The Book Thickness of 1-Planar Graphs is Constant

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.