Some doubly semi-equivelar maps on the plane and the torus

Abstract

A vertex $v$ in a map $M$ has the face-sequence $(p_1 ^{n_1}. \ldots. p_k^{n_k})$, if there are $n_i$ numbers of $p_i$-gons incident at $v$ in the given cyclic order, for $1 \leq i \leq k$. A map $M$ is called a semi-equivelar map if each of its vertex has same face-sequence. Doubly semi-equivelar maps are a generalization of semi-equivelar maps which have precisely 2 distinct face-sequences. In this article, we enumerate the types of doubly semi-equivelar maps on the plane and torus which have combinatorial curvature 0. Further, we present classification of doubly semi-equivelar maps on the torus and illustrate this classification for those doubly semi-equivelar maps which comprise of face-sequence pairs $\{(3^6), (3^3.4^2)\}$ and $\{(3^3.4^2), (4^4)\}$.