Skip to main content
Log in

On interval Δ-coloring of bipartite graphs

  • System Analysis and Operations Research
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

There exists an important class of problems where the minimal-length scheduling comes to regular coloring of a bipartite graph with the least possible number of colors, and the scheduling without downtimes, to the interval coloring of a bipartite graph. Consideration was given to the problem of interval Δ-coloring of the bipartite multigraph. An example was built of the (6, 3)-biregular graph having no interval Δ-coloring.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Garey, M.L. and Johnson, D.S., Computers and Intractability: A Guide to the Theory of NP-Completness, San Francisco: Freeman, 1979. Translated under the title Vychislitel’nye machiny i trudnoreshaemye zadachi, Moscow: Mir, 1982.

    Google Scholar 

  2. Vizing, V.G., On Estimation of the Chromatic Class of the p-graph, Diskret. Anal., 1964, vol. 3, pp. 25–30.

    MathSciNet  Google Scholar 

  3. Lovász, L. and Plummer, M.D., Matching Theory, Budapest: Akadémiai Kiadó-North Holland, 1986. Translated under the title Prikladnye zadachi teorii grafov. Teoriya parosochetanii v matematike, fizike, khimii, Moscow: Mir, 1998.

    MATH  Google Scholar 

  4. Holyer, I., The NP-completeness of Edge-coloring, SIAM J. Comput., 1981, vol. 10, no. 4, pp. 718–720.

    Article  MATH  MathSciNet  Google Scholar 

  5. Asratyan, A.S. and Kamalyan, R.R., Interval Coloring of the Multigraph Edges, Prikl. Mat., Erevan: Erevan. Univ., 1987, vol. 5, pp. 25–34.

    Google Scholar 

  6. Magomedov, A.M., Raskraska grafa s nepreryvnym spektrom (Coloring of Continuous-spectrum Graph), Available from VINITI, 1985, Moscow, no. 478–485.

    Google Scholar 

  7. Even, S., Itai, A., and Shamir, A., On the Complexity of Timetable and Integral Multi-commodity Flow Problems, SIAM J. Comput., 1976, vol. 5, no. 4, pp. 691–703.

    Article  MATH  MathSciNet  Google Scholar 

  8. Sevast’yanov, S.V., On Interval Colorability of the Edges of Bipartite Graph, Metody Diskret. Anal., 1990, vol. 50, pp. 61–72.

    MATH  MathSciNet  Google Scholar 

  9. Hansen, H.M., Scheduling with Minimum Waiting Periods, Master Thesis, Odense: Odense Univ., Denmark, 1992.

    Google Scholar 

  10. Magomedov, A.M., K voprosu ob usloviyakh uplotneniya matritsy iz 6 stolbtsov (On Conditions for Compression of the Six-column Matrix), Available from VINITI, 1991, Moscow.

    Google Scholar 

  11. Hanson, D., Loten, C.O.M., and Toft, B., On Interval Colourings of Bi-regular Bipartite Graphs, Ars Combinat., 1998, vol. 50, pp. 23–32.

    MATH  MathSciNet  Google Scholar 

  12. Giaro, K., Compact Task Scheduling on Dedicated Processors with noWaiting Period, PhD Dissertation, Gdansk: Technical Univ. of Gdansk, 1999.

    Google Scholar 

  13. Magomedov, A.M., Usloviya i algoritm uplotneniya matritsy iz 4 stolbtsov (Conditions and Algorithm to Compress the Four-column Matrix), Available from VINITI, 1992, Moscow, no. 175-V92.

    Google Scholar 

  14. Giaro, K., The Complexity of Consecutive Δ-coloring of Bipartite Graphs: 4 Is Easy, 5 Is Hard, Ars Combin., 1997, vol. 47, pp. 287–298.

    MATH  MathSciNet  Google Scholar 

  15. Asratian, A.S. and Casselgren, C.J., Some Results on Interval Edge Colorings of (α, β)-biregular Bipartite Graphs, in Department Math., Linköping: Linköping Univ., 2007, S-581 83.

    Google Scholar 

  16. Giaro, K., Kubale, M., and Malafiejcki, M., On the Deficiency of Bipartite Graphs, Discrete Appl. Math., 1999, vol. 94, pp. 193–203.

    Article  MATH  MathSciNet  Google Scholar 

  17. Tanaev, V.S., Sotskov, Yu.N., and Strusevich, V.A., Teoriya raspisanii. Mnogostadiinye sistemy (Theory of Schedules. Multistage Systems), Moscow: Nauka, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. M. Magomedov.

Additional information

Original Russian Text © A.M. Magomedov, 2015, published in Avtomatika i Telemekhanika, 2015, No. 1, pp. 101–109.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Magomedov, A.M. On interval Δ-coloring of bipartite graphs. Autom Remote Control 76, 80–87 (2015). https://doi.org/10.1134/S0005117915010075

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117915010075

Keywords

Navigation