Abstract
We investigate the maximum number of simple cycles and the maximum number of Hamiltonian cycles in a planar graph G with n vertices. Using the transfer matrix method we construct a family of graphs which have at least 2.4262n simple cycles and at least 2.0845n Hamilton cycles.
Based on counting arguments for perfect matchings we prove that 2.3404n is an upper bound for the number of Hamiltonian cycles. Moreover, we obtain upper bounds for the number of simple cycles of a given length with a face coloring technique. Combining both, we show that there is no planar graph with more than 2.8927n simple cycles. This reduces the previous gap between the upper and lower bound for the exponential growth from 1.03 to 0.46.
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References
Alt, H., Fuchs, U., Kriegel, K.: On the number of simple cycles in planar graphs. Combinatorics, Probability & Computing 8(5), 397–405 (1999)
Sharir, M., Welzl, E.: On the number of crossing-free matchings (cycles, and partitions). In: Proc. 17th ACM-SIAM Sympos. Discrete Algorithms, pp. 860–869. ACM Press, New York (2006)
Sharir, M., Welzl, E.: Random triangulations of planar point sets. In: 22nd Annu. ACM Sympos. Comput. Geom., pp. 273–281. ACM Press, New York (2006)
Barequet, G., Moffie, M.: The complexity of Jensen’s algorithm for counting polyominoes. In: Proc. 1st Workshop on Analytic Algorithmics and Combinatorics, pp. 161–169 (2004)
Conway, A., Guttmann, A.: On two-dimensional percolation. J. Phys. A: Math. Gen. 28(4), 891–904 (1995)
Jensen, I.: Enumerations of lattice animals and trees. J. Stat. Phys. 102(3–4), 865–881 (2001)
Stanley, R.P.: Enumerative combinatorics, vol. 2. Cambridge University Press, Cambridge (1999)
Graham, A.: Nonnegative Matrices and Applicable Topics in Linear Algebra. John Wiley & Sons, New York (1987)
Stoffel, A.: Software for Counting Cycles on the Twisted Cylinder, http://page.mi.fu-berlin.de/~schulza/cyclecount/
Lovasz, L., Plummer, M.: Matching theory. Elsevier, Amsterdam (1986)
Aichholzer, O., Hackl, T., Vogtenhuber, B., Huemer, C., Hurtado, F., Krasser, H.: On the number of plane graphs. In: Proc. 17th ACM-SIAM Sympos. Discrete Algorithms, pp. 504–513. ACM Press, New York (2006)
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Buchin, K., Knauer, C., Kriegel, K., Schulz, A., Seidel, R. (2007). On the Number of Cycles in Planar Graphs. In: Lin, G. (eds) Computing and Combinatorics. COCOON 2007. Lecture Notes in Computer Science, vol 4598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73545-8_12
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DOI: https://doi.org/10.1007/978-3-540-73545-8_12
Publisher Name: Springer, Berlin, Heidelberg
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