Abstract
A series-parallel graph is a graph that does not contain a complete graph with four vertices as a minor. An explicit formula for the number of labeled series-parallel tricyclic graphs with a given number of vertices is obtained, and the corresponding asymptotics for the number of such graphs with a large number of vertices is found. We prove that under a uniform probability distribution, the probability that the labeled tricyclic graph is a series-parallel graph is asymptotically equal to 13/15.
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References
M. Bodirsky, O. Gimenez, M. Kang, and M. Noy, “Enumeration and limit laws of series-parallel graphs,” Eur. J. Combin., 28, No. 8, 2091–2105 (2007).
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, New York (1990).
C. McDiarmid, A. Scott, “Random graphs from a block-stable class,” Eur. J. Combin., 58, 96–106 (2016).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 2 [in Russian], Nauka, Moscow (1983).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 3 [in Russian], Nauka, Moscow (1986).
S. Radhavan, “Low-connectivity network design on series-parallel graphs,” Networks, 43, No. 3, 163–176 (2004).
J. Riordan, Combinatorial Identities, Wiley, New York (1982).
V. A. Voblyi, “Enumeration of labeled connected graphs with given order and number of edges,” J. Appl. Industr. Math., 10, No. 2, 302–310 (2016).
V. A. Voblyi, “The number of labeled tetracyclic series-parallel blocks,” Prill. Diskr. Mat., No. 47, 57–61 (2020).
V. A. Voblyi, “The second Riddel relation and its consequences,” 2019, 13, No. 1, 168–174.
V. A. Voblyi, “The number of labeled outerplanar k-cyclic graphs,” Mat. Zametki, 103, No. 5, 657–666 (2018).
V. A. Voblyi and A. M. Meleshko, “On the number of labeled series-parallel tricyclic blocks,” in: Proc. XV Int. Conf. “Algebra, Number Theory, and Discrete Geometry. Contemporary Problems and Applications” (Tula, May 28-31, 2018), Tula, Tula Pedagogical State Univ., pp. 168–170.
E. M. Wright, “The number of connected sparsely edged graphs,” J. Graph Theory., 1, No. 4 (1977), pp. 317–330.
E. M. Wright, “The number of connected sparsely edged graphs,” J. Graph Theory., 2, No. 4 (1978), pp. 299–305.
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 177, Algebra, 2020.
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Voblyi, V.A. Enumeration of Labeled Series-Parallel Tricyclic Graphs. J Math Sci 275, 778–782 (2023). https://doi.org/10.1007/s10958-023-06720-7
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DOI: https://doi.org/10.1007/s10958-023-06720-7