Skip to main content
SpringerLink
Log in

Vertex Coloring of a Graph for Memory Constrained Scenarios

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

Given an undirected graph \(G=(V,E)\), where V is a set of n vertices and E is a set of m edges, the vertex coloring problem consists in assigning colors to the graph vertices such that no two adjacent vertices share the same color. The vertex coloring problem has several practical applications, for instance, resource scheduling, compiler register allocation, pattern matching, puzzle solving, exam timetabling, among others. It is known that the problem of vertex k-coloring of a graph, for any \(k \ge 3\), is NP-complete. In this work, we focus on an approximate solution that can be implemented on simple electronic equipments that do not require a complete set of operations present in common microprocessors. The solution is suitable for sensors and other devices present in several applications for collecting and measuring data.

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
Fig. 3

Similar content being viewed by others

References

  1. Appel, K., Haken, W.: The solution of the four-color-map problem. Sci. Am. 237, 108–121 (1977)

    Article  MathSciNet  Google Scholar 

  2. Arumugam, S., Premalatha, K., Bača, M., Semaničová-Feňovčíková, A.: Local antimagic vertex coloring of a graph. Gr. Comb. 33(2), 275–285 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barba, L., Cardinal, J., Korman, M., Langerman, S., Van Renssen, A., Roeloffzen, M., Verdonschot, S.: Dynamic graph coloring. In: Workshop on Algorithms and Data Structures, pp. 97–108. Springer (2017)

  4. Beigel, R., Eppstein, D.: 3-Coloring in time \(0(1.3446^n)\): a no-MIS algorithm. In: 36th Annual Symposium on Foundations of Computer Science, pp. 444–452. IEEE (1995)

  5. Bhattacharya, S., Chakrabarty, D., Henzinger, M., Nanongkai, D.: Dynamic algorithms for graph coloring. In: 29th Annual ACM-SIAM symposium on discrete algorithms, pp. 1–20. Society for Industrial and Applied Mathematics (2018)

  6. Bollobás, B.: Modern Graph Theory. Graduate Texts in Mathematics. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  7. Boman, E.G., Bozdağ, D., Catalyurek, U., Gebremedhin, A.H., Manne, F.: A scalable parallel graph coloring algorithm for distributed memory computers. In: European Conference on Parallel Processing, pp. 241–251. Springer (2005)

  8. Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. The Macmillan Press Ltd. (1976)

  9. Bonomo, F., Chudnovsky, M., Maceli, P., Schaudt, O., Stein, M., Zhong, M.: Three-coloring and list three-coloring of graphs without induced paths on seven vertices. Combinatorica 38(4), 779–801 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brélaz, D.: New methods to color the vertices of a graph. Commun. ACM 22(4), 251–256 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  11. Burjons, E., Hromkovič, J., Královič, R., Královič, R., Muñoz, X., Unger, W.: Online graph coloring against a randomized adversary. Int. J. Found. Comput. Sci. 29(04), 551–569 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Byskov, J.M.: Enumerating maximal independent sets with applications to graph colouring. Oper. Res. Lett. 32(6), 547–556 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, L., Peng, J., Ralescu, D.A.: Uncertain vertex coloring problem. Soft Comput. 23(4), 1337–1346 (2019)

    Article  MATH  Google Scholar 

  14. Coleman, T.F., Moré, J.J.: Estimation of sparse Jacobian matrices and graph coloring blems. SIAM J. Numer. Anal. 20(1), 187–209 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dao, H.T., Kim, S.: Vertex graph-coloring-based pilot assignment with location-based channel estimation for massive MIMO systems. IEEE Access 6, 4599–4607 (2018)

    Article  Google Scholar 

  16. Dharwadker, A.: A new proof of the four colour theorem. Can. Math. Soc. 221, 1–34 (2000)

    Google Scholar 

  17. Diks, K.: A fast parallel algorithm for six-colouring of planar graphs. In: International Symposium on Mathematical Foundations of Computer Science, pp. 273–282. Springer (1986)

  18. Eppstein, D.: Improved algorithms for 3-coloring, 3-edge-coloring, and constraint satisfaction. In: 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 329–337. Society for Industrial and Applied Mathematics (2001)

  19. Eppstein, D.: Small maximal independent sets and faster exact graph coloring. In: Workshop on Algorithms and Data Structures, pp. 462–470. Springer (2001)

  20. Garey, M.R., Johnson, D.S.: Computers and Intractability, vol. 29. W.H. Freeman and Company, New York (2002)

    Google Scholar 

  21. Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete problems. In: 6th Annual ACM Symposium on Theory of Computing, pp. 47–63. ACM (1974)

  22. Gonthier, G.: Formal proof: the four-color theorem. Not. AMS 55(11), 1382–1393 (2008)

    MathSciNet  MATH  Google Scholar 

  23. Grech, N., Kastrinis, G., Smaragdakis, Y.: Efficient reflection string analysis via graph coloring. In: 32nd European Conference on Object-Oriented Programming, p. 25. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2018)

