Nonorientable triangular embeddings of complete graphs with arbitrarily large looseness Original Research Article

The looseness of a triangular embedding of a complete graph in a closed surface is the minimum integer image such that for every assignment of image colors to the vertices of the embedding (such that all image colors are used) there is a face incident with vertices of three distinct colors. In this paper we show that for every image there is a nonorientable triangular embedding of a complete graph with looseness at least image.