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Enumeration of unrooted maps of a given genus. (English) Zbl 1102.05033
Authors’ abstract: Let \({\mathcal N}_g(f)\) denote the number of rooted maps of genus \(g\) having \(f\) edges. An exact formula for \({\mathcal N}_g(f)\) is known for \(g= 0\) [W. T. Tutte, Can. J. Math. 15, 249–271 (1963; Zbl 0115.17305)], \(g=1\) [D. Arquès, J. Comb. Theory, Ser B 43, 253–274 (1987; Zbl 0628.05040)], \(g=2,3\) [E. A. Bender and E. R. Canfield, J. Comb. Theory, Ser. B 53, No. 2, 293–299 (1991; Zbl 0751.05052)]. In the present paper we derive an enumeration formula for the number \(\Theta_\gamma(e)\) of unrooted maps on an orientable surface \(S_\gamma\) of a given genus \(\gamma\) and with a given number of edges \(e\). It has a form of a linear combination \(\sum_{i,j} c_{i,j}{\mathcal N}_{g_j}(f_i)\) of numbers of rooted maps \({\mathcal N}_{g_j}(f_i)\) for some \(g_j\leq\gamma\) and \(f_i\leq e\). The coefficients \(c_{i,j}\) are functions of \(\gamma\) and \(e\). We consider the quotient \(S_\gamma/ Z_\ell\) of \(S_\gamma\) by a cyclic group of automorphisms \(Z_\ell\) as a two-dimensional orbifold \(O\). The task of determining \(c_{i,j}\) requires solving the following two subproblems:
(a) to compute the number \(Epi_{O}(\Gamma,Z_\ell)\) of order-preserving epimorphisms from the fundamental group \(\Gamma\) of the orbifold \(O=S_\gamma/Z_\ell\) onto \(Z_\ell\);
(b) to calculate the number of rooted maps on the orbifold \(O\) which lifts along the branched covering \(S_\gamma\to S_\gamma/ Z_\ell\) to maps on \(S_\gamma\) with the given number \(e\) of edges.
The number \(Epi_{O}(\Gamma,Z_\ell)\) is expressed in terms of classical number-theoretical functions. The other problem is reduced to the standard enumeration problem of determining the numbers \({\mathcal N}_g(f)\) for some \(g\leq\gamma\) and \(f\leq e\). It follows that \(\Theta_\gamma(e)\) can be calculated whenever the numbers \({\mathcal N}_g(f)\) are known for \(g\leq\gamma\) and \(f\leq e\). In the end of the paper the above approach is applied to derive the functions \(\Theta_\gamma(e)\) explicitly for \(\gamma\leq 3\). We note that the function \(\Theta_\gamma(e)\) was known only for \(\gamma=0\) [V. A. Liskovets, Colloq. Math. Soc. János Bolyai 25, 479–494 (1981; Zbl 0519.05040)]. Tables containing the numbers of isomorphism classes of maps with up to 30 edges for genus \(\gamma=1,2,3\) are presented.

05C30 Enumeration in graph theory
05C10 Planar graphs; geometric and topological aspects of graph theory
Full Text: DOI
[1] Apostol, T.M., Introduction to analytical number theory, (1976), Springer Berlin
[2] Arques, D., Relations fonctionelles et dénombrement des Cartes poinées sur le tore, J. combin. theory ser. B, 43, 253-274, (1987), (in French) · Zbl 0628.05040
[3] Arques, D.; Giorgetti, A., Énumération des Cartes pointées sur une surface orientable de genre quelconque en fonction des nombres de sommets et de faces, J. combin. theory ser. B, 77, 1-24, (1999) · Zbl 1029.05073
[4] Bender, E.A.; Canfield, E.A., The number of rooted maps on an orientable surface, J. combin. theory ser. B, 53, 293-299, (1991) · Zbl 0751.05052
[5] Bender, E.A.; Wormald, N.C., The number of loopless planar maps, Discrete math., 54, 235-237, (1985) · Zbl 0563.05032
[6] Bender, E.A.; Canfield, E.A.; Robinson, R.W., The enumeration of maps on the torus and on the projective plane, Canad. math. bull., 31, 257-271, (1988) · Zbl 0617.05036
[7] Bogopol’skii, O.V., Classifying the action of finite groups on oriented surface of genus 4, Siberian adv. math., 7, 4, 9-38, (1997)
[8] Bousquet, M.; Labelle, G.; Leroux, P., Enumeration of planar two-face maps, Discrete math., 222, 1-25, (2000) · Zbl 0993.05089
[9] Broughton, S.A., Classifying finite group actions on surface of low genus, J. pure appl. algebra, 69, 233-270, (1990) · Zbl 0722.57005
[10] Bujalance, E.; Etayo, J.J.; Gamboa, J.M.; Gromadzki, G., Automorphism groups of compact bordered Klein surfaces, (1990), Springer Berlin · Zbl 0709.14021
[11] Chalambides, Ch.A., Enumerative combinatorics, (2002), Chapman & Hall/CRC Boca Raton, FL
[12] Comtet, L., Advanced combinatorics, (1974), Reidel Dordrecht
[13] R. Cori, Bijective census of rooted planar maps: A survey, in: A. Barlotti, M. Delest, R. Pinzani (Eds.), Proceedings of the Fifth Conference on Formal Power Series and Algebraic Combinatorics, Florence, 1993, pp. 131-141
[14] Gross, J.L.; Tucker, T.W., Topological graph theory, (1987), Wiley New York · Zbl 0621.05013
[15] Harary, F.; Tutte, W.T., The number of plane trees with given partition, Mathematika, 11, 99-101, (1964) · Zbl 0193.53401
[16] Harary, F.; Prins, G.; Tutte, W.T., The number of plane trees, Indag. math., 26, 319-329, (1964) · Zbl 0126.19002
[17] Harvey, W.J., Cyclic groups of automorphisms of a compact Riemann surface, Quart. J. math. Oxford, 17, 86-97, (1966) · Zbl 0156.08901
[18] Jones, G.A., Enumeration of homomorphisms and surface-coverings, Quart. J. math. Oxford, 46, 2, 485-507, (1995) · Zbl 0859.57001
[19] Jones, G.A.; Singerman, D., Theory of maps on orientable surfaces, Proc. London math. soc., 37, 273-307, (1978) · Zbl 0391.05024
[20] Jordan, C., Traité des substitutions et des équationes algébriques, (1989), Éditions J. Gabay Paris, xvi+669 p · Zbl 0828.01011
[21] Kuribayashi, I.; Kuribayashi, A., Automorphism groups of compact Riemann surfaces of genera three and four, J. pure appl. algebra, 65, 277-292, (1990) · Zbl 0709.30036
[22] Labelle, G.; Leroux, P., Enumeration of (uni- or bicolored) plane trees according to their degree distribution, Discrete math., 157, 227-240, (1996) · Zbl 0868.05030
[23] Liskovets, V.A., A census of nonisomorphic planar maps, (), 479-494
[24] Liskovets, V.A., Enumeration of nonisomorphic planar maps, Selecta math. sovietica, 4, 303-323, (1985) · Zbl 0578.05033
[25] Liskovets, V.A., A reductive technique for enumerating non-isomorphic planar maps, Discrete math., 156, 197-217, (1996) · Zbl 0857.05044
[26] Liskovets, V.A., Reductive enumeration under mutually orthogonal group actions, Acta appl. math., 52, 91-120, (1998), (Section 7) · Zbl 0908.05003
[27] Liskovets, V.A.; Walsh, T.R., The enumeration of non-isomorphic 2-connected planar maps, Canad. J. math., 35, 417-435, (1983) · Zbl 0519.05041
[28] Liu, Y.P., Some enumerating problems of maps with vertex partition, Kexue tongbao sci. bul., English ed., 31, 1009-1014, (1986) · Zbl 0604.05023
[29] Liu, Y.P., Enumerative theory of maps, (1999), Kluwer Academic London · Zbl 0990.05070
[30] Malnič, A.; Nedela, R.; Škoviera, M., Regular homomorphisms and regular maps, European J. combin., 23, 449-461, (2002) · Zbl 1007.05062
[31] Nicol, C.A.; Vandiver, H.S., A von sterneck arithmetical function and restricted partitions with respect to modulus, Proc. natl. acad. sci. USA, 40, 825-835, (1954) · Zbl 0056.04001
[32] Sloane, N.J.A.; Plouffe, S., The encyclopedia of integer sequences, (1995), Academic Press San Diego · Zbl 0845.11001
[33] Schulte, J., Über die jordansche verallgemeinerung der eulerschen funktion, Results math., 36, 354-364, (1999) · Zbl 0942.11006
[34] Tutte, W.T., A census of planar triangulations, Canad. J. math., 14, 21-38, (1962) · Zbl 0103.39603
[35] Tutte, W.T., A census of Hamiltonian polygons, Canad. J. math., 14, 402-417, (1962) · Zbl 0105.17601
[36] Tutte, W.T., A census of slicings, Canad. J. math., 14, 708-722, (1962) · Zbl 0111.35202
[37] Tutte, W.T., A census of planar maps, Canad. J. math., 15, 249-271, (1963) · Zbl 0115.17305
[38] Tutte, W.T., The number of planted plane trees with a given partition, Amer. math. monthly, 71, 64-74, (1964) · Zbl 0117.17403
[39] Walsh, T.R.S., Generating nonisomorphic maps without storing them, SIAM J. algebraic discrete methods, 4, 161-178, (1983) · Zbl 0521.05034
[40] Walsh, T.R.S.; Lehman, A.B., Counting rooted maps by genus I, J. combin. theory ser. B, 13, 192-218, (1972) · Zbl 0228.05108
[41] Walsh, T.R.S.; Lehman, A.B., Counting rooted maps by genus II, J. combin. theory ser. B, 13, 122-141, (1973) · Zbl 0228.05109
[42] Wong, C.K., A uniformization theorem for arbitrary Riemann surfaces with signature, Proc. amer. math. soc., 28, 2, 489-495, (1971) · Zbl 0218.30020
[43] Wormald, N.C., Counting unrooted planar maps, Discrete math., 36, 205-225, (1981) · Zbl 0467.05034
[44] Wormald, N.C., On the number of planar maps, Canad. J. math., 33, 1, 1-11, (1981) · Zbl 0408.57006
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