Graphs with Eulerian unit spheres

Abstract

d-spheres in graph theory are inductively defined as graphs for which all unit spheres S(x) are (d-1)-spheres and that the removal of one vertex renders the graph contractible. Eulerian d-spheres are geometric d-spheres which are d+1 colorable. We prove here that G is an Eulerian sphere if and only if the degrees of all the (d-2)-dimensional sub-simplices in G are even. This generalizes a Kempe-Heawood result for d=2 and is work related to the conjecture that all d-spheres have chromatic number d+1 or d+2 which is based on the geometric conjecture that every d-sphere can be embedded in an Eulerian (d+1)-sphere. For d=2, such an embedding into an Eulerian 3-sphere would lead to a geometric proof of the 4 color theorem, allowing to see "why 4 colors suffice". To achieve the goal of coloring a d-sphere G with d+2 colors, we hope to embed it into a (d+1)-sphere and refine or thin out the later using special homotopy deformations without touching the embedded sphere. Once rendered Eulerian and so (d+2)-colorable, it colors the embedded graph G. In order to define the degree of a simplex, we introduce a notion of dual graph H' of a subgraph H in a general finite simple graph G. This leads to a natural sphere bundle over the simplex graph. We look at geometric graphs which admit a unique geodesic flow: their unit spheres must be Eulerian. We define Platonic spheres graph theoretically as d-spheres for which all unit spheres S(x) are graph isomorphic Platonic (d-1)-spheres. Gauss-Bonnet allows a classification within graph theory: all spheres are Platonic for d=1, the octahedron and icosahedron are the Platonic 2-spheres, the sixteen and six-hundred cells are the Platonic 3-spheres. The cross polytop is the unique Platonic d-sphere for d>3. It is Eulerian.