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arXiv · 2003.04280

Adjacency Labelling for Planar Graphs (and Beyond)

Vida Dujmović·Louis Esperet·Gwenaël Joret·Cyril Gavoille·Piotr Micek·Pat Morin
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Abstract

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an $n$-vertex planar graph $G$ is assigned a $(1+o(1))\log_2 n$-bit label and the labels of two vertices $u$ and $v$ are sufficient to determine if $uv$ is an edge of $G$. This is optimal up to the lower order term and is the first such asymptotically optimal result. An alternative, but equivalent, interpretation of this result is that, for every $n$, there exists a graph $U_n$ with $n^{1+o(1)}$ vertices such that every $n$-vertex planar graph is an induced subgraph of $U_n$. These results generalize to bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and $k$-planar graphs.

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Vida Dujmović, Louis Esperet, Gwenaël Joret, Cyril Gavoille, Piotr Micek, Pat Morin. 2020-03-09. Adjacency Labelling for Planar Graphs (and Beyond). https://doi.org/10.1145/3477542

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