Abstract
The classical four color reduction process takes on a new appearance in the light of the recently begun theory of open sets of colorings. In this paper we show that specific configurations and clusters can be simply classified as either reducible or irreducible, without appealing to the truth or falsity of the Four Color Conjecture (4CC). By treating irreducibility and reducibility together, we hope to round out the theory and gain a better understanding of why clusters do or do not reduce.
The new methods are illustrated by giving a condensed proof of irreducibility for all reasonable candidates on the order of the six-ring or less. The principle tools are the union and splicing properties of open sets, and the rotation of "antiset" pairs (Lemma 1). Complete details and extension to higher rings will come in a later paper or papers.
At the conclusion it is shown that the 4CC is equivalent to the set of irreducible clusters (in our definition) being infinite!
Keywords
- Plane Graph
- Interior Vertex
- Minimal Graph
- Proper Ring
- Splice Diagram
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your 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
F. Allaire and E.R. Swart, A systematic approach to the determination of reducible configurations in the four-color conjecture, to appear in Journ. Comb. Theory (B).
K. Appel and W. Haken, An unavoidable set of configurations in planar triangulations, to appear in Journ. Comb. Theory (B).
A. Bernhart, Six-rings in minimal five-color maps, Amer. Journ. Math. 64 (1947), 391–412.
F. Bernhart, Splicing and the four color conjecture, to appear.
F. Bernhart, Topics in Graph Theory Related to the Five Color Conjecture, dissertation, Kansas State University, 1974.
G.D. Birkhoff, The reducibility of maps, Amer. Journal Math. 35 (1913), 114–128.
D. Cohen, Small Rings in Critical Maps, thesis, Harvard, 1975.
P. J. Heawood, Map-colour theorem, Quart. Journ. Math. 24 (1890), 322–338.
H. Heesch, Untersuchungen zum Vierfarbenproblem, Bibliog. Instit. AG, Mannheim, 1969.
A.B. Kempe, On the geographical problem of the four colours, Amer. Journ. Math. 2 (1879), 193–204.
O. Ore and J. Stemple, Numerical calculations on the four-color problem, Journ. Comb. Theory (B) 8 (1970), 65–78.
W. Stromquist, Some Aspects of the Four Color Problem, thesis, Harvard, 1975.
H. Whitney and W.T. Tutte, Kempe chains and the four color conjecture, Utilitas Math. 2 (1972), 241–281.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1978 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bernhart, F.R. (1978). Irreducible configurations and the four color conjecture. In: Alavi, Y., Lick, D.R. (eds) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol 642. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0070363
Download citation
DOI: https://doi.org/10.1007/BFb0070363
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-08666-6
Online ISBN: 978-3-540-35912-8
eBook Packages: Springer Book Archive
