# Sparse universal graphs for planarity

We show that for every integer n≥ 1 there exists a graph G_n with (1+o(1))n vertices and n^1 + o(1) edges such that every n-vertex planar graph is isomorphic to a subgraph of G_n. The best previous bound on the number of edges was O(n^3/2), proved by Babai, Chung, Erdős, Graham, and Spencer in 1982. We then show that for every integer n≥ 1 there is a graph U_n with n^1 + o(1) vertices and edges that contains induced copies of every n-vertex planar graph. This significantly reduces the number of edges in a recent construction of the authors with Dujmović, Gavoille, and Micek.

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