Skip to main content
Springer Nature Link
Log in

Monochromatic disconnection: Erdős-Gallai-type problems and product graphs

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

For an edge-colored graph G, we call an edge-cut M of G monochromatic if the edges of M are colored with a same color. The graph G is called monochromatically disconnected if any two distinct vertices of G are separated by a monochromatic edge-cut. The monochromatic disconnection number, denoted by md(G), of a connected graph G is the maximum number of colors that are allowed to make G monochromatically disconnected. In this paper, we solve the Erdős-Gallai-type problems for the monochromatic disconnection, and give the monochromatic disconnection numbers for four graph products, i.e., Cartesian, strong, lexicographic, and tensor products.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Bondy JA, Murty USR (2008) Graph theory, GTM 244, Springer

  • Bonsma P, Farley AM, Proskurowski A (2012) Extremal graphs having no matching cuts. J. Graph Theory 69:206–222

    Article  MathSciNet  Google Scholar 

  • Cai Q, Li X, Wu D (2017) Erdős-Gallai-type results for colorful monochromatic connectivity of a graph. J. Comb. Optim. 33(1):123–131

    Article  MathSciNet  Google Scholar 

  • Cai Q, Li X, Wu D (2018) Some extremal results on the colorful monochromatic vertex-connectivity of a graph. J. Comb. Optim. 35:1300–1311

    Article  MathSciNet  Google Scholar 

  • Caro Y, Yuster R (2011) Colorful monochromatic connectivity. Discrete Math. 311:1786–1792

    Article  MathSciNet  Google Scholar 

  • Chartrand G, Devereaux S, Haynes TW, Hedetniemi ST, Zhang P (2018) Rainbow disconnection in graphs. Discuss. Math. Graph Theory 38(4):1007–1021

    Article  MathSciNet  Google Scholar 

  • Gu R, Li X, Qin Z, Zhao Y (2017) More on the colorful monochromatic connectivity. Bull. Malays. Math. Sci. Soc. 40(4):1769–1779

    Article  MathSciNet  Google Scholar 

  • Jiang H, Li X, Zhang Y (2019) Erdős-Gallai-type results for total monochromatic connection of graphs. Discuss. Math. Graph Theory 39(4):775–785

    Article  MathSciNet  Google Scholar 

  • Jin Z, Li X, Wang K (2020) The monochromatic connectivity of graphs. Taiwanese J. Math. 24(4):785–815

    Article  MathSciNet  Google Scholar 

  • Li P, Li X (2021) Monochromatic disconnection of graphs. Discrete Appl. Math. 288:171–179

    Article  MathSciNet  Google Scholar 

  • Li X, Wu D (2018) A survey on monochromatic connections of graphs. Theory Appl. Graphs 0(1), Art. 4

  • Mao Y, Wang Z, Yanling F, Ye C (2016) Monochromatic connectivity and graph products. Discrete Math. Algorithms Appl. 8(01):1650011

    Article  MathSciNet  Google Scholar 

  • Gonzlez-Moreno D, Guevara M, Montellano-Ballesteros JJ (2017) Monochromatic connecting colorings in strongly connected oriented graphs. Discrete Math. 340(4):578–584

    Article  MathSciNet  Google Scholar 

  • Weichsel PM (1963) The Kronecker product of graphs. Proc. Amer. Math. Soc. 13:47–52

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Combinatorics and LPMC, Nankai University, Tianjin, 300071, China

    Ping Li & Xueliang Li

Authors
  1. Ping Li
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Xueliang Li
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Xueliang Li.

Additional information

Publisher's Note

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

Supported by NSFC No.12131013 and 11871034.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, P., Li, X. Monochromatic disconnection: Erdős-Gallai-type problems and product graphs. J Comb Optim 44, 136–153 (2022). https://doi.org/10.1007/s10878-021-00820-3

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-021-00820-3

Keywords

Mathematics Subject Classification