E_9\!+\!e$ family and the graph $H_{12}\! +\! e$ in the $H_9\!+\!e$ family.
Comments(15 pages, 9 figures)
