Sparse universal graphs for planarity

Louis Esperet (1) , Gwenaël Joret , Pat Morin

1 G-SCOP_OC - Optimisation Combinatoire

We show that for every integer $n\geq 1$ there exists a graph $G_n$ with $n^{1 + o(1)}$ 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, Erd\H{o}s, Chung, Graham, and Spencer in 1982.

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Louis Esperet, Gwenaël Joret, Pat Morin. Sparse universal graphs for planarity. Journal of the London Mathematical Society, 2023, 108 (4), pp.1333-1357. ⟨10.1112/jlms.12781⟩. ⟨hal-02965331⟩