Enumeration of unrooted odd-valent regular planar maps Zhicheng Gao ∗ Faculty of Business Administration University of Macau Macau China zcgao@umac.mo Valery A. Liskovets † Institute of Mathematics National Academy of Sciences Minsk, 220072 Belarus liskov@im.bas-net.by Nicholas Wormald ‡ Department of Combinatorics and Optimization University of Waterloo Waterloo ON Canada N2L 3G1 nwormald@uwaterloo.ca June 10, 2005 Abstract We derive closed formulae for the numbers of rooted maps with a fixed number of vertices of the same odd degree except for the root vertex and one other vertex of degree 1. A similar result, but without the vertex of degree 1, was obtained by the first author and Rahman. These formulae are combined with results of the second author to count unrooted regular maps of odd degree. We succeed in finding, for each even n, a closed formula f n (r) for the number of unrooted maps (up to orientation-preserving homeomorphisms) with n vertices and odd degree r, provided r is an odd prime or gcd(r, n − 2) = 1 or n = 2. The functions f n become more cumbersome as n increases, but for n> 2 each has a bounded number of terms independent of r. 2000 Mathematics Subject Classification. Primary: 05C30, Secondary: 05A15. Key words. odd degree, unrooted map, rooted planar map, regular map, rotation, quotient map, closed formula * Supported by University of Macau. † Supported by the Belarusian RFFR (grant No. F05-227). ‡ Supported by the Canada Research Chairs program, NSERC and the University of Macau. 1