面完全图的一个分类目录
A Catalog of Facially Complete Graphs
AI总结：
本文提出面完全图的完整分类（共七类），用新方法证明了面完全平面图顶点数的上界，实现了2-连通面完全图的计数，还证明了特定平面图加4面对角线后是5-可着色的。
摘要：
当地图中的区域只要有非空交集就被视为相邻（区别于传统要求交集为线段的观点）时，便引出了面完全图（facially complete graph）的概念：一个平面图，若在每两个共面的顶点之间都添加边，就会成为完全图。本文给出了面完全图的完整分类目录：它们可分为七类。由此得到一个结论：若q是面完全平面图G中最大面的大小，则G的顶点数至多为Floor[3/2 q]。该上界此前已被提出，但我们的证明与Chen、Grigni和Papadimitriou在1998年提出的方法完全不同。我们的方法还实现了对n个顶点的2-连通面完全图的计数。我们还证明，若一个平面图至多有两个大小为4的面且没有更大的面，则给每个4面都添加两条对角线后得到的图是5-可着色的。
英文摘要：
Considering regions in a map to be adjacent when they have nonempty intersection (as opposed to the traditional view requiring intersection in a linear segment) leads to the concept of a facially complete graph: a plane graph that becomes complete when edges are added between every two vertices that lie on a face. Here we present a complete catalog of facially complete graphs: they fall into seven types. A consequence is that if q is the size of the largest face in a plane graph G that is facially complete, then G has at most Floor[3/2 q] vertices. This bound was known, but our proof is completely different from the 1998 approach of Chen, Grigni, and Papadimitriou. Our method also yields a count of the 2-connected facially complete graphs with n vertices. We also show that if a plane graph has at most two faces of size 4 and no larger face, then the addition of both diagonals to each 4-face leads to a graph that is 5-colorable.
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。
