Adjacency labelling for proper minor-closed graph classes
arXiv:2605.0661640.3
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AI Analysis
This solves a fundamental open problem in graph theory and labeling schemes for all proper minor-closed classes.
The paper proves that every proper minor-closed graph class has a (1+o(1)) log₂ n-bit adjacency labeling scheme, which is optimal up to lower-order terms.
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.