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Non-rainbow colorings of 3-, 4- and 5-connected plane graphs

✍ Scribed by Zdeněk Dvořák; Daniel Král'; Riste Škrekovski

Publisher
John Wiley and Sons
Year
2009
Tongue
English
Weight
166 KB
Volume
63
Category
Article
ISSN
0364-9024

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✦ Synopsis

Abstract

We study vertex‐colorings of plane graphs that do not contain a rainbow face, i.e., a face with vertices of mutually distinct colors. If G is a 3 ‐connected plane graph with n vertices, then the number of colors in such a coloring does not exceed
$\lfloor{{7n-8}\over {9}}\rfloor$. If G is 4 ‐connected, then the number of colors is at most
$\lfloor {{5n-6}\over {8}}\rfloor$, and for n≡3(mod8), it is at most $\lfloor{{5n-6}\over {8}}\rfloor-1$. Finally, if G is 5 ‐connected, then the number of colors is at most
$\lfloor{{25}\over{58}}{\rm n}-{{22} \over {29}}\rfloor$. The bounds for 3 ‐connected and 4 ‐connected plane graphs are the best possible as we exhibit constructions of graphs with colorings matching the bounds. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 129–145, 2010

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