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 polyhedral graph formed as the Schlegel diagram of a regular dodecahedron.
Schlegel diagram of truncated icosidodecahedral graph

Characterization

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). 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.[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]

Hamiltonicity and shortness

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 graph with the fewest vertices and edges is the 11-vertex and 18-edge Herschel graph,[4] and there also exists an 11-vertex non-Hamiltonian polyhedral graph in which all faces are triangles, the Goldner–Harary graph.[5]

More strongly, there exists a constant α < 1 (the shortness exponent) and an infinite family of polyhedral graphs such that the length of the longest simple path of an n-vertex graph in the family is O(nα).[6][7]

Combinatorial enumeration

Duijvestijn provides a count of the polyhedral graphs with up to 26 edges;[8] The number of these graphs with 6, 7, 8, ... edges is

1, 0, 1, 2, 2, 4, 12, 22, 58, 158, 448, 1342, 4199, 13384, 43708, 144810, ... (sequence A002840 in the OEIS).

One may also enumerate the polyhedral graphs by their numbers of vertices: for graphs with 4, 5, 6, ... vertices, the number of polyhedral graphs is

1, 2, 7, 34, 257, 2606, 32300, 440564, 6384634, 96262938, 1496225352, ... (sequence A000944 in the OEIS).

Special cases

A polyhedral graph is the graph of a simple polyhedron if it is cubic (every vertex has three edges), and it is the graph of a simplicial polyhedron if it is a maximal planar graph. 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.

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References
  1. Lectures on Polytopes, by Günter M. Ziegler (1995) ISBN 0-387-94365-X , Chapter 4 "Steinitz' Theorem for 3-Polytopes", p.103.
  2. Grünbaum, Branko (2003), Convex Polytopes, Graduate Texts in Mathematics, 221 (2nd ed.), Springer-Verlag, ISBN 978-0-387-40409-7.
  3. Tutte, W. T. (1963), "How to draw a graph", Proceedings of the London Mathematical Society, 13: 743–767, doi:10.1112/plms/s3-13.1.743, MR 0158387.
  4. Weisstein, Eric W. "Herschel Graph". MathWorld..
  5. Weisstein, Eric W. "Goldner-Harary Graph". MathWorld..
  6. Weisstein, Eric W. "Shortness Exponent". MathWorld..
  7. Grünbaum, Branko; Motzkin, T. S. (1962), "Longest simple paths in polyhedral graphs", Journal of the London Mathematical Society, s1-37 (1): 152–160, doi:10.1112/jlms/s1-37.1.152.
  8. Duijvestijn, A. J. W. (1996), "The number of polyhedral (3-connected planar) graphs" (PDF), Mathematics of Computation, 65: 1289–1293, doi:10.1090/S0025-5718-96-00749-1.
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