Skip to main content
SpringerLink
Log in

Some of My Favorite Coloring Problems for Graphs and Digraphs

  • Chapter
  • First Online:
Graph Theory

Part of the book series: Problem Books in Mathematics ((PBM))

  • 4009 Accesses

Abstract

One of the central notions in graph theory is that of a coloring–a partition of the vertices where each part induces a graph with some given property. The most studied property is that of inducing an empty graph–a graph without any edges. Changing the property slightly creates interesting variations. In this paper I will discuss a few of my favorite coloring problems and variations. This discussion is not meant to be comprehensive. The field is so massive that attempts to catalog all important developments were abandoned many years ago. So I will restrict this to a very small set of problems that reflect my personal interests and perhaps nothing more.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 29.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 39.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 49.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Albertson, M., Jameson, R., Hedetniemi, S.: The subchromatic number of a graph. Graph colouring and variations. Discret. Math. 74 (1–2), 33–49 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alon, N., Krivelevich, M., Sudakov, B.: Subgraphs with a large cochromatic number. J. Graph Theory 25 (4), 295–297 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alon, N., Ding, G., Oporowski, B., Vertigan, D.: Partitioning into graphs with only small components. J. Comb. Theory Ser. B 87 (2), 231–243 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barnette, D.: Coloring polyhedra manifolds. In: Discrete Geometry and Convexity (New York, 1982). Annals of the New York Academy of Sciences, vol. 440, pp. 192–195. New York Academy of Sciences, New York (1985)

    Google Scholar 

  5. Borodin, O., Glebov, A., Jensen, T.: A step towards the strong version of Havel’s three color conjecture. J. Comb. Theory Ser. B 102 (6), 1295–1320 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Campos, V., Klein, S., Sampaio, R., Silva, A.: Fixed-parameter algorithms for the cocoloring problem. Discret. Appl. Math. 167, 52–60 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chappell, G., Gimbel, J.: On subgraphs without large components. Preprint

    Google Scholar 

  8. Coulson, D.: A 15-colouring of 3-space omitting distance one. Discret. Math. 256, 83–90 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chudnovsky, M., Seymour, P.: Extending the Gyárfás-Sumner conjecture. J. Comb. Theory Ser. B 105, 11–16 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Croft, H., Falconer, K., Guy, R.: Unsolved problems in geometry. In: Unsolved Problems in Intuitive Mathematics, II. Springer, New York (1991)

    Book  MATH  Google Scholar 

  11. Cowen, L., Cowen, R., Woodall, D.: Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency. J. Graph Theory 10 (2), 187–195 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cowen, L., Goddard, W., Jerusem, C.: Defective coloring revisited. J. Graph Theory 24 (3), 205–219 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. de Bruijn, N.G., Erdös, P.: A colour problem for infinite graphs and a problem in the theory of relations. Ned. Akad. Wet. Proc. A 54 (13), 369–373 (1951)

    MathSciNet  MATH  Google Scholar 

  14. Domke, G., Laskar, R., Hedetniemi, S.: The edge subchromatic number of a graph. 250th Anniversary Conference on Graph Theory (Fort Wayne, IN, 1986). Congressus Numerantium, vol. 64, pp. 95–104. Utilitas Mathematica Publishing, Winnipeg (1988)

    Google Scholar 

  15. Dvořák, Z., Král, D., Thomas, R.: Coloring Planar Graphs with Triangles Far Apart. Citeseer (2009)

    Google Scholar 

  16. Dvořak, Z., Kawarabayashi, K.-I., Thomas, R.: Three-coloring triangle-free planar graphs in linear time. ACM Trans. Algoritm. (TALG) 7 (4), 41 (2011)

    Google Scholar 

  17. Ekim, T., Gimbel, J.: Some defective parameters in graphs. Graphs Comb. 29 (2), 213–224 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Erdös, P.: Problems and results in number theory and graph theory. Proceedings of the Ninth Manitoba Conference on Numerical Mathematics and Computing (University of Manitoba, Winnipeg, 1979). Congressus Numerantium, vol. XXVII, pp. 3–21. Utilitas Mathematica, Winnipeg (1980)

    Google Scholar 

  19. Erdös, P., Gimbel, J.: Some problems and results in cochromatic theory. In: Quo Vadis, Graph Theory? Annals of Discrete Mathematics, vol. 55, pp. 261–264. North-Holland, Amsterdam (1993)

    Google Scholar 

  20. Erdös, P., Gimbel, J., Kratsch, D.: Some extremal results in cochromatic and dichromatic theory. J. Graph Theory 15 (6), 579–585 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Esperet, L., Joret, G.: Colouring planar graphs with three colours and no large monochromatic components. Comb. Probab. Comput. 23 (4), 551–570 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fink, J., Jacobson, M.: On n-domination and n-dependence and forbidden subgraphs. In: Alavi, Y., Schwenk, A. (eds.) Graph Theory with Applications to Algorithms and Computer Science (Kalamazoo, 1984), pp. 301–311. Wiley, New York (1985)

    Google Scholar 

  23. Fisk, S., Mohar, B.: Coloring graphs without short non-bounding cycles. J. Comb. Theory Ser. B 60, 268–276 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Foldes, S., Hammer, P.: Split graphs. Technical Report. Department of Combinatorics and Optimization, University of Waterloo. CORR 76–3 (1975)

    Google Scholar 

  25. Foldes, S., Hammer, P.: Split graphs. In: Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State University, Baton Rouge, 1977), Congressus Numerantium, vol. XIX, pp. 311–315. Utilitas Mathematica, Winnipeg (1977)

    Google Scholar 

  26. Gardner, M.: Mathematical games. Sci. Am. 203, 172–180 (1960)

    Article  Google Scholar 

  27. Garey, M., Johnson, D., Stockmeyer, L.: Some simplified NP-complete problems. Theor. Comp. Sci. 1 (3), 237–267 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gimbel, J., Hartman, C.: Subcolorings and the subchromatic number of a graph. Discret. Math. 272 (2–3), 139–154 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Grötzsch, H.: Zur Theorie der diskreten Gebilde, VII: Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel, Wiss. Z. Martin-Luther-U., Halle-Wittenberg, Math.- Nat. Reihe 8, 109–120 (1959)

    Google Scholar 

  30. Grünbaum, B.: Grötzsch’s theorem on 3-colorings. Mich. Math. J. 10, 303–310 (1963)

    Article  MATH  Google Scholar 

  31. Gyárfás, A., Lehel, J.: A Helly-type problem in trees. In: Combinatorial Theory and Its Applications, II (Proceedings of the Colloquium, Balatonfüred, 1969), pp. 571–584. North-Holland, Amsterdam (1970)

    Google Scholar 

  32. Hadwiger, H.: Überdeckung des euklidischen Raumes durch kongruente Mengen. Port. Math. 4, 238–242 (1945)

    MathSciNet  MATH  Google Scholar 

  33. Hadwiger, H.: Ungelöste probleme, Nr 11. Elemente der Mathematik 16, 103–106 (1961)

    MathSciNet  Google Scholar 

  34. Havel, I.: The coloring of planar graphs by three colors. In: (Czech) Mathematics (Geometry and Graph Theory), pp. 89–91. University Karlova, Prague (1970)

    Google Scholar 

  35. Haxell, P., Szabo, T., Tardos, G.: Bounded size components, partitions and transversals. J. Comb. Theory Ser. B 88 (2), 281–297 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Heawood, P.: Map-colour theorem. Q. J. Pure Appl. Math. 24, 332–338 (1890)

    MATH  Google Scholar 

  37. Jacob, H., Meyniel, H.: Extension of Turán’s and Brooks’ theorems and new notions of stability and coloring in digraphs. In: Combinatorial Mathematics (Marseille-Luminy, 1981), vol. 75, pp. 365–370. North-Holland, Amsterdam (1983)

    Google Scholar 

  38. Jensen, T., Toft, B.: Graph Coloring Problems. Wiley-Interscience Series in Discrete Mathematics and Optimization, xxii+295 pp. Wiley, New York (1995). ISBN:0-471-02865-7

    Google Scholar 

  39. Lesniak, L., Straight, H.J.: The cochromatic number of a graph. Ars Comb. 3, 39–45 (1977)

    MathSciNet  MATH  Google Scholar 

  40. Martinov, N.: 3-colorable planar graphs. Serdica 3 (1), 11–16 (1977)

    MathSciNet  Google Scholar 

  41. Moser, L., Moser, W.: Solution to problem 10. Can. Math. Bull. 4, 187–189 (1961)

    MATH  Google Scholar 

  42. Nešetřil, J., Raspaud, A., Sopena, E.: Colorings and girth of oriented planar graphs. Discret. Math. 165/166, 519–530 (1997)

    Google Scholar 

  43. Neumann-Lara, V.: The dichromatic number of a digraph. J. Comb. Theory Ser. B 33 (3), 265–270 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  44. Neumann-Lara, V.: Vertex colorings in digraphs. Technical Report, University of Waterloo (1985)

    Google Scholar 

  45. Neumann-Lara, V., Urrutia, J.: Vertex critical r-dichromatic tournaments. Discret. Math. 49 (1), 83–87 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ore, O.: The Four-Color Problem. Pure and Applied Mathematics, vol. 27. Academic, New York (1967)

    Google Scholar 

  47. Radoičić, R., Tóth, G.: Note on the chromatic number of space. In: Discrete and Computational Geometry: the Goodman-Pollack Festschrift, vol. 25, pp. 695–698. Springer, New York (2002)

    Google Scholar 

  48. Soifer, A.: The Mathematical Coloring Book. Mathematics of Coloring and the Colorful Life of Its Creators. Springer, New York (2009)

    MATH  Google Scholar 

  49. Steinberg, R.: The state of the three color problem. In: Quo Vadis, Graph Theory? Annals of Discrete Mathematics, vol. 55. pp. 211–248. North-Holland, Amsterdam (1993)

    Google Scholar 

  50. Székely, L., Wormald, N.: Bounds on the measurable chromatic number of \(\mathbb{R}^{n}\). Graph theory and combinatorics. Discret. Math. 75 (1–3), 343–372 (1989)

    MathSciNet  MATH  Google Scholar 

  51. Thomassen, C.: Grötzsch’s 3-color theorem and its counterparts for the torus and the projective plane. J. Comb. Theory Ser. B 62 (2), 268–279 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  52. Thomassen, C.: Color-critical graphs on a fixed surface. J. Comb. Theory Ser. B 70 (1), 67–100 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  53. Thomassen, C.: A short list color proof of Grötzsch’s theorem. J. Comb. Theory Ser. B 88, 189–192 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  54. Wormald, N.: A 4-chromatic graph with a special plane drawing. J. Aust. Math. Soc. Ser A 28 (1), 1–8 (1979)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematic and Statistics, University of Alaska Fairbanks, Fairbanks, AK, 99775-6660, USA

    John Gimbel

Authors
  1. John Gimbel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John Gimbel .

Editor information

Editors and Affiliations

  1. Department of Applied Mathematics, Naval Postgraduate School, Monterey, California, USA

    Ralucca Gera

  2. School of Computing, Clemson University, Clemson, South Carolina, USA

    Stephen Hedetniemi

  3. Department of Mathematics, Virginia Commonwealth University, Richmond, Virginia, USA

    Craig Larson

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Gimbel, J. (2016). Some of My Favorite Coloring Problems for Graphs and Digraphs. In: Gera, R., Hedetniemi, S., Larson, C. (eds) Graph Theory. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-31940-7_7

Download citation

Publish with us

Policies and ethics

Search

Navigation