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

Abstract

The 2-uniform tilings of the plane provide doubly semi-equivelar maps on torus, as the 1uniform tilings provide semi-equivelar maps. There are twenty distinct 2-uniform tilings of the plane. In this article, we give a construction to classify and enumerate doubly semi-equivelar maps on the surface of torus corresponding to the 2-uniform tilings [3 6 : 3

Hamiltonicity of doubly semi-equivelar maps on the torus

“…To explore a Hamiltonian cycle in M , we describe its planar representation, called an M (i, j, k) representation, which is obtained by cutting M along any two non-homologous cycles (defined in Subsection 3.1) at a vertex. In [14], such representations are described for the DSEMs of types [3 6…”

“…map with finite vertex set), for a maximal path (a path of maximum length) P of type A α , there is an edge e in M such that P ∪ e is a cycle, say of type A α , where α ∈ {1, 2}. By [ [14], Lemma 3.1.3], one can see easily that these cycles are non-contractible.…”

“…This gives another representation say M (i, j, k) representation of the (i, j) representation of M . For detailed description, see [ [14], P. 5].…”

“…Proceeding similarly as in the previous section, we get its M (i, j, k)representation, shown in the below figures. One can see their detailed description in [14]. x 2(i/3)…”

“…In this article, we deal with the 2-uniform tilings [3 3 .4 4 .6] 1 , [3 6 : 3 4 .6] 2 , [3 3 .4 2 : 3.4.6.4], [3 6 : 3 2 .4.12], [3.4.3.12 : 3.12 2 ], [3.4.6.4 : 4.6.12] and [3.4 2 .6 : 3.4.6.4], which induce infinitely many doubly semi-equivelar maps on torus, see [14,19]. A study of doubly semi-equivelar maps on torus and Klein bottle is also carried out in [15,16].…”