Skip to main content
SpringerLink
Log in

On the Complexity of Deciding Whether the Regular Number is at Most Two

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

Abstract

The regular number of a graph \(G\) denoted by \(reg(G)\) is the minimum number of subsets into which the edge set of \(G\) can be partitioned so that the subgraph induced by each subset is regular. In this work we answer to the problem posed as an open problem in Ganesan et al. (J Discrete Math Sci Cryptogr 15(2–3):49–157, 2012) about the complexity of determining the regular number of graphs. We show that computation of the regular number for connected bipartite graphs is NP-hard. Furthermore, we show that, determining whether \( reg(G) = 2 \) for a given connected \(3\)-colorable graph \( G \) is NP-complete. Also, we prove that a new variant of the Monotone Not-All-Equal 3-Sat problem is NP-complete.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Bryś, K., Lonc, Z.: Polynomial cases of graph decomposition: a complete solution of Holyer’s problem. Discrete Math. 309(6), 1294–1326 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dor, D., Tarsi, M.: Graph decomposition is NP-complete: a complete proof of Holyer’s conjecture. SIAM J. Comput. 26(4), 1166–1187 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ganesan, A., Iyer, R.R.: The regular number of a graph. J. Discrete Math. Sci. Cryptogr. 15(2–3), 149–157 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Garey, M.R., Johnson, D.S.: Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  5. Holyer, I.: The NP-completeness of some edge-partition problems. SIAM J. Comput. 10(4), 713–717 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kochol, M., Krivo\(\check{{\rm n}}\)áková, N., Smejová, S., Šranková, K.: Complexity of approximation of 3-edge-coloring of graphs. Inf. Process. Lett. 108(4), 238–241 (2008)

  7. Kulli, V.R., Janakiram, B., Iyer, R.R.: Regular number of a graph. J. Discrete Math. Sci. Cryptogr. 4(1), 57–64 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Leven, D., Galil, Z.: NP-completeness of finding the chromatic index of regular graphs. J. Algorithms 4(1), 35–44 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lonc, Z.: On the complexity of some edge-partition problems for graphs. Discrete Appl. Math. 70(2), 177–183 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Péroche, B.: Complexity of the linear arboricity of a graph. RAIRO Rech. Opér. 16(2), 125–129 (1982). (in French)

    MATH  Google Scholar 

  11. Plummer, M.D.: Graph factors and factorization: 1985–2003: a survey. Discrete Math. 307(7–8), 791–821 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Vizing, V.G.: On an estimate of the chromatic class of a p-graph. Diskret. Analiz 3, 25–30 (1964)

    MathSciNet  Google Scholar 

  13. Douglas, B.: West. Introduction to Graph Theory. Prentice Hall Inc., Upper Saddle River (1996)

    Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous referee for his/her useful comments and suggestions, which helped to improve the presentation of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran

    Ali Dehghan & Mohammad-Reza Sadeghi

  2. Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

    Arash Ahadi

Authors
  1. Ali Dehghan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammad-Reza Sadeghi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Arash Ahadi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad-Reza Sadeghi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dehghan, A., Sadeghi, MR. & Ahadi, A. On the Complexity of Deciding Whether the Regular Number is at Most Two. Graphs and Combinatorics 31, 1359–1365 (2015). https://doi.org/10.1007/s00373-014-1446-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-014-1446-9

Keywords

Mathematics Subject Classification (2000)

Search

Navigation