Abstract
For a graph G of order n and an integer k with \(1\le k\le n\), let \(\pi (G,k)\) denote the minimum number of edges spanned by k vertices and call \((\pi (G,1),\pi (G,2),\ldots ,\pi (G,n))\) the sparse sequence of G. In 2022, Katona proposed a problem: Can we find necessary and sufficient conditions for a function f(k) under which a graph G exists such that \(\pi (G,k)=f(k)\)? In this paper, we solve this problem partly and give a sufficient condition for a sequence to be the sparse sequences of a general graph and a tree, respectively. For any tree T and \(1\le k\le n-2\), we show that \(\pi (T,k+2)-\pi (T,k+1)\ge \pi (T,k+1)-\pi (T,k)-1\). Finally, we introduce the subgraph-size polynomial of a graph and establish a recursive relation for graphs with a cut edge, based on which we give a recursive algorithm for determining the sparse sequences of a tree and obtain the subgraph-size polynomial of spider graphs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Ahlswede, R.: Simple hypergraphs with maximal number of adjacent pairs of edges. J. Comb. Theory (B) 28, 164–167 (1980)
Asahiro, Y., Iwama, K., Tamaki, H., Tokuyama, T.: Greedily finding a dense subgraph. J. Algorithm 34, 203–221 (2000)
Bhaskara,A., Charikar,M., Chlamtac,E., Feige,U., Vijayaraghavan,A.: Detecting high log-densities an \(o(n^{\frac{1}{4}})\) approximation for densest \(k\)-subgraph (2010). CoRR, arXiv:1001.2891
Billionnet, A.: Different formulations for solving the heaviest \(k\)-subgraph problem. Inf. Syst. Oper. Res. 43(3), 171–186 (2005)
Bonomo, F., Maranco, J., Saban, D., Stier-Moses, N.: A polyhedral study of the maximum edge subgraph problem. Discrete Appl. Math. 160(18), 2573–2590 (2012)
Corneil, D., Perl, Y.: Clustering and domination in perfect graphs. Discrete Appl. Math. 9, 27–39 (1984)
Das, S., Gan, W., Sudakov, B.: The minimum number of disjoint pairs in set systems and related problems. Combinatorica 36(6), 623–660 (2016)
Erdős, P., Ko, C., Rado, R.: Intersection theorems for systems of finite sets. Q. J. Math. Oxf. (2) 12, 313–320 (1961)
Feige,U., Seltser,M.: On the densest \(k\)-subgraph problem. Technical report CS97-16, Weizmann Institute (1997)
Feige, U., Kortsarz, G., Peleg, D.: The dense \(k\)-subgraph problem. Algorithmica 29, 410–421 (2001)
Katona, G.: A generalization of the independence number. Discrete Appl. Math. 321, 1–3 (2001)
Keil, J.M., Brecht, T.B.: The complexity of clustering in planar graphs. J. Comb. Math. Comb. Comput. 9, 155–159 (1991)
Kleitman,D.: A conjecture of Erdős-Katona on commensurable pairs among subsets of an \(n\)-set. In: Theory of Graphs: Proc. Colloq. Tihany, pp. 215–218 (1966)
Kortsarz, G., Peleg, D.: On choosing a dense subgraph. In: Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science, pp. 692–701. IEEE, Piscataway (1993)
Qian, J., Engel, K., Xu, W.: A generalization of Sperner’s theorem and an application to graph orientations. Discrete Appl. Math. 157, 2170–2176 (2009)
Sperner, E.: Ein Satz ber Untermengen einer endlichen Menge. Math. Z. 27, 544–548 (1928)
Acknowledgements
This work was supported by the National Natural Science Foundation of China [Grant Number 12361070].
Funding
This work was supported by the National Natural Science Foundation of China [Grant Number 12361070]. Sumin Huang has received research support from China Scholarship Council.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. All authors contributed to the study conception and design.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Research supported by NSFC [12361070].
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Huang, S., Qian, J. The sparse sequences of graphs. Graphs and Combinatorics 40, 119 (2024). https://doi.org/10.1007/s00373-024-02860-y
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00373-024-02860-y