publication . Preprint . 2015

Graphs with Eulerian unit spheres

Knill, Oliver;
Open Access English
  • Published: 12 Jan 2015
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-spher...
free text keywords: Mathematics - Combinatorics, Computer Science - Discrete Mathematics, 05C15, 05C10, 57M15
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