On the minimal nonzero distance between triangular embeddings of a complete graph. (Q1402074)

A triangular embedding of a graph is a 2-cell embedding of the graph in a surface such that only a face is triangular. Let \(t\), \(t'\) be two triangular embeddings of a complete graph \(K\) and \(\varphi: V(K)\to V(K)\) a bijection. The authors denote by \(M(\varphi)\) the set of faces \((n,y,z)\) such that \((\varphi(n),\varphi(y),\varphi(z))\) is not a face of \(f'\). By traking the minimal value of \(| M(\varphi)|\) (under some conditions) as distance between \(f\) and \(f'\), the authors give some results about minimal nonzero distance between \(f\) and \(f'\) in several cases.