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Planar graphs (and related families of graphs) have \((1+o(1))\log n\)-bit adjacency labelling schemes. Equivalently, there is a graph with \(n^{1+o(1)}\) vertices that contains every \(n\)-vertex planar graph as an induced subgraph.

arxiv.org/abs/2003.04280

The proof combines our year-old product structure theorem with binary search trees!

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