Abstract
In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given cubic graph and constructions for cubic graphs with some special properties of their polyhedral embeddings. Some key results are that even cubic graphs with a polyhedral embedding on the torus can also have polyhedral embeddings in arbitrarily high genus, in fact in a genus {\em close} to the theoretical maximum for that number of vertices, and that there is no bound on the number of genera in which a cubic graph can have a polyhedral embedding. While these results suggest a large variety of polyhedral embeddings, computations for up to 28 vertices suggest that by far most of the cubic graphs do not have a polyhedral embedding in any genus and that the ratio of these graphs is increasing with the number of vertices.
Highlights
Three-dimensional convex polyhedra, or short polyhedra, as geometric objects have already been a subject of mathematical research in ancient Greece and played an important role ever since
While the geometric characterization cannot directly be generalized to graphs on surfaces of higher genus, some key properties of polyhedra can be used to generalize the concept to polyhedral embeddings of higher genus: Definition 1 A polyhedral embedding of a graph G = (V, E) in an orientable surface is an embedding so that each facial walk is a simple cycle and an intersection of two faces is either empty, a single vertex, or a single edge
A generalization of Whitney’s theorem that polyhedra have a unique embedding in the plane, shows that for each genus g, there is a number ζ(2g) such that no graph can have more than ζ(2g) polyhedral embeddings in the orientable surface of genus g Mohar and Robertson (2001)
Summary
Three-dimensional convex polyhedra, or short polyhedra, as geometric objects have already been a subject of mathematical research in ancient Greece and played an important role ever since. A generalization of Whitney’s theorem that polyhedra have a unique embedding in the plane, shows that for each genus g, there is a number ζ(2g) such that no graph can have more than ζ(2g) polyhedral embeddings in the orientable surface of genus g Mohar and Robertson (2001). Vk be a maximal subsequence of vertices of a directed cycle C that all have no edges on the left (right) of C If this sequence contains all vertices, C (C−1) is a face. Note that in a cubic graph the number of left and right facial subpaths of a directed cycle is. Due to Whitney’s famous theorem, plane embeddings of 3-connected planar graphs are – up to mirror images – unique.
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