A Sublinear Bound on the Page Number of Upward Planar Graphs

TL;DR

This paper establishes a sublinear upper bound on the page number of upward planar graphs, improving understanding of their structural complexity and addressing a long-standing open problem.

Contribution

It introduces the first asymptotic upper bound on the page number of upward planar graphs, relating it to graph width and height.

Findings

Page number of upward planar graphs is bounded by their width and height.

Every n-vertex upward planar graph has page number O(n^{2/3} log(n)^{2/3}).

Improves the lower bound to 5 for the page number of upward planar graphs.

Abstract

The page number of a directed acyclic graph $G$ is the minimum $k$ for which there is a topological ordering of $G$ and a $k$-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological ordering. We address the long-standing open problem asking for the largest page number among all upward planar graphs. We improve the best known lower bound to $5$ and present the first asymptotic improvement over the trivial $O(n)$ upper bound, where $n$ denotes the number of vertices in $G$. Specifically, we first prove that the page number of every upward planar graph is bounded in terms of its width, as well as its height. We then combine both approaches to show that every $n$-vertex upward planar graph has page number $O(n^{2/3} \log(n)^{2/3})$.