Four Pages Are Indeed Necessary for Planar Graphs

TL;DR

This paper proves that the minimum number of pages needed to embed any planar graph without crossings is four, resolving a long-standing open problem in graph theory.

Contribution

It establishes that the book thickness of planar graphs is exactly four, providing the first known examples requiring four pages.

Findings

Planar graphs can require four pages in their book embedding.

The book thickness of planar graphs is exactly four.

This resolves the open problem about the minimum pages needed for planar graphs.

Abstract

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.