哪些三次图具有四边形球面浸入？
Which Cubic Graphs have Quadrangulated Spherical Immersions?
摘要：
我们考虑球面四边形-多重图的球面嵌入，可能有循环，使得每个面都有长度为4的边界行走-其中所有顶点都有3或4度。将每个4度顶点解释为交叉，这些嵌入也可以被认为是三次图的横向浸入，我们称之为{ \it提取图}。我们还考虑了圆盘的四边形，其中内部顶点的度为3或4，边界顶点的度为2或3。首先，我们分类所有这样的四边形的磁盘。然后，我们提供了四种方法来构建球面四边形，其中两个使用四边形的磁盘作为输入。其中两种方法提供了一个参数家庭的四边形，我们证明了序列的同构类型提取的图形是周期性的。我们关闭的计算机计算，产生了球面四边形，但三个立方重八顶点的描述。
英文摘要：
We consider spherical quadrangulations -- spherical embeddings of multigraphs, possibly with loops, so that every face has boundary walk of length 4 -- in which all vertices have degree 3 or 4. Interpreting each degree 4 vertex as a crossing, these embeddings can also be thought of as transversal immersions of cubic graphs which we refer to as the {\it extracted graphs}. We also consider quadrangulations of the disk in which interior vertices have degree 3 or 4 and boundary vertices have degree 2 or 3. First, we classify all such quadrangulations of the disk. Then, we provide four methods for constructing spherical quadrangulations, two of which use quadrangulations of the disk as input. Two of these methods provide one-parameter families of quadrangulations, for which we prove that the sequence of isomorphism types of extracted graphs is periodic. We close with a description of computer computations which yielded spherical quadrangulations for all but three cubic multigraphs on eight vertices.
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。
