Schnyder's theorem

In graph theory, Schnyder's theorem is a characterization of planar graphs in terms of the order dimension of their incidence posets. It is named after Walter Schnyder, who published its proof in 1989.

The incidence poset P(G) of an undirected graph G with vertex set V and edge set E is the partially ordered set of height 2 that has V ∪ E as its elements. In this partial order, there is an order relation x < y when x is a vertex, y is an edge, and x is one of the two endpoints of y.

The order dimension of a partial order is the smallest number of total orderings whose intersection is the given partial order; such a set of orderings is called a realizer of the partial order. Schnyder's theorem states that a graph G is planar if and only if the order dimension of P(G) is at most three.

This theorem has been generalized by Brightwell and Trotter (1993, 1997) to a tight bound on the dimension of the height-three partially ordered sets formed analogously from the vertices, edges and faces of a convex polyhedron, or more generally of an embedded planar graph: in both cases, the order dimension of the poset is at most four. However, this result cannot be generalized to higher-dimensional convex polytopes, as there exist four-dimensional polytopes whose face lattices have unbounded order dimension.

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