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.

arXiv commentv5: corrects an error in the proof of Lemma 20

References in corpus (2)

• Planar graphs have bounded queue-number
• Adjacency Labelling for Planar Graphs (and Beyond)

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