Skip to main content
SpringerLink
Log in

Maximum Cuts in \(\mathscr {H}\)-Free Graphs

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

For a graph G, let f(G) be the maximum number of edges in a cut of G. For a positive integer m and a set of graphs \(\mathscr {H}\), let \(f(m,\mathscr {H})\) be the minimum possible cardinality of f(G), as G ranges over all graphs on m edges which contain no graphs in \(\mathscr {H}\). Suppose that \(r\ge 8\) is an even integer and \(\mathscr {H}=\{C_{5},C_{6},\ldots ,C_{r-1}\}\). In this paper, we prove that \(f(m,\mathscr {H})\ge m/2+cm^{\frac{r}{r+1}}\) for some real \(c>0\), which improves a result of Alon et al. We also give a general lower bound on \(f(m,\mathscr {H})\) when \(\mathscr {H}=\{K_{s}+\overline{K_{t}}\}\). This extends a result of Zeng and Hou.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alon, N.: Bipartite subgraphs. Combinatorica 16, 301–311 (1996)

    Article  MathSciNet  Google Scholar 

  2. Alon, N., Bollobás, B., Krivelevich, M., Sudakov, B.: Maximum cuts and judicious partitions in graphs without short cycles. J. Combin. Theory, Ser. B 88, 329–346 (2003)

    Article  MathSciNet  Google Scholar 

  3. Alon, N., Krivelevich, M., Sudakov, B.: Maxcut in \(H\)-free graphs. Combin. Probab. Comput. 14, 629–647 (2005)

    Article  MathSciNet  Google Scholar 

  4. Bollobás, B., Scott, A.D.: Exact bounds for judicious partitions of graphs. Combinatorica 19, 473–486 (1999)

    Article  MathSciNet  Google Scholar 

  5. Bollobás, B., Scott, A.D.: Problems and results on judicious partitions. Random Struct. Alg. 21, 414–430 (2002)

    Article  MathSciNet  Google Scholar 

  6. Edwards, C.S.: Some extremal properties of bipartite graphs. Can. J. Math. 3, 475–485 (1973)

    Article  Google Scholar 

  7. Edwards, C.S.: An improved lower bound for the number of edges in a largest bipartite subgraph. In: Proceedings of 2nd Czechoslovak Symposium on Graph Theory, pp. 167–181 (1975)

  8. Erdös, P.: Problems and results in graph theory and combinatorial analysis. Graph Theory and Related Topics (Proc. Conf. Waterloo, 1977), pp. 153–163. Academic Press, New York (1979)

    Google Scholar 

  9. Erdös, P., Gallai, T.: Maximal paths and circuits in graphs. Acta Math. Acad. Sci. Hungar. 10, 337–356 (1959)

    Article  MathSciNet  Google Scholar 

  10. Fan, G., Hou, J., Zeng, Q.: A bound for judicious \(k\)-partitions of graphs. Discret. Appl. Math. 179, 86–99 (2014)

    Article  MathSciNet  Google Scholar 

  11. Li, Y., Rousseau, C., Zang, W.: Asymptotic upper bounds for Ramsey functions. Graphs Combin. 17, 123–128 (2001)

    Article  MathSciNet  Google Scholar 

  12. Jensen, T.R., Toft, B.: Graph Coloring Problems, Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1995)

    Google Scholar 

  13. Ma, J., Yen, P., Yu, X.: On several partitioning problems of Bollobás and Scott. J. Combin. Theory, Ser. B 100, 631–649 (2010)

    Article  MathSciNet  Google Scholar 

  14. Ma, J., Yu, X.: Bounds for pairs in partitions of graphs. Discret. Math. 310, 2069–2081 (2010)

    Article  MathSciNet  Google Scholar 

  15. Poljak, S., Tuza, Z.: Bipartite subgraph of triangle-free graphs. SIAM J. Discret. Math. 7, 307–313 (1994)

    Article  MathSciNet  Google Scholar 

  16. Shearer, J.: A note on bipartite subgraphs of triangle-free graphs. Random Struct. Alg. 3, 223–226 (1992)

    Article  MathSciNet  Google Scholar 

  17. Shearer, J.: On the independence number of sparse graphs. Random Struct. Alg. 7, 269–271 (1995)

    Article  MathSciNet  Google Scholar 

  18. Xu, B., Yu, X.: Better bounds for \(k\)-partitions of graphs. Comb. Probab. Comput. 20, 631–640 (2011)

    Article  MathSciNet  Google Scholar 

  19. Xu, B., Yu, X.: On judicious bisections of graphs. J. Comb. Theory, Ser. B 106, 30–69 (2014)

    Article  MathSciNet  Google Scholar 

  20. Yannakakis, M.: Node- and edge-deletion NP-complete problems. STOC 20, 253–264 (1978)

    MathSciNet  MATH  Google Scholar 

  21. Zeng, Q., Hou, J.: Bipartite subgraphs of \(H\)-free graphs. Bull. Aust. Math. Soc. 96, 1–13 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Colledge of Mathematics and Computer Science, Yan’an University, Shanxi, 716000, China

    Huawen Ma

  2. Center for Discrete Mathematics, Fuzhou University, Fujian, 350003, China

    Huawen Ma

Authors
  1. Huawen Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huawen Ma.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by YDBK (No. 2019-71).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ma, H. Maximum Cuts in \(\mathscr {H}\)-Free Graphs. Graphs and Combinatorics 36, 1503–1516 (2020). https://doi.org/10.1007/s00373-020-02189-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-020-02189-2

Keywords

Search

Navigation