Sparse universal graphs for planarity

Abstract

We show that for every integer $n\ge 1$ 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 $n\ge 1$ 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.

v4: minor change. v3: revised following referee's comments. v2: added new result about induced-universal graphs

Cited by in corpus (5)

• Separating layered treewidth and row treewidth
• Product structure of graphs with an excluded minor
• Universality in minor-closed graph classes
• Renaming in distributed certification
• The grid-minor theorem revisited