Polyhedral graph

In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected planar graphs.

The Schlegel diagram of a convex polyhedron represents its vertices and edges as points and line segments in the Euclidean plane, forming a subdivision of an outer convex polygon into smaller convex polygons. It has no crossings, so every polyhedral graph is also a planar graph. Additionally, by Balinski's theorem, it is a 3-vertex-connected graph.

According to Steinitz's theorem, these two graph-theoretic properties are enough to completely characterize the polyhedral graphs: they are exactly the 3-vertex-connected planar graphs. That is, whenever a graph is both planar and 3-vertex-connected, there exists a polyhedron whose vertices and edges form an isomorphic graph. Given such a graph, a representation of it as a subdivision of a convex polygon into smaller convex polygons may be found using the Tutte embedding.

Tait conjectured that every cubic polyhedral graph (that is, a polyhedral graph in which each vertex is incident to exactly three edges) has a Hamiltonian cycle, but this conjecture was disproved by a counterexample of W. T. Tutte, the polyhedral but non-Hamiltonian Tutte graph. If one relaxes the requirement that the graph be cubic, there are much smaller non-Hamiltonian polyhedral graphs; the one with the fewest vertices and edges is the 11-vertex and 18-edge Herschel graph, and there also exists an 11-vertex non-Hamiltonian polyhedral graph in which all faces are triangles, the Goldner–Harary graph.

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