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. According to Steinitz's theorem, the polyhedral graphs may also be characterized in purely graph-theoretic terms, as the 3-vertex-connected planar graphs.[1][2]
Tait conjectured that every polyhedral graph has a Hamiltonian cycle, but this conjecture was disproved by a counterexample of W. T. Tutte, the polyhedral but non-Hamiltonian Tutte graph. 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).[3][4]
References
- ^ Lectures on Polytopes, by Günter M. Ziegler (1995) ISBN 038794365X , Chapter 4 "Steinitz' Theorem for 3-Polytopes", p.103.
- ^ Branko Grünbaum, Convex Polytopes, 2nd edition, prepared by Volker Kaibel, Victor Klee, and Gunter M. Ziegler, 2003, ISBN 0387404090, ISBN 978-0387404097, 466pp.
- ^ Weisstein, Eric W. "Shortness Exponent". MathWorld..
- ^ 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.