Adjacency labelling for proper minor-closed graph classes

Abstract

We show that every proper minor-closed class of graphs admits a $(1+o(1)){\log }_{2}n$-bit adjacency labelling scheme. Equivalently, for every proper minor-closed class $\mathcal{G}$ and every positive integer $n$ there exists an $n^{1+o(1)}$-vertex graph $U$ such that every $n$-vertex graph in $\mathcal{G}$ is isomorphic to an induced subgraph of $U$. Both results are optimal up to the lower order term. They generalize the corresponding results for planar graphs and apex-minor-free classes (Dujmović et al., J.~ACM 2021) to all proper minor-closed classes, answering the open question raised in that paper and anticipated earlier by Bonamy, Gavoille, and Pilipczuk (SODA 2020).

Improved writing throughout. Fixed bug related to injectivity of clique labels