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 $O(n^{2/3} \log(n)^{2/3})$.

翻译：设 $G$ 为一个有向无环图，其页码定义为最小的整数 $k$，使得存在 $G$ 的一个拓扑序及边的一种 $k$-着色，且没有两条同色的边在拓扑序中交替出现（即边的端点交替排列）。本文针对所有向上平面图中最大页码这一长期未决的公开问题展开研究。我们将已知的下界改进至 $5$，并首次在渐近意义上改进了平凡上界 $O(n)$，其中 $n$ 表示图 $G$ 的顶点数。具体而言，我们首先证明每个向上平面图的页码均受其宽度和高度的共同约束。随后，我们结合这两种方法，证明每个包含 $n$ 个顶点的向上平面图的页码为 $O(n^{2/3} \log(n)^{2/3})$。