Abstract
We address the problem of counting geometric graphs on point sets. Using analytic combinatorics we show that the so-called double chain point configuration of N points has Ω* (12.31N) non-crossing spanning trees and Ω* (13.40N) non-crossing forests. This improves the previous lower bounds on the maximum number of non-crossing spanning trees and of non-crossing forests among all sets of N points in general position given by Dumitrescu, Schulz, Sheffer and Tóth in 2011. A new upper bound of O* (22.12N) for the number of non-crossing spanning trees of the double chain is also obtained.
Supported by Projects MTM2012-30951, DGR2009-SGR1040, and EuroGIGA, CRP ComPoSe: grant EUI-EURC-2011-4306.
Supported by Projects MTM2011-24097 and DGR2009-SGR1040.
We use the O*-, Θ* -, and Ω*-notation to describe the asymptotic growth of the number of geometric graphs as a function of the number N of points, neglecting polynomial factors.
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© 2013 Scuola Normale Superiore Pisa
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Huemer, C., de Mier, A. (2013). An improved lower bound on the maximum number of non-crossing spanning trees. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_46
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DOI: https://doi.org/10.1007/978-88-7642-475-5_46
Publisher Name: Edizioni della Normale, Pisa
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Online ISBN: 978-88-7642-475-5
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