The Book Thickness of 1-Planar Graphs is Constant

TL;DR

This paper proves that all 1-planar graphs can be embedded in a book with a constant number of pages, significantly improving previous bounds which were proportional to the square root of the number of vertices.

Contribution

It establishes that the book thickness of 1-planar graphs is constant, advancing understanding of their structural properties.

Findings

Every 1-planar graph admits a constant-page book embedding.

Improves previous upper bounds from O(√n) to a constant.

Enhances theoretical understanding of graph embedding complexity.

Abstract

In a book embedding, the vertices of a graph are placed on the spine of a book and the edges are assigned to pages, so that edges on the same page do not cross. In this paper, we prove that every $1$-planar graph (that is, a graph that can be drawn on the plane such that no edge is crossed more than once) admits an embedding in a book with constant number of pages. To the best of our knowledge, the best non-trivial previous upper-bound is $O(\sqrt{n})$, where $n$ is the number of vertices of the graph.