×

zbMATH — the first resource for mathematics

Counting rooted 4-regular unicursal planar maps. (English) Zbl 1378.05091
Summary: A map is 4-regular unicursal if all its vertices are 4-valent except two odd-valent vertices. This paper investigates the number of rooted 4-regular unicursal planar maps and presents some formulae for such maps with four parameters: the number of edges, the number of inner faces and the valencies of the two odd vertices.
MSC:
05C30 Enumeration in graph theory
05C10 Planar graphs; geometric and topological aspects of graph theory
05C45 Eulerian and Hamiltonian graphs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bousquet-Mélou, M.; Schaeffer, G., Enumeration of planar constellations, Adv. Appl. Math., 24, 337-368, (2000) · Zbl 0955.05004
[2] Brown, W.G., Enumeration of quadrangular dissections of the disc, Canad. J. Math., 17, 302-317, (1965) · Zbl 0138.19104
[3] Cai, J.L.; Liu, Y.P., The number of nearly 4-regular planar maps, Utilitas Mathematica, 58, 243-254, (2000) · Zbl 0964.05032
[4] Cai, J.L.; Liu, Y.P., Enumerating rooted Eulerian planar maps, Acta Math. Sci., 21, 289-294, (2001) · Zbl 0989.05055
[5] Cai, J.L., The number of rooted loopless Eulerian planar maps, Acta Math. Appl. Sin., 29, 210-216, (2006)
[6] Cai, J.L.; Liu, Y.P., The number of rooted Eulerian planar maps, Science in China Series A: Mathematics, 51, 2005-2012, (2008) · Zbl 1246.05008
[7] Good, I.J., Generalizations to several varibles of lagrange’s expansion, with applications to stochastic processes, Proc. Camb. Phil. Soc., 56, 367-380, (1960) · Zbl 0135.18802
[8] Koganov, L.M.; Liskovets, V.A.; Walsh, T.R.S., Total vertex enumeration in rooted planar maps, Ars Combin., 54, 149-160, (2000) · Zbl 0993.05087
[9] Liskovets, V.A., Enumeration of nonisomorphic planar maps, Selecta Math. Soviet, 4, 304-323, (1985) · Zbl 0578.05033
[10] Liskovets, V.A., A pattern of asymptotic vertex valency distributions in planar maps, J. Combin. Theory Ser. B, 75, 116-133, (1999) · Zbl 0930.05050
[11] Liskovets, V.A.; Walsh, T.R.S., Enumeration of Eulerian and unicursal planar maps, Discrete Math., 282, 209-221, (2004) · Zbl 1051.05049
[12] Liskovets, V.A., Walsh, T.R.S. Enumeration of unrooted maps on the plane, Rapport technique. UQAM, No. 2005-01, Montreal, Canada, 2005
[13] Liu, Y.P., On functional equation arising from map enumerations, Discrete Math., 123, 93-109, (1993) · Zbl 0792.05074
[14] Liu, Y.P., On the number of Eulerian planar maps, Acta Math. Sci., 12, 418-423, (1992) · Zbl 0774.05047
[15] Liu, Y.P., Enumeration of rooted bipartite planar maps, Acta Math. Sci., 9, 21-28, (1989)
[16] Liu, Y.P. Enumerative Theory of Maps. Kluwer, Boston, 1999 · Zbl 0990.05070
[17] Liu, Y.P., A polyhedral theory on graphs, Acta Math. Sin., 10, 136-142, (1994) · Zbl 0812.05017
[18] Liu, Y.P., Combinatorial invariants on graphs, Acta Math. Sin., 11, 211-220, (1995) · Zbl 0838.05038
[19] Liu, Y.P., On the number of rooted c-nets, J. Combin. Theory Ser. B, 36, 118-123, (1984) · Zbl 0538.05042
[20] Long, S.D.; Cai, J.L., Enumeration of rooted unicursal planar maps (in Chinese), Acta Math. Sin., 53, 9-16, (2010) · Zbl 1224.05285
[21] Mullin, R.C.; Schellenberg, P.J., The enumeration of c-nets via quadrangulations, J. Combin. Theory, 4, 256-276, (1964) · Zbl 0183.52403
[22] Ren, H.; Liu, Y.P., The number of loopless 4-regular maps on the projective plane, J. Combin. Theory Ser. B, 84, 84-99, (2002) · Zbl 1018.05046
[23] Ren, H.; Liu, Y.P.; Li, Z.X., Enumeration of 2-connected loopless 4-regular maps on the plane, European J. Combin., 23, 93-111, (2002) · Zbl 1005.05025
[24] Tutte, W.T., A census of slicings, Canad. J. Math., 14, 708-722, (1962) · Zbl 0111.35202
[25] Tutte, W.T., A census of planar maps, Canad. J. Math., 15, 249-271, (1963) · Zbl 0115.17305
[26] Walsh, T.R.S., Hypermaps versus bipartite maps, J. Combin. Theory Ser. B, 18, 155-163, (1975) · Zbl 0302.05101
[27] Whittaker, E.T., Watson, G.N. A course of modern analysis. Cambridge, 1940 · JFM 45.0433.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.