Skip to main content
SpringerLink
Log in

Acyclically 3-colorable planar graphs

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

In this paper we study the acyclic 3-colorability of some subclasses of planar graphs. First, we show that there exist infinite classes of cubic planar graphs that are not acyclically 3-colorable. Then, we show that every planar graph has a subdivision with one vertex per edge that is acyclically 3-colorable and provide a linear-time coloring algorithm. Finally, we characterize the series-parallel graphs for which every 3-coloring is acyclic and provide a linear-time recognition algorithm for such graphs.

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.

Similar content being viewed by others

References

  • Albertson MO, Berman D (1973) Every planar graph has an acyclic 7-coloring. Isr J Math 14:390–408

    Article  Google Scholar 

  • Alon N, McDiarmid C, Reed BA (1991) Acyclic coloring of graphs. Random Struct Algorithms 2(3):277–288

    Article  MathSciNet  MATH  Google Scholar 

  • Angelini P, Frati F (2010) Acyclically 3-colorable planar graphs. In: Rahman MdS, Fujita S (eds) Workshop on algorithms and computation (WALCOM ’10). LNCS, vol 5942, pp 113–124

    Chapter  Google Scholar 

  • Appel K, Haken W (1977) Every planar map is 4-colorable. I. Discharging. Ill J Math 21(3):429–490

    MathSciNet  MATH  Google Scholar 

  • Appel K, Haken W, Koch J (1977) Every planar map is 4-colorable. II. Reducibility. Ill J Math 21(3):491–567

    MathSciNet  MATH  Google Scholar 

  • Borodin OV (1979) On acyclic colourings of planar graphs. Discrete Math 25:211–236

    Article  MathSciNet  MATH  Google Scholar 

  • Borodin OV, Kostochka AV, Woodall DR (1999) Acyclic colourings of planar graphs with large girth. J Lond Math Soc 60(2):344–352

    Article  MathSciNet  Google Scholar 

  • Brook RL (1941) On coloring the nodes of a network. Proc Camb Philos Soc 37:194–197

    Article  Google Scholar 

  • Coleman F, Cai J-Y (1986) The cyclic coloring problem and estimation of sparse Hessian matrices. SIAM J Algebr Discrete Methods 7(2):221–235

    Article  MathSciNet  MATH  Google Scholar 

  • Dujmović V, Pór A, Wood DR (2004) Track layouts of graphs. Discrete Math Theor Comput Sci 6(2):497–522

    MathSciNet  MATH  Google Scholar 

  • Dujmović V, Morin P, Wood DR (2005) Layout of graphs with bounded tree-width. SIAM J Comput 34(3):553–579

    Article  MathSciNet  MATH  Google Scholar 

  • Felsner S, Liotta G, Wismath SK (2003) Straight-line drawings on restricted integer grids in two and three dimensions. J Graph Algorithms Appl 7(4):363–398

    MathSciNet  MATH  Google Scholar 

  • Gebremedhin AH, Manne F, Pothen A (2002) Graph coloring in optimization revisited. Technical Report 226, University of Bergen, Norway

  • Gebremedhin AH, Tarafdar A, Pothen A, Walther A (2009) Efficient computation of sparse Hessians using coloring and automatic differentiation. INFORMS J Comput 21(2):209–223

    Article  MathSciNet  Google Scholar 

  • Grünbaum B (1973) Acyclic colorings of planar graphs. Isr J Math 14:390–408

    Article  MATH  Google Scholar 

  • Heath LS, Rosenberg AL (1992) Laying out graphs using queues. SIAM J Comput 21(5):927–958

    Article  MathSciNet  MATH  Google Scholar 

  • Heath LS, Leighton FT, Rosenberg AL (1992) Comparing queues and stacks as mechanisms for laying out graphs. SIAM J Discrete Math 5(3):398–412

    Article  MathSciNet  MATH  Google Scholar 

  • Kostochka AV (1976) Acyclic 6-coloring of planar graphs. Metody Diskret Anal 28:40–56

    MATH  Google Scholar 

  • Kostochka AV (1978) Upper bounds of chromatic functions of graphs. PhD thesis, University of Novosibirsk (in Russian)

  • Kostochka AV, Melnikov LS (1976) Note to the paper of B. Grünbaum on acyclic colorings. Discrete Math 14:403–406

    Article  MathSciNet  MATH  Google Scholar 

  • Lyons A (2009) Restricted coloring problems and forbidden induced subgraphs. Preprint ANL/MCS-P1611-0409, Mathematics and Computer Science Division, Argonne National Laboratory

  • Mitchem J (1974) Every planar graph has an acyclic 8-coloring. Duke Math J 41:177–181

    Article  MathSciNet  MATH  Google Scholar 

  • Ochem P (2005) Negative results on acyclic improper colorings. In: Felsner S (ed) European conference on combinatorics, graph theory and applications (EuroComb ’05). DMTCS, vol AE, pp 357–362

    Google Scholar 

  • Robertson N, Sanders DP, Seymour PD, Thomas R (1996) Efficiently four-coloring planar graphs. In: ACM symposium on theory of computing (STOC ’96), pp 571–575

    Google Scholar 

  • Skulrattanakulchai S (2004) Acyclic colorings of subcubic graphs. Inf Process Lett 92(4):161–167

    Article  MathSciNet  MATH  Google Scholar 

  • Valdes J, Tarjan RE, Lawler EL (1982) The recognition of series parallel digraphs. SIAM J Comput 11(2):298–313

    Article  MathSciNet  MATH  Google Scholar 

  • Wood DR (2005) Acyclic, star and oriented colourings of graph subdivisions. Discrete Math Theor Comput Sci 7(1):37–50

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica e Automazione, Università Roma Tre, Rome, Italy

    Patrizio Angelini & Fabrizio Frati

Authors
  1. Patrizio Angelini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabrizio Frati
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabrizio Frati.

Additional information

This work is partially supported by the Italian Ministry of Research, Grant number RBIP06BZW8, FIRB project “Advanced tracking system in intermodal freight transportation” and by MIUR of Italy under project AlgoDEEP prot. 2008TFBWL4. A preliminary version of this paper appeared at the 4th Workshop on Algorithms and Computation (WALCOM ’10) (Angelini and Frati 2010).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Angelini, P., Frati, F. Acyclically 3-colorable planar graphs. J Comb Optim 24, 116–130 (2012). https://doi.org/10.1007/s10878-011-9385-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-011-9385-3

Keywords

Search

Navigation