# 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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https://hal.archives-ouvertes.fr/hal-02265219
Contributor : Cyril Gavoille <>
Submitted on : Thursday, August 8, 2019 - 6:53:28 PM
Last modification on : Saturday, August 10, 2019 - 1:08:43 AM

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### Identifiers

• HAL Id : hal-02265219, version 1
• ARXIV : 1908.03341

### Citation

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

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