Desargues Graph - Constructions

Constructions

There are several different ways of constructing the Desargues graph:

• It is the generalized Petersen graph G(10, 3). To form the Desargues graph in this way, connect ten of the vertices into a regular decagon, and connect the other ten vertices into a ten-pointed star that connects pairs of vertices at distance three in a second decagon. The Desargue graph consists of the 20 edges of these two polygons together with an additional 10 edges connecting points of one decagon to the corresponding points of the other.
• It is the Levi graph of the Desargues configuration. This configuration consists of ten points and ten lines describing two perspective triangles, their center of perspectivity, and their axis of perspectivity. The Desargues graph has one vertex for each point, one vertex for each line, and one edge for every incident point-line pair. Desargues' theorem, named after 17th-century French mathematician Gérard Desargues, describes a set of points and lines forming this configuration, and the configuration and the graph take their name from it.
• It is the bipartite double cover of the Petersen graph, formed by replacing each Petersen graph vertex by a pair of vertices and each Petersen graph edge by a pair of crossed edges.
• It is the bipartite Kneser graph H_5,2. Its vertices can be labeled by the ten two-element subsets and the ten three-element subsets of a five-element set, with an edge connecting two vertices when one of the corresponding sets is a subset of the other.
• The Desargues graph is Hamiltonian and can be constructed from the LCF notation: 5