Geodesic
A Generalization of the Regular Maps of Type{4, 4}b, c and {3, 6}b, c
Canadian mathematical bulletin, Tome 12 (1969) no. 3, pp. 293-297

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In [1], Coxeter gave a complete enumeration of the regular maps on a torus. The maps consist of two families of type {4, 4}b, c and {3, 6}b, c (and their duals). b and c are non-negative integers, which determine the maps uniquely. The maps are irreflexible if and only if bc(b - c) ≠ 0.On surfaces of genus h > 1, irreflexible regular maps are rather exceptional. The simplest surface of negative characteristic which admits irreflexible regular maps is the orientable surface of genus 7. This was shown by the author [4, Theorem 3. 1 ]. The corresponding map was discovered by J. R. Edmonds [2, p. 388]. 
Garbe, Dietmar. A Generalization of the Regular Maps of Type{4, 4}b, c and {3, 6}b, c. Canadian mathematical bulletin, Tome 12 (1969) no. 3, pp. 293-297. doi: 10.4153/CMB-1969-037-6
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[1] 1. Coxeter, H.S.M., Configurations and maps. Reports of a Math. Colloq. (2) 8 (1948) 18–38. Google Scholar

[2] 2. Coxeter, H.S.M., Introduction to geometry. (New York, 1961.) Google Scholar

[3] 3. Coxeter, H.S.M. and Moser, W. O. J., Generators and relations for discrete groups. (2nd éd., Berlin, 1965.) Google Scholar

[4] 4. Garbe, D., Über die regulären Zerlegungen geschlossener orientierbarer Flachen. J. reine angew. Math, (to appear). Google Scholar

[5] 5. Sherk, F. A., A family of regular maps of type {6, 6}. Canad. Math. Bull. 5 (1962) 13–20. Google Scholar

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