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Proceedings of the American Mathematical Society

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A three and five color theorem


Author: Frank R. Bernhart
Journal: Proc. Amer. Math. Soc. 52 (1975), 493-498
MSC: Primary 05C15
DOI: https://doi.org/10.1090/S0002-9939-1975-0373944-2
MathSciNet review: 0373944
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Abstract: Let $ f$ be a face of a plane graph $ G$. The Three and Five Color Theorem proved here states that the vertices of $ G$ can be colored with five colors, and using at most three colors on the boundary of $ f$. With this result the well-known Five Color Theorem for planar graphs can be strengthened, and a relative coloring conjecture of Kainen can be settled except for a single case which happens to be a paraphrase of the Four Color Conjecture. Some conjectures are presented which are intermediate in strength to the Four Color Conjecture and the Three and Five Color Theorem.


References [Enhancements On Off] (What's this?)

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  • [2] -, A generalization of the $ 5$-color theorem, Proc. Amer. Math. Soc. 45 (1974), 450-453. MR 0345861 (49:10591)
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DOI: https://doi.org/10.1090/S0002-9939-1975-0373944-2
Article copyright: © Copyright 1975 American Mathematical Society

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