Enumeration of Doubly Semi-Equivelar Maps on the Klein Bottle

A vertex v in a map M has the face-sequence (p_1^n_1. p_2^n_2. … . p_k^n_k) , if consecutive n_i numbers of p_i -gons are incident at v in the given cyclic order for 1 ≤ i ≤ k . A map is called semi-equivelar if the face-sequence of each vertex is same throughout the map. A doubly semi-equivelar map is a generalization of semi-equivelar map which has precisely 2 distinct face-sequences. In this article, we determine all the types of doubly semi-equivelar maps of combinatorial curvature 0 on the Klein bottle. We present classification of doubly semi-equivelar maps on the Klein bottle and illustrate this classification for those doubly semi-equivelar maps which comprise of face-sequence pairs {(3^6), (3^3.4^2)} and {(3^3.4^2), (4^4)} .