Graphs with Eulerian unit spheres

d-spheres are defined graph theoretically and inductively as the empty graph in dimension d = −1 and d-dimensional graphs for which all unit spheres S(x) are (d−1)-spheres and such that for d ≥ 0 the removal of one vertex renders the graph contractible. Eulerian d-spheres are geometric d-spheres which can be colored with d + 1 colors. They are Eulerian graphs in the classical sense and for d ≥ 2, all unit spheres of an Eulerian sphere are Eulerian spheres. 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 result of Kempe-Heawood 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 Ĥ 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. GaussBonnet allows a straightforward classification within graph theory independent of the classical Schläfli-Schoute-Coxeter classification: 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 polytope is the unique Platonic d-sphere for d > 3. It is Eulerian. Date: January 11, 2015. 1991 Mathematics Subject Classification. Primary: 05C15, 05C10, 57M15 .