Abstract
A coloring of the vertices of a graph byk colors is called acyclic provided that no circuit is bichromatic. We prove that every planar graph has an acyclic coloring with nine colors, and conjecture that five colors are sufficient. Other results on related types of colorings are also obtained; some of them generalize known facts about “point-arboricity”.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Bosák,Hamiltonian lines in cubic graphs, Proc. Int. Symp. Theory of Graphs (Rome 1966), P. Rosenstiehl (ed.), Gordon and Breach, New York; Dunod, Paris, 1967, pp. 35–46.
G. Chartrand, D. P. Geller and S. Hedetniemi,Graphs with forbidden subgraphs, J. Combinatorial Theory10 (1971), 12–41.
G. Chartrand and H. V. Kronk,The point-arboricity of planar graphs, J. London Math. Soc.44 (1969), 612–616.
G. Chartrand, H. V. Kronk and C. E. Wall,The point-arboricity of a graph, Israel J. Math.6 (1968), 169–175.
P. Erdös,Graph theory and probability, Canad. J. Math.11 (1959), 34–38.
P. Franklin,The four color problem, Amer. J. Math.44 (1922), 225–236.
H. Grötzsch,Zur Theorie der diskreten Gebilde. VII. Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math. Natur. Reihe.8 (1958/59), 109–120.
B. Grünbaum,Grötzsch's theorem on 3-colorings, Michigan Math. J.10 (1963), 303–310.
B. Grünbaum,Convex Polytopes, Interscience, New York, 1967.
B. Grünbaum,A problem in graph coloring, Amer. Math. Monthly77 (1970), 1088–1092.
B. Grünbaum and H. Walther,Shortness exponents of families of graphs, J. Combinatorial Theory (to appear.)
S. Hedetniemi,On partitioning planar graphs, Canad. Math. Bull.11 (1968), 203–211.
A. Kotzig,Prispevok k teórii Eulerovských polyédrov, (Slovak. Russian summary) Mat. Časopis Sloven. Akad. Vied.5 (1955), 101–113.
H. V. Kronk,An analogue to the Heawood map-coloring problem, J. London Math. Soc. (2)1 (1969), 750–752.
H. V. Kronk and A. T. White,A 4-color theorem for toroidal graphs, Proc. Amer Math. Soc.34 (1972), 83–86.
H. Lebesgue,Quelques conséquences simples de la formule d'Euler, J. Math. Pures Appl. (9)19 (1940), 27–43.
J. Lederberg,Hamilton circuits of convex trivalent polyhedra (up to 18 vertices), Am er Math. Monthly74 (1967), 522–527.
D. R. Lick,A class of point partition numbers, Recent Trends in Graph Theory, M. Capobianco et al. (eds.), Springer-Verlag, New York 1971, pp. 185–190.
D. R. Lick and A. T. White,The point partition numbers of closed 2-manifolds, J London Math. Soc. (2)4 (1972), 577–583.
T. S. Motzkin,Colorings, cocolorings, and determinant terms, Proc. Int. Symp. Theory of Graphs (Rome 1966), P. Rosenstiehl (ed.), Gordon and Breach, New York; Dunod, Paris, 1967, pp. 253–254.
O. Ore,The Four-Color Problem, Academic Press, New York 1967.
H. Sachs,Einführung in die Theorie der endlichen Graphen. Teil II, Teubner, Leipzig, 1972.
S. K. Stein,B-sets and coloring problems, Bull. Amer. Math. Soc.76 (1970), 805–806.
P. Wernicke,Über den kartographischen Vierfarbensatz, Math. Ann.58 (1904), 413–426.
Author information
Authors and Affiliations
Additional information
Research supported in part by the Office of Naval Research under Grant N00014-67-A-0103-0003.
Rights and permissions
About this article
Cite this article
Grünbaum, B. Acyclic colorings of planar graphs. Israel J. Math. 14, 390–408 (1973). https://doi.org/10.1007/BF02764716
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02764716