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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Boundary values in the four color problem


Authors: Michael O. Albertson and Herbert S. Wilf
Journal: Trans. Amer. Math. Soc. 181 (1973), 471-482
MSC: Primary 05C15
MathSciNet review: 0319794
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Abstract: Let $ G$ be a planar graph drawn in the plane so that its outer boundary is a $ k$-cycle. A four coloring of the outer boundary $ \gamma $ is admissible if there is a four coloring of $ G$ which coincides with $ \gamma $ on the boundary. If $ \psi $ is the number of admissible boundary colorings, we show that the 4CC implies $ \psi \geqslant 3 \cdot {2^k}$ for $ k = 3, \cdots ,6$. We conjecture this to be true for all $ k$ and show $ \psi $ is $ \geqslant c{((1 + {5^{1/2}})/2)^k}$.

A graph is totally reducible (t.r.) if every boundary coloring is admissible. There are triangulations of the interior of a $ k$-cycle which are t.r. for anv $ k$. We investigate a class of graphs called annuli, characterize t.r. annuli and show that annuli satisfy the above conjecture.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0319794-7
PII: S 0002-9947(1973)0319794-7
Article copyright: © Copyright 1973 American Mathematical Society