Nonisomorphic Orientable Pentagonal Embeddings and Colorings of Complete Graphs

In this paper, we construct the current graph and pentagonal rule of the complete graph pentagonal embedding, and prove that the complete graph K 31 has at least 663552 nonisomorphic orientable pentagonal embeddings. Each embedding has 720 5-edge colorings, and each coloring appears on each face of the orientable pentagonal embeddings. For all pentagonal embeddings in the orientable surface 𝒮 125 , these embeddings determine polyhedra with a minimal number of pentagons. We also establish similar results for the complete graph K 11 .