  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.
