Abstract
A graph is path k-colorable if it has a vertex k-coloring in which the subgraph induced by each color class is a disjoint union of paths. A graph is path k-choosable if, whenever each vertex is assigned a list of k colors, such a coloring exists in which each vertex receives a color from its list.
It is known that every planar graph is path 3-colorable [Poh90, God91] and, in fact, path 3-choosable [Har97]. We investigate which planar graphs are path 2-colorable or path 2-choosable. We seek results of a “threshold” nature: on one side of a threshold, every graph is path 2-choosable, and there is a fast coloring algorithm; on the other side, determining even path 2-colorability is NP-complete.
We first consider maximum degree. We show that every planar graph with maximum degree at most 4 is path 2-choosable, while for k ≥ 5 it is NP-complete to determine whether a planar graph with maximum degree k is path 2-colorable.
Next we consider girth. We show that every planar graph with girth at least 6 is path 2-choosable, while for k ≤ 4 it is NP-complete to determine whether a planar graph with girth k is path 2-colorable. The case of girth 5 remains open.
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
J. Akiyama, H. Era, S. V. Gervacio, and M. Watanabe, Path chromatic numbers of graphs, J. Graph Theory 13(5) (1989), 569–575.
J. Andrews, M. Jacobson, On a generalization of chromatic number, Congressus Numer. 47 (1985), 33–48.
K. Appel and W. Haken, Every planar map is four colorable. Part I. Discharging, Illinois J. Math. 21 (1977), 429–490.
K. Appel, W. Haken, and J. Koch, Every planar map is four colorable. Part II. Reducibility, Illinois J. Math. 21 (1977), 491–567.
O. V. Borodin, A. V. Kostochka, B. Toft, Variable degeneracy: extensions of Brooks’ and Gallai’s theorems, Discrete Math. 214 (2000), no. 1–3, 101–112.
G. Chartrand, L. Lesniak, Graphs & Digraphs, 3rd ed., Chapman & Hall, London, 1996.
L. Cowen, W. Goddard, C.E. Jesurum, Defective coloring revisited, J. Graph Theory 24 (1997), 205–219.
L. Cowen, R. H. Cowen, D.R. Woodall, Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency, J. Graph Theory 10 (1986), 187–195.
R. Cowen, Some connections between set theory and computer science, in Computational logic and proof theory (Brno, 1993), Lecture Notes in Comput. Sci., 713, Springer, Berlin, 1993, pp. 14–22.
A. Farrugia, Vertex-partitioning into fixed additive induced-hereditary properties is NP-hard, Elec. J. Combin. 11 (2004), no. 1, research paper 46, 9 pp.
A. Galluccio, L. A. Goddyn, P. Hell, High-girth graphs avoiding a minor are nearly bipartite. J. Combin. Theory Ser. B 83 (2001), no. 1, 1–14.
J. Gimbel, C. Hartman, Subcolorings and the subchromatic number of a graph, Discrete Math. 272 (2003), no. 2–3, 139–154.
W. Goddard, Acyclic colorings of planar graphs, Discrete Math. 91 (1991), 91–94.
C. Hartman, Extremal Problems in Graph Theory, Ph.D. Thesis, University of Illinois, 1997.
K. S. Poh, On the linear vertex-arboricity of a planar graph, J. Graph Theory 14 (1990), 73–75.
N. Robertson, D. P. Sanders, P. D. Seymour, and R. Thomas, The four colour theorem, J. Combin. Theory Ser. B. 70 (1997), 2–44.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chappell, G.G., Gimbel, J., Hartman, C. (2006). Thresholds for Path Colorings of Planar Graphs. In: Klazar, M., Kratochvíl, J., Loebl, M., Matoušek, J., Valtr, P., Thomas, R. (eds) Topics in Discrete Mathematics. Algorithms and Combinatorics, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-33700-8_21
Download citation
DOI: https://doi.org/10.1007/3-540-33700-8_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-33698-3
Online ISBN: 978-3-540-33700-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)