# Shorter Labeling Schemes for Planar Graphs

## TL;DR

This paper presents a new adjacency labeling scheme for planar graphs with shorter labels of about (4/3) log n bits, enabling efficient adjacency queries and improving bounds on universal graphs.

## Contribution

It introduces a shorter labeling scheme for planar graphs, reducing label size from (2+o(1)) log n to (4/3+o(1)) log n bits, and extends to graphs of bounded Euler genus.

## Key findings

- Labels of length (4/3+o(1)) log n bits for planar graphs.
- Constant-time adjacency decision from labels.
- Construction of universal graphs with fewer vertices.

## 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.