In the mathematical field of graph theory, the Dürer graph is an undirected graph with 12 vertices and 18 edges. It is named after Albrecht Dürer, whose 1514 engraving Melencolia I includes a depiction of Dürer's solid, a convex polyhedron having the Dürer graph as its skeleton. Dürer's solid is one of only four well-covered simple convex polyhedra.
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Dürer's solid is combinatorially equivalent to a cube with two opposite vertices truncated,[1] although Dürer's depiction of it is not in this form but rather as a truncated rhombohedron.[2] The exact geometry of the solid depicted by Dürer is a subject of some academic debate.[3] Richter (1957) claims that the rhombi of the rhombohedron from which this shape is formed have 5:6 as the ratio between their short and long diagonals, from which the acute angles of the rhombi would be approximately 80°. Schröder (1980) and Lynch (1982) instead conclude that the ratio is √3:2 and that the angle is approximately 82°. MacGillavry (1981) measures features of the drawing and finds that the angle is approximately 79°. Schrieber (1999) argues based on the writings of Dürer that all vertices of Dürer's solid lie on a common sphere, and further claims that the rhombus angles are 72°. Weitzel (2004) analyzes a 1510 sketch by Dürer of the same solid, from which he confirms Schrieber's hypothesis that the shape has a circumsphere but with rhombus angles of approximately 79.5°.
| Dürer graph | |
|---|---|
The Dürer graph |
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| Named after | Albrecht Dürer |
| Vertices | 12 |
| Edges | 18 |
| Diameter | 4 |
| Girth | 3 |
| Automorphisms | 12 (D6) |
| Chromatic number | 3 |
| Chromatic index | 3 |
| Properties | Cubic Planar well-covered |
The Dürer graph is the graph formed by the vertices and edges of the Dürer solid. It is a cubic graph of girth 3 and diameter 4. As well as its construction as the skeleton of Dürer's solid, it can be obtained by applying a Y-Δ transform to the opposite vertices of a cube graph, or as the generalized Petersen graph G(6,2). As with any graph of a convex polyhedron, the Dürer graph is a 3-vertex-connected simple planar graph.
The Dürer graph is a well-covered graph, meaning that all of its maximal independent sets have the same number of vertices, four. It is one of four well-covered cubic polyhedral graphs and one of seven well-covered 3-connected cubic graphs. The only other three well-covered simple convex polyhedra are the tetrahedron, triangular prism, and pentagonal prism.[4]
The Dürer graph is Hamiltonian, with LCF notation [-4,5,2,-4,-2,5;-].[5] More precisely, it has exactly six Hamiltonian cycles, each pair of which may be mapped into each other by a symmetry of the graph.[6]
The automorphism group both of the Dürer graph and of the Dürer solid (in either the truncated cube form or the form shown by Dürer) is isomorphic to the dihedral group of order 12 : D6.