Polyhedral graph

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 (a convex drawing of the graph of the polyhedron).

[1][2] 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.

[3] 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.

(the shortness exponent) and an infinite family of polyhedral graphs such that the length of the longest simple path of an

The Halin graphs, graphs formed from a planar embedded tree by adding an outer cycle connecting all of the leaves of the tree, form another important subclass of the polyhedral graphs.

The polyhedral graph formed as the Schlegel diagram of a regular dodecahedron .
Orthographic projections and Schlegel diagrams with Hamiltonian cycles of the vertices of the five platonic solids – only the octahedron has an Eulerian path or cycle, by extending its path with the dotted one