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Sparse universal graphs for planarity

par Esperet, Louis;Joret, Gwenaël ;Morin, Pat
Référence Journal of the London Mathematical Society
Publication Publié, 2023-06-29

Article révisé par les pairs

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Résumé :

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