Advertisement

Acyclic Colorings of Graph Subdivisions

  • Debajyoti Mondal
  • Rahnuma Islam Nishat
  • Sue Whitesides
  • Md. Saidur Rahman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7056)

Abstract

An acyclic coloring of a graph G is a coloring of the vertices of G, where no two adjacent vertices of G receive the same color and no cycle of G is bichromatic. An acyclic k-coloring of G is an acyclic coloring of G using at most k colors. In this paper we prove that any triangulated plane graph G with n vertices has a subdivision that is acyclically 4-colorable, where the number of division vertices is at most 2n − 6. We show that it is NP-complete to decide whether a graph with degree at most 7 is acyclically 4-colorable or not. Furthermore, we give some sufficient conditions on the number of division vertices for acyclic 3-coloring of subdivisions of partial k-trees and cubic graphs.

Keywords

Acyclic coloring Subdivision Triangulated plane graph 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Albertson, M.O., Berman, D.M.: Every planar graph has an acyclic 7-coloring. Israel Journal of Mathematics 28(1-2), 169–174 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Angelini, P., Frati, F.: Acyclically 3-colorable planar graphs. In: Rahman, M. S., Fujita, S. (eds.) WALCOM 2010. LNCS, vol. 5942, pp. 113–124. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    Borodin, O.V.: On acyclic colorings of planar graphs. Discrete Mathematics 306(10-11), 953–972 (2006)CrossRefzbMATHGoogle Scholar
  4. 4.
    Coleman, T.F., Cai, J.: The cyclic coloring problem and estimation of sparse hessian matrices. SIAM Journal on Algebraic and Discrete Methods 7(2), 221–235 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dhandapani, R.: Greedy drawings of triangulations. Discrete & Computational Geometry 43(2), 375–392 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dujmović, V., Morin, P., Wood, D.R.: Layout of graphs with bounded tree-width. SIAM Journal of Computing 34, 553–579 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gebremedhin, A.H., Tarafdar, A., Pothen, A., Walther, A.: Efficient computation of sparse hessians using coloring and automatic differentiation. INFORMS Journal on Computing 21, 209–223 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grünbaum, B.: Acyclic colorings of planar graphs. Israel Journal of Mathematics 14(4), 390–408 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kostochka, A.V.: Acyclic 6-coloring of planar graphs. Diskretn. Anal. 28, 40–56 (1976)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Mitchem, J.: Every planar graph has an acyclic 8-coloring. Duke Mathematical Journal 41(1), 177–181 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Miura, K., Azuma, M., Nishizeki, T.: Canonical decomposition, realizer, Schnyder labeling and orderly spanning trees of plane graphs. International Journal of Foundations of Computer Science 16(1), 117–141 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nishizeki, T., Rahman, M.S.: Planar Graph Drawing. World Scientific (2004)Google Scholar
  13. 13.
    Ochem, P.: Negative results on acyclic improper colorings. In: European Conference on Combinatorics (EuroComb 2005), pp. 357–362 (2005)Google Scholar
  14. 14.
    Skulrattanakulchai, S.: Acyclic colorings of subcubic graphs. Information Processing Letters 92(4), 161–167 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wood, D.R.: Acyclic, star and oriented colourings of graph subdivisions. Discrete Mathematics & Theoretical Computer Science 7(1), 37–50 (2005)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Debajyoti Mondal
    • 1
  • Rahnuma Islam Nishat
    • 2
  • Sue Whitesides
    • 2
  • Md. Saidur Rahman
    • 3
  1. 1.Department of Computer ScienceUniversity of ManitobaCanada
  2. 2.Department of Computer ScienceUniversity of VictoriaCanada
  3. 3.Department of Computer Science and EngineeringBangladesh University of Engineering and Technology (BUET)Bangladesh

Personalised recommendations