Adjacency Labelling for Planar Graphs (and Beyond)

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
$(1+o(1))\text{log}_{2}n$
-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 positive integer
$n$
, 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 a number of other graph classes, including bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and
$k$
-planar graphs.