  24. Grötzsch, H.: Zur Theorie der Diskreten Gebilde. VII. Ein Dreifarbensatz far Dreikreisfreie Netze auf der Kugel. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-Nat. Reihe 8, 109–120 (1958/1959)

  25. Grünbaum, B.: A problem in graph coloring. Am. Math. Mon. 77(10), 1088–1092 (1970)

    Article  MathSciNet  Google Scholar 

  26. Irving, R.W.: NP-completeness of a family of graph-colouring problems. Discret. Appl. Math. 5(1), 111–117 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  27. Janczewski, R., Kubale, M., Manuszewski, K., Piwakowski, K.: The smallest hard-to-color graph for algorithm DSATUR. Discret. Math. 236(1), 151–165 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jones, M.T., Plassmann, P.E.: A parallel graph coloring heuristic. SIAM J. Sci. Comput. 14(3), 654–669 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kučera, L.: The greedy coloring is a bad probabilistic algorithm. J. Algorithms 12(4), 674–684 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kuratowski, C.: Sur le Probleme des Courbes Gauches en Topologie. Fundamenta Mathematicae 15(1), 271–283 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lawler, E.L.: A note on the complexity of the chromatic number problem. Inf. Process. Lett. 5(3), 66–67 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  32. van Lint, J., Wilson, R.: A Course in Combinatorics. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  33. Lozin, V.V., Malyshev, D.S.: Vertex coloring of graphs with few obstructions. Discret. Appl. Math. 216, 273–280 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Luby, M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15(4), 1036–1053 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  35. Matgraph: Toolbox for Working with Simple, Undirected Graphs. https://www.mathworks.com/matlabcentral/mlc-downloads/downloads/submissions/19218/versions/1/previews/matgraph/html/matgraph/@graph/color.html (2019)

  36. Moalic, L., Gondran, A.: Variations on memetic algorithms for graph coloring problems. J. Heurist. 24(1), 1–24 (2018)

    Article  Google Scholar 

  37. Mustafa, H., Schilken, I., Karasikov, M., Eickhoff, C., Rätsch, G., Kahles, A.: Dynamic compression schemes for graph coloring. Bioinformatics 35(3), 407–414 (2018)

    Article  Google Scholar 

  38. Mycielski, J.: Sur le Coloriage des Graphs. Colloquium Mathematicae 3(2), 161–162 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  39. Naor, J.: A fast parallel coloring of planar graphs with five colors. Inf. Process. Lett. 25(1), 5 1–53 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  40. Orden, D., Gimenez-Guzman, J., Marsa-Maestre, I., de la Hoz, E.: Spectrum graph coloring and applications to Wi-Fi channel assignment. Symmetry 10(3), 65 (2018)

    Article  MATH  Google Scholar 

  41. Petersen, J.: Die Theorie der Regulären Graphs. Acta Mathematica 15(1), 193–220 (1891)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ramsey, F.: On a problem of formal logic. In: Classic Papers in Combinatorics, pp. 1–24. Springer (2009)

  43. Scheinerman, E.: Coloring Graphs in Matgraph. https://www.mathworks.com/matlabcentral/fileexchange/19218-matgraph/content/matgraph/samples/html/coloring.html (2019)

  44. Şeker, O., Ekim, T., Taşkın, Z.C.: A decomposition approach to solve the selective graph coloring problem in some perfect graph families. Networks 73(2), 145–169 (2019)

    Article  MathSciNet  Google Scholar 

  45. Silva, E., Guedes, A., Todt, E.: Independent spanning trees on systems-on-chip hypercubes routing. Int. Conf. Electron. Circuits Syst. 75, 93–96 (2013)

    Google Scholar 

  46. Tarjan, R.E., Trojanowski, A.E.: Finding a maximum independent set. SIAM J. Comput. 6(3), 537–546 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  47. Welsh, D.J., Powell, M.B.: An upper bound for the chromatic number of a graph and its application to timetabling problems. Comput. J. 10(1), 85–86 (1967)

    Article  MATH  Google Scholar 

  48. Woeginger, G.J.: Exact algorithms for NP-hard problems: a survey. In: Combinatorial Optimization, pp. 185–207. Springer (2003)

  49. Zhou, Y., Duval, B., Hao, J.K.: Improving probability learning based local search for graph coloring. Appl. Soft Comput. 65, 542–553 (2018)

    Article  Google Scholar 

Download references

Acknowledgements

The authors are thankful to FAPESP (Grants #2014/12236-1 and #2017/12646-3) and CNPq (Grant #309330/2018-1) for their financial support.

Author information

Authors and Affiliations

  1. Institute of Computing, University of Campinas, Campinas, SP, 13083-852, Brazil

    Eduardo Sant’Ana da Silva & Helio Pedrini

Authors
  1. Eduardo Sant’Ana da Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Helio Pedrini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Helio Pedrini.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

da Silva, E.S., Pedrini, H. Vertex Coloring of a Graph for Memory Constrained Scenarios. Math.Comput.Sci. 14, 9–17 (2020). https://doi.org/10.1007/s11786-019-00409-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-019-00409-4

Keywords

Mathematics Subject Classification

Search

Navigation