平面图确实需要四页
Four Pages Are Indeed Necessary for Planar Graphs
摘要：
一个图在书中的嵌入由它的顶点沿着书脊的线性顺序和它的边到书的页面的分配组成，使得在同一页面上没有两条边交叉。图的书厚是所有书嵌入的最小页数。因此，一类图的书厚是其所有成员上的最大书厚。在本文中，我们解决了一个长期存在的公开问题，关于确切的书厚的类平面图，这是以前已知的是三个或四个。我们解决这个问题，证明平面图，需要四页在任何他们的书嵌入，从而建立平面图类的书厚度是4。
英文摘要：
An embedding of a graph in a book consists of a linear order of its vertices along the spine of the book and of an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. Accordingly, the book thickness of a class of graphs is the maximum book thickness over all its members. In this paper, we address a long-standing open problem regarding the exact book thickness of the class of planar graphs, which previously was known to be either three or four. We settle this problem by demonstrating planar graphs that require four pages in any of their book embeddings, thus establishing that the book thickness of the class of planar graphs is four.
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。
