We show how to assign labels of size $\tilde O(n^{1/3})$ to the vertices of a face $F$ in an $n$-vertex unweighted undirected planar graph $G$ s.t one can deduce the distance between any two vertices of $F$ given their labels alone. This matches (up to log factors) the $Ω(n^{1/3})$ lower bound of Gavoille, Peleg, Pérennes, Raz from 2004. Prior to our work, the smallest labels were of size $O(n^{1/2})$.
We show that planar graphs admit adjacency labels of size $\log n+O(\log\log n)$: every vertex gets a label of this many bits, and the adjacency of two vertices is decided from their labels alone, in constant time. This improves the $\log n+O(\sqrt{\log n\log\log n})$ of Dujmović et al. (J. ACM 2021) and the $\log n+O(\sqrt{\log n})$ of Gawrychowski and Janczewski (SOSA 2022). Equivalently, $n$-vertex planar graphs have an induced-universal graph with $n(\log n)^{O(1)}$ vertices, down from $n2^{O(\sqrt{\log n})}$. The same holds for graphs of bounded genus, apex-minor-free graphs and $k$-planar graphs. For every proper minor-closed class we get $\log n+O(\sqrt{\log n\log\log n})$, down from the $\log n+O((\log n)^{3/4})$ of Dujmović et al. (2026). Both results come from tight B-trees: nodes are split into parts of nearly equal size, so every node but the topmost stays within a factor $1\pm O(1/\log n)$ of its target size, and a root-to-leaf path loses only $O(\log\log n)$ bits.
Gavoille and Labourel (ESA 2007) gave an adjacency labeling scheme of size $\log n+O(k\log\log n)$ for graphs of treewidth at most $k$, and asked whether $\log n+O(k)$ bits suffice. We give a labeling scheme of size $\log n+O(\log\log^* n)$ for every fixed $k$, the first improvement of the second-order term for any $k\ge 2$. Equivalently, there is an induced-universal graph with $n(\log^* n)^{O_k(1)}$ vertices for this family. We work with laminar families of intervals, in which a vertex finds each earlier neighbor as the owner of an interval of a given nesting depth around it. As long as every subtree occupies consecutive positions, the nesting depth must be logarithmic, and $\log\log n$ bits per neighbor follow. We give up contiguity. Heavy subtrees are moved out, which brings the depth down to $O(\log^* n)$; the intervals broken by the moves are repaired with $2^{O(k^3)}$ additional intervals per vertex; and the bits needed to find these intervals are paid for by the moved subtrees.