On the enumeration of a class of toroidal graphs

Ashish Kumar Upadhyay, Dipendu Maity

Abstract


We present enumerations of a class of toroidal graphs which are called semi-equivelar maps. Semi-equivelar maps are generalizations of equivelar maps. There are eight non-isomorphic types of semi-equivelar maps on the torus: $\{3^{3}, 4^{2}\}$, $\{3^{2}, 4, 3, 4\}$, $\{3, 6, 3, 6\}$, $\{3^{4}, 6\}$, $\{4, 8^{2}\}$, $\{3, 12^{2}\}$, $\{4, 6, 12\}$, $\{3, 4, 6, 4\}$. We attempt to classify all these maps.

Keywords


Toroidal Graphs, Semi-Equivelar Maps, Cycles

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References


A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217.

J. A. Bondy and U. S. R. Murthy, Graph theory, Graduate Texts in Mathematics, vol. 244, Springer, 2008.

U. Brehm and W. Kuhnel, Equivelar maps on the torus, European J. Combin. 29 (2008), 1843-1861.

U. Brehm and E. Schulte, Handbook of discrete and computational geometry, ch. Polyhedral maps, pp. 345-358, CRC Press, NY, 1997.

W. Kurth, Enumeration of platonic maps on the torus, Discrete Math. 61 (1986), 71-83.

S. Negami, Uniqueness and faithfulness of embedding of toroidal graphs, Discrete Math. 44 (1983), 161-180.

A. K. Tiwari and A. K. Upadhyay, An enumeration of semi-equivelar maps on torus and klein bottle, to appear.

A. K. Upadhyay, A. K. Tiwari, and D. Maity, Semi-equivelar maps, Beitr. Algebra Geom. 55 (2014), 229-242.




PID: http://hdl.handle.net/10515/sy5gq6rk3

Contributions to Discrete Mathematics. ISSN: 1715-0868