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The page number of a directed acyclic graph $G$ is the minimum $k$ for whichthere is a topological ordering of $G$ and a $k$-coloring of the edges suchthat no two edges of the same color cross, i.e., have alternating endpointsalong the topological ordering. We address the long-standing open problemasking for the largest page number among all upward planar graphs. We improvethe best known lower bound to $5$ and present the first asymptotic improvementover the trivial $O(n)$ upper bound, where $n$ denotes the number of verticesin $G$. Specifically, we first prove that the page number of every upwardplanar graph is bounded in terms of its width, as well as its height. We thencombine both approaches to show that every $n$-vertex upward planar graph haspage number $O(n^{2/3} \log(n)^{2/3})$.

A Sublinear Bound on the Page Number of Upward Planar Graphs