Book Embeddings of k-Map Graphs

TL;DR

This paper establishes new upper bounds on the number of pages needed for book embeddings of k-map graphs, improving previous results and providing bounds for related graph classes.

Contribution

It proves that every k-map graph has a book embedding in at most 6*floor(k/2)+5 pages, and offers improved bounds for 1-planar and 2-planar graphs.

Findings

Every k-map graph has a book embedding in at most 6*floor(k/2)+5 pages.

Some k-map graphs require at least floor(3k/4) pages.

Improved upper bounds of 11 pages for 1-planar graphs and 17 pages for optimal 2-planar graphs.

Abstract

A map is a partition of the sphere into regions that are labeled as countries or holes. The vertices of a map graph are the countries of a map. There is an edge if and only if the countries are adjacent and meet in at least one point. For a k-map graph, at most k countries meet in a point. A graph is k-planar if it can be drawn in the plane with at most k crossings per edge. A p-page book embedding of a graph is a linear ordering of the vertices and an assignment of the edges to p pages, so that there is no conflict for edges assigned to the same page. The minimum number of pages is the book thickness of a graph, also known as stack number or page number. We show that every k-map graph has a book embedding in $6\lfloor k/2 \rfloor+5$ pages, which, for n-vertex graphs, can be computed in O(kn) time from its map. Our result improves the best known upper bound. Towards a lower bound, it is shown that some k-map graphs need $\lfloor 3k/4 \rfloor$ pages. In passing, we obtain an improved upper bound of eleven pages for 1-planar graphs, which are subgraphs of 4-map graphs, and of 17 pages for optimal 2-planar graphs.