Nonorientable triangular embeddings of complete graphs with arbitrarily large looseness

The looseness   of a triangular embedding of a complete graph in a closed surface is the minimum integer mm such that for every assignment of mm colors to the vertices of the embedding (such that all mm colors are used) there is a face incident with vertices of three distinct colors. In this paper we show that for every p⩾3p⩾3 there is a nonorientable triangular embedding of a complete graph with looseness at least pp.