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A Sublinear Bound on the Page Number of Upward Planar Graphs

Jungeblut, Paul ORCID iD icon 1; Merker, Laura 1; Ueckerdt, Torsten 1
1 Institut für Theoretische Informatik (ITI), Karlsruher Institut für Technologie (KIT)

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


Volltext §
DOI: 10.5445/IR/1000147771
Veröffentlicht am 08.06.2022
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Theoretische Informatik (ITI)
Publikationstyp Forschungsbericht/Preprint
Publikationsdatum 09.05.2022
Sprache Englisch
Identifikator KITopen-ID: 1000147771
Nachgewiesen in Dimensions
arXiv
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