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Enumeration of Unrooted Odd-Valent Regular Planar Maps
Zhicheng Gao1, Valery A. Liskovets2, and Nicholas Wormald3
1School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S 5B6, Canada
zgao@math.carleton.ca
2Institute of Mathematics, National Academy of Sciences, Minsk 220072, Belarus
liskov@im.bas-net.by
3Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario,\break\phantom{\raisebox{1.1mm}{\footnotesize 1}}N2L 3G1, Canada
nwormald@uwaterloo.ca
Annals of Combinatorics 13 (2) pp.233-259 June, 2009
AMS Subject Classification: 05C05, 92D15
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 exceptional vertex of degree 1. The same applies to the generating functions for these numbers. Similar results, but without the vertex of degree 1, were obtained by the first author and Rahman. We also show, by manipulating a recursion of Bouttier, Di Francesco and Guitter, that there are closed formulae when the exceptional vertex has arbitrary degree. We combine these formulae with results of the second author to count unrooted regular maps of odd degree. In this way we obtain, for each even n, a closed formula for the function fn whose value at odd positive integers $r$ is the number of unrooted maps (up to orientation-preserving homeomorphisms) with n vertices and degree $r$. The formula for fn becomes more cumbersome as $n$ increases, but for n> 2 each has a bounded number of terms independent of r.
Keywords: odd degree, unrooted map, rooted planar map, regular map, rotation, quotient map, closed formula

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