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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Albertson, M.O., Berman, D.M.: Every planar graph has an acyclic 7-coloring. Israel Journal of Mathematics 28(1-2), 169–174 (1977)
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)
Borodin, O.V.: On acyclic colorings of planar graphs. Discrete Mathematics 306(10-11), 953–972 (2006)
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)
Dhandapani, R.: Greedy drawings of triangulations. Discrete & Computational Geometry 43(2), 375–392 (2010)
Dujmović, V., Morin, P., Wood, D.R.: Layout of graphs with bounded tree-width. SIAM Journal of Computing 34, 553–579 (2005)
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)
Grünbaum, B.: Acyclic colorings of planar graphs. Israel Journal of Mathematics 14(4), 390–408 (1973)
Kostochka, A.V.: Acyclic 6-coloring of planar graphs. Diskretn. Anal. 28, 40–56 (1976)
Mitchem, J.: Every planar graph has an acyclic 8-coloring. Duke Mathematical Journal 41(1), 177–181 (1974)
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)
Nishizeki, T., Rahman, M.S.: Planar Graph Drawing. World Scientific (2004)
Ochem, P.: Negative results on acyclic improper colorings. In: European Conference on Combinatorics (EuroComb 2005), pp. 357–362 (2005)
Skulrattanakulchai, S.: Acyclic colorings of subcubic graphs. Information Processing Letters 92(4), 161–167 (2004)
Wood, D.R.: Acyclic, star and oriented colourings of graph subdivisions. Discrete Mathematics & Theoretical Computer Science 7(1), 37–50 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mondal, D., Nishat, R.I., Whitesides, S., Rahman, M.S. (2011). Acyclic Colorings of Graph Subdivisions. In: Iliopoulos, C.S., Smyth, W.F. (eds) Combinatorial Algorithms. IWOCA 2011. Lecture Notes in Computer Science, vol 7056. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25011-8_20
Download citation
DOI: https://doi.org/10.1007/978-3-642-25011-8_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25010-1
Online ISBN: 978-3-642-25011-8
eBook Packages: Computer ScienceComputer Science (R0)