On The Enumeration of Polyhedral Embeddings of Cartesian Products of Cycles

We define a new embedding invariant, namely \(n\)-polyhedrality, and we propose a program of research in which the objective is to enumerate the \(n\)-polyhedral embeddings of a given graph for various values of \(n\). We begin the program for the cartesian products of cycles by showing that \(C_3 \times C_n\) has exactly one \(3\)-polyhedral embedding.