Colorings and Homomorphisms of Minor Closed Classes

  • Jaroslav Nešetřil
  • Patrice Ossona de Mendez
Part of the Algorithms and Combinatorics book series (AC, volume 25)


We relate acyclic (and star) chromatic number of a graph to the chromatic number of its minors and as a consequence we show that the set of all triangle free planar graphs is homomorphism bounded by a triangle free graph. This solves a problem posed in [[15]]. It also improves the best known bound for the star chromatic number of planar graphs from 80 to 30. Our method generalizes to all minor closed classes and puts Hadwiger conjecture in yet another context.


Planar Graph Chromatic Number Proper Coloring Mixed Graph Acyclic Orientation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. Alon, T. H. Marshall: Homomorphisms of edge-colored graphs and Coxeter groups, J. Algebraic Comb.8 (1998), 5–13.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    N. Alon, C. J. H. McDiarmid, B. A. Reed: Acyclic Coloring of Graphs, Random Structures and Algorithms 2 (1991), 343–365.MathSciNetCrossRefGoogle Scholar
  3. 3.
    N. Alon, B. Mohar, D. P. Sanders: On acyclic colorings of graphs on surfaces,Israel J. Math. 94 (1996), 273–283.MathSciNetzbMATHGoogle Scholar
  4. P. Dreyer, Ch. Malon, J. Ne¡¦set¡¦ril: Universal H-colorable graphs without a given configuration, Discrete Math.(in press)Google Scholar
  5. 5.
    P. Erdös, A. Hajnal: Chromatic number of finite and infinite graphs and hypergraphs, Discrete Math. 53 (1985), 281–285.MathSciNetzbMATHCrossRefGoogle Scholar
  6. G. Fertin, A. Raspaud, B. Reed: On Star Coloring of graphs. In: Proccedings of GW’01, LNCS, Springer VerlagGoogle Scholar
  7. J. Fiala, J. Kratochv´ıl, A. Proskurowski: Partial covers of graphs, to appear in Discussiones Mathematicae Graph Theory.Google Scholar
  8. 8.
    S.L. Hakimi: On the degree of the vertices of a directed graph, J. Franklin Inst. 279 (1965), 4.Google Scholar
  9. 9.
    R. Häggkvist, P. Hell: Universality of A-mote graphs, European J. Combinatorics 14 (1993), 23–27.zbMATHCrossRefGoogle Scholar
  10. 10.
    T. Jensen, B. Toft: Graph coloring problems, Willey, 1995.Google Scholar
  11. 11.
    A. Kostochka: On the minimum of the Hadwiger number for graphs with given average degree, Metody Diskret. Analiz., 38(1982), Novosibirsk, 37–58, in Russian,English translation: AMS Translations (2), 132(1986), 37–58.Google Scholar
  12. 12.
    A. Kostochka, E. Sopena, X. Zhu: Acyclic and oriented chromatic number of graphs, J. of Graph Th. 24 (1997), 331–340.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    W. Mader: Homomorphiesätze für Graphen, Math. Ann. 178 (1968), 154–168.MathSciNetzbMATHCrossRefGoogle Scholar
  14. T. H. Marshall, R. Nasraser, J. Ne¡¦set¡¦ril: Homomorphism Bounded Classes of Graphs (to appear in European J. Comb.)Google Scholar
  15. 15.
    J. Nešetřil: Aspects of Structural Combinatorics, Taiwanese J. Math. 3, 4 (1999), 381 - 424.MathSciNetzbMATHGoogle Scholar
  16. 16.
    J. Nešetřil: Homomorphisms of derivative graphs, Discrete Math., 1, 3 (1971),257–268.MathSciNetzbMATHCrossRefGoogle Scholar
  17. J. Nešetřil, P. Ossona de Mendez: Foldings (submitted)Google Scholar
  18. 18.
    E. Sopena, A. Raspaud: Good and semi-strong colorings of oriented planar graphs, Inf. Processing Letters 51 (1994), 171–174.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    J. Nešetřil, A. Raspaud: Colored homomorphisms of colored mixed graphs, J.Comb. Th. B, 80 (2000), 147–155.zbMATHGoogle Scholar
  20. 20.
    D. Seinsche: On a property of the class of n-colorable graphs, J. Comb. Th.B, 16 (1974), 191–193.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    A. Thomason: An extremal function for contractions of graphs, Math. Proc.Cambridge Philos. Soc. 95 (1984), 261–265.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    A. Thomason: The extremal function for complete minors, J. Comb. Th. B 81 (2001), 318–338.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    C. Thomassen: Gr¡§otsch’s 3-color theorem and its counterparts for torus and the projective plane, J. Comb. Th. B, 62 (1994), 268–279.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    E. S. Wolk: A note on the comparability graph of a tree, Proc. Amer. Math.Soc. 16 (1965), 17–20.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Jaroslav Nešetřil
    • 1
  • Patrice Ossona de Mendez
    • 2
  1. 1.Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI)Charles UniversityPraha 1Czech Republic
  2. 2.Centre d’Analyse et de Mathématiques Sociales (UMR 8557), CNRSParisFrance

Personalised recommendations