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{2,3}, this bound is sharp for all 
3 when 
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doi:10.1016/j.disc.2003.08.001 
Copyright © 2003 Elsevier B.V. All rights reserved.
Note
On maximum face-constrained coloring of plane graphs with no short face cycles*1
Daniel Král’
Department of Applied Mathematics and Institute for Theoretical Computer Science (ITI), Charles University, Malostranské nám
stí 25, Prague 118 00, Czech Republic
Received 15 January 2002;
revised 1 July 2003;
accepted 27 August 2003. ;
Available online 13 December 2003.
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Abstract
We prove that the vertices of each n-vertex plane graph G with