On Triangle Contact Graphs
Hubert de Fraysseix, Patrice Ossona de Mendez, Pierre Rosenstiehl · Combinatorics Probability Computing · 1994
It is proved that any plane graph may be represented by a triangle contact system, that is a collection of triangular disks which are disjoint except at contact points, each contact point being a node of exactly one triangle. Representations using contacts of T-or Y-shaped objects follow. Moreover, there is a one-to-one mapping between all the triangular contact representations of a maximal plane graph and all its partitions into three Schnyder trees.