Adjacency labelling for proper minor-closed graph classes

2026-05-07

AI Summary

This work investigates adjacency labeling schemes for hereditary subgraph-closed graph classes, aiming to encode vertex adjacency relations using the fewest possible bits. By constructing a universal graph $U$ with $n^{1+o(1)}$ vertices such that every $n$-vertex graph in any proper subgraph-closed class embeds as an induced subgraph of $U$, the authors achieve adjacency labels of length $(1+o(1))\log_2 n$ bits. This result constitutes the first asymptotically optimal labeling scheme for such graph classes and establishes an equivalence between the existence of compact adjacency labelings and universal graphs. The study thus overcomes a fundamental theoretical bottleneck in the efficient encoding of these graph structures.

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.

Problem

• adjacency labelling
• minor-closed graph classes
• induced subgraph
• universal graph
• graph encoding

Innovation

• adjacency labelling
• minor-closed graph classes
• universal graphs
• graph encoding
• induced subgraphs