Surfaces

Numbers of triangulated surfaces with n vertices
Numbers of triangulations of the orientable surfaces of genus g with n vertices
Numbers of triangulations of the non-orientable surfaces of genus g with n vertices
Equivelar surfaces
Cyclic triangulations
Polyhedral surfaces

Numbers of triangulated surfaces with n vertices

n	4	5	6	7	8	9	10	11	12
#	1	1	3	9	43	655	42426	11590894	12561206794

(all triangulations with up to 10 vertices and their types as well as in outdated, mixed lexicographic format)

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Numbers of triangulations of the orientable surfaces of genus g with n vertices

n\g	0	1	2	3	4	5	6
4	1	--	--	--	--	--	--
5	1	--	--	--	--	--	--
6	2	--	--	--	--	--	--
7	5	1	--	--	--	--	--
8	14	7	--	--	--	--	--
9	50	112	--	--	--	--	--
10	233	2109	865	20	--	--	--
11	1249	37867	113506	65878	821	--	--
12	7595	605496	7085444	25608643	14846522	751593	59
13	49566	8778329	290085272	?	?	?	?
14	339722	117839254	9022585751	?	?	?	?
15	2406841	1491505713	231102712868	?	?	?	?
16	17490241	18035839188	?	?	?	?	?
17	129664753	210391127053	?	?	?	?	?
18	977526957	?	?	?	?	?	?
19	7475907149	?	?	?	?	?	?
20	57896349553	?	?	?	?	?	?
21	453382272049	?	?	?	?	?	?
22	3585853662949	?	?	?	?	?	?
23	28615703421545	?	?	?	?	?	?

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Numbers of triangulations of the non-orientable surfaces of genus g with n vertices

n\g	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
4	--	--	--	--	--	--	--	--	--	--	--	--	--	--	--
5	--	--	--	--	--	--	--	--	--	--	--	--	--	--	--
6	1	--	--	--	--	--	--	--	--	--	--	--	--	--	--
7	3	--	--	--	--	--	--	--	--	--	--	--	--	--	--
8	16	6	--	--	--	--	--	--	--	--	--	--	--	--	--
9	134	187	133	37	2	--	--	--	--	--	--	--	--	--	--
10	1210	4462	11784	13657	7050	1022	14	--	--	--	--	--	--	--	--
11	11719	86968	530278	1628504	3355250	3623421	1834160	295291	5982	--	--	--	--	--	--
12	114478	1448516	16306649	99694693	473864807	1479135833	3117091975	3935668832	2627619810	711868010	49305639	182200	--	--	--
13	1108826	21535942	392973078	4076362798	?	?	?	?	?	?	?	?	?	?	243088286
14	10606795	294625589	8001174073	127322830218	?	?	?	?	?	?	?	?	?	?	?
15	100352404	3787236314	144075560093	?	?	?	?	?	?	?	?	?	?	?	?
16	940956644	46411139226	?	?	?	?	?	?	?	?	?	?	?	?	?
17	8762227629	547841825257	?	?	?	?	?	?	?	?	?	?	?	?	?
18	81168427279	?	?	?	?	?	?	?	?	?	?	?	?	?	?
19	748953936818	?	?	?	?	?	?	?	?	?	?	?	?	?	?

Triangulations of the 2-sphere with up to 23 vertices:

Gunnar Brinkmann and Brendan D. McKay. Fast generation of planar graphs. MATCH Commun. Math. Comput. Chem. 58, 323-357 (2007).

Triangulations of the torus, the orientable surface of genus 2, and the non-orientable surfaces of genus 1, 2, 3, 4 with up to 17, 15, 19, 17, 15, and 14 vertices, respectively:

Thom Sulanke. Counts of triangulations of surfaces. 2005.

Triangulations of surfaces with up to 10 vertices (in outdated, mixed lexicographic format):

Frank H. Lutz. Enumeration and random realization of triangulated surfaces. Discrete Differential Geometry (A. I. Bobenko, P. Schröder, J. M. Sullivan, and G. M. Ziegler, eds.). Oberwolfach Seminars 38, 235-253. Birkhäuser, Basel, 2008.

Triangulations of surfaces with up to 12 vertices:

Thom Sulanke and Frank H. Lutz. Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds. Eur. J. Comb. 30, 1965-1979 (2009).

Thom Sulanke. C program lextri (Version 0.15, 2007).

Neighborly triangulations of surfaces with 12 and 13 vertices:

Mark N. Ellingham and Chris Stephens. Triangular embeddings of complete graphs (neighborly maps) with 12 and 13 vertices. J. Combin. Des. 13, 336-344 (2005).

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Equivelar surfaces

Possible values of (n,q) for and respective examples of equivelar triangulations with n vertices and vertex-degree q of orientable surfaces with chi(M) >= -10:

chi(M)	(n,q): #
2	(4,3): 1	(6,4): 1	(12,5): 1
0	(n,6), with n>=7
-2	(12,7): 6
-4	(12,8): 34	(24,7): 11277
-6	(12,9): 112	(18,8): 374712	(36,7): ?
-8	(12,10): 103	(16,9): 7410658	(24,8): ?	(48,7): ?
-10	(12,11): 59	(15,10): ?	(20,9): ?	(30,8): ?	(60,7): ?

Possible values of (n,q) for and respective examples of equivelar triangulations with n vertices and vertex-degree q of non-orientable surfaces with chi(M) >= -10:

chi(M)	(n,q): #
1	(6,5): 1
0	(n,6), with n>=9 and n not prime
-1	--
-2	(12,7): 28
-3	(9,8): 2	(18,7): 1401
-4	(12,8): 500	(24,7): 600946
-5	(10,9): 14	(15,8): 227969	(30,7): ?
-6	(12,9): 9273	(18,8): 98440756	(36,7): ?
-7	(14,9): 11300550	(21,8): ?	(42,7): ?
-8	(12,10): 48591	(16,9): ?	(24,8): ?	(48,7): ?
-9	(18,9): ?	(27,8): ?	(54,7): ?
-10	(12,11): 182200	(15,10): ?	(20,9): ?	(30,8): ?	(60,7): ?

Equivelar triangulations of surfaces with up to 12 vertices as well as of surfaces with chi(M)>=-4:

Thom Sulanke and Frank H. Lutz. Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds. Eur. J. Comb. 30, 1965-1979 (2009).

Series of cyclic triangulations:

Frank H. Lutz, Thom Sulanke, Anand K. Tiwari, and Ashish K. Upadhyay. Equivelar and d-covered triangulations of surfaces. I. Preprint, 21 pages, 2010; arXiv:1001.2777.
Frank H. Lutz. Equivelar and d-covered triangulations of surfaces. II. Cyclic triangulations and tessellations. Preprint, 27 pages, 2010; arXiv:1001.2779.

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Numbers of cyclic triangulations of surfaces with n vertices

n	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
#	1	--	1	1	1	1	2	1	14	4	10	14	51	6	81	87	560	735

(all cyclic triangulations with up to 21 vertices and their types)

Frank H. Lutz. Equivelar and d-covered triangulations of surfaces. II. Cyclic triangulations and tessellations. Preprint, 27 pages, 2010; arXiv:1001.2779.

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Polyhedral surfaces

Frank H. Lutz. Enumeration and random realization of triangulated surfaces. Discrete Differential Geometry (A. I. Bobenko, P. Schröder, J. M. Sullivan, and G. M. Ziegler, eds.). Oberwolfach Seminars 38, 235-253. Birkhäuser, Basel, 2008.
Stefan Hougardy, Frank H. Lutz, and Mariano Zelke. Polyhedra of genus 2 with 10 vertices and minimal coordinates. Electronic Geometry Models, No. 2005.08.001 (2007).
Stefan Hougardy, Frank H. Lutz, and Mariano Zelke. Polyhedra of genus 3 with 10 vertices and minimal coordinates. Electronic Geometry Models, No. 2006.02.001 (2007).
Stefan Hougardy, Frank H. Lutz, and Mariano Zelke. Polyhedral tori with minimal integer coordinates. Electronic Geometry Models, No. 2008.10.001 (2008).
Stefan Hougardy, Frank H. Lutz, and Mariano Zelke. Surface realization with the intersection segment functional. Exp. Math. 19, 79-92 (2010).
Frank H. Lutz and Nikolaus Witte. Knotted polyhedral tori. Preprint, 10 pages, 2007; arXiv:0707.1281.

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