We show that for every integer $n\geq 1$ 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, Erd\H{o}s, Graham, and Spencer in 1982. We then show that for every integer $n\geq 1$ 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 Dujmovi\'c, Gavoille, and Micek.

翻译：我们证明，对于每个整数 $n\geq 1$，存在一个图 $G_n$，其顶点数为 $(1+o(1))n$，边数为 $n^{1+o(1)}$，使得每个 $n$ 顶点平面图都同构于 $G_n$ 的一个子图。此前关于边数的最佳上界是 $O(n^{3/2})$，由 Babai、Chung、Erdős、Graham 和 Spencer 于 1982 年证明。接着，我们证明对于每个整数 $n\geq 1$，存在一个图 $U_n$，其顶点数和边数均为 $n^{1+o(1)}$，并且包含每个 $n$ 顶点平面图作为导出子图。这显著减少了作者与 Dujmović、Gavoille 和 Micek 近期构造中的边数。