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Preprints, Working Papers, ... Year : 2019

Shorter Labeling Schemes for Planar Graphs

Abstract

An \emph{adjacency labeling scheme} for a given class of graphs is an algorithm that for every graph $G$ from the class, assigns bit strings (labels) to vertices of $G$ so that for any two vertices $u,v$, whether $u$ and $v$ are adjacent can be determined by a fixed procedure that examines only their labels. It is known that planar graphs with $n$ vertices admit a labeling scheme with labels of bit length $(2+o(1))\log{n}$. In this work we improve this bound by designing a labeling scheme with labels of bit length $(\frac{4}{3}+o(1))\log{n}$. In graph-theoretical terms, this implies an explicit construction of a graph on $n^{4/3+o(1)}$ vertices that contains all planar graphs on $n$ vertices as induced subgraphs, improving the previous best upper bound of $n^{2+o(1)}$. Our scheme generalizes to graphs of bounded Euler genus with the same label length up to a second-order term. All the labels of the input graph can be computed in polynomial time, while adjacency can be decided from the labels in constant time.
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Dates and versions

hal-02265219 , version 1 (08-08-2019)
hal-02265219 , version 2 (10-04-2020)

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Marthe Bonamy, Cyril Gavoille, Michał Pilipczuk. Shorter Labeling Schemes for Planar Graphs. 2019. ⟨hal-02265219v2⟩

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