Adjacency labelling for proper minor-closed graph classes

TL;DR

This paper proves that all proper minor-closed graph classes have adjacency labelling schemes with near-optimal label size, enabling efficient graph representation.

Contribution

It establishes the existence of nearly optimal adjacency labelling schemes for all proper minor-closed graph classes, matching lower bounds up to lower order terms.

Findings

Every proper minor-closed class admits a $(1+o(1)) ext{log}_2 n$-bit adjacency labelling scheme.

For each such class, there exists an $n^{1+o(1)}$-vertex universal graph containing all $n$-vertex graphs in the class as induced subgraphs.

Both results are proven to be optimal up to lower order terms.

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.