Abstract
In this work, we study the problem of finding a weighted graph with exactly k minimum spanning trees (MSTs, in short) while minimizing the number of vertices. While finding a graph with k MSTs is easy, finding such a graph with the minimum number of vertices remains an interesting open problem. Recently, Stong [15] proved an upper bound within \(\log k\) multiplicative factor of the minimum. In this work, we prove the following results which make further progress on this problem:
-
1.
Large weights do not help in constructing a minimal weighted graph with prime number of spanning trees.
-
2.
For \(n\ge 6\) and \(1 \le k \le n^2\), n vertices suffice for constructing a graph with k spanning trees.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Azarija, J., Škrekovski, R.: Euler’s idoneal numbers and an inequality concerning minimal graphs with a prescribed number of spanning trees. Mathematica Bohemica (2012)
Bogdanowicz, Z.R.: Formulas for the number of spanning trees in a fan. Appl. Math. Sci. 2(16), 781–786 (2008)
Cheng, C.S.: Maximizing the total number of spanning trees in a graph: two related problems in graph theory and optimum design theory. J. Comb. Theor. Ser. B 31(2), 240–248 (1981)
Gilbert, B., Myrvold, W.: Maximizing spanning trees in almost complete graphs. Netw. Int. J. 30(1), 23–30 (1997)
Grenet, B., Kaltofen, E.L., Koiran, P., Portier, N.: Symmetric determinantal representation of weakly-skew circuits. In: Schwentick, T., Dürr, C. (eds.) 28th International Symposium on Theoretical Aspects of Computer Science, STACS 2011, March 10-12, 2011, Dortmund, Germany. LIPIcs, vol. 9, pp. 543–554. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011). https://doi.org/10.4230/LIPIcs.STACS.2011.543
Grenet, B., Monteil, T., Thomassé, S.: Symmetric determinantal representations in characteristic 2. CoRR abs/1210.5879 (2012). http://arxiv.org/abs/1210.5879
Kelmans, A.K.: On graphs with the maximum number of spanning trees. Random Struct. Algorithms 9(1–2), 177–192 (1996)
Mokhlissi, R., El Marraki, M.: Enumeration of spanning trees in a closed chain of fan and wheel. Appl. Math. Sci. 8(82), 4053–4061 (2014)
Nebeskỳ, L.: On the minimum number of vertices and edges in a graph with a given number of spanning trees. Časopis pro pěstování matematiky 98(1), 95–97 (1973)
Petingi, L., Boesch, F., Suffel, C.: On the characterization of graphs with maximum number of spanning trees. Discret. Math. 179(1–3), 155–166 (1998)
Petingi, L., Rodriguez, J.: A new technique for the characterization of graphs with a maximum number of spanning trees. Discret. Math. 244(1–3), 351–373 (2002)
Rote https://mathoverflow.net/users/30800/gwith a prescribed number of spanning trees.MathOverflow. https://mathoverflow.net/q/122093
Sedláček, J.: On the minimal graph with a given number of spanning trees. Can. Math. Bull. 13(4), 515–517 (1970)
Shier, D.R.: Maximizing the number of spanning trees in a graph with n nodes and m edges. J. Res. National Bur. Stan. Sect. B 78(193–196), 3 (1974)
Stong, R.: Minimal graphs with a prescribed number of spanning trees. Australas. J. Comb. 82(2), 182–196 (2022)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Dutta, A., Muthu, R., Tawari, A., Sunitha, V. (2024). Exactly k MSTs: How Many Vertices Suffice?. In: Wu, W., Guo, J. (eds) Combinatorial Optimization and Applications. COCOA 2023. Lecture Notes in Computer Science, vol 14461. Springer, Cham. https://doi.org/10.1007/978-3-031-49611-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-031-49611-0_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-49610-3
Online ISBN: 978-3-031-49611-0
eBook Packages: Computer ScienceComputer Science (R0)