Enumeration of doubly semi-equivelar maps on the Klein bottle

A vertex v in a map M has the face-sequence (p(1)(n1) . p(2)(n2).....p(k)(nk)), 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))}.