Application of Plantri Graph:All Combinatorial Structure of Orderable And Deformable Compact Coxeter Hyperbolic Polyhedra

By Andreev's theorem and Choi's theorem, we proved that the degree of each vertex is three and the number of vertices of orderable compact Coxeter polyhedral is at most 10. Therefore a combinatorial polyhedron is a 3-connected planner graph. From the Plantri program, we found that the number of 3-connected planner graphs with at most 10 vertices of degree 3 is 9. We find that only five planner graphs among these 9 graphs satisfy the properties of orderable compact Coxeter polyhedra. Then we verify the polyhedra which are associated with these 5 planner graphs are orderable. Therefore the number of combinatorial polyhedra of orderable and deformable compact hyperbolic Coxeter polyhedra is five up to symmetry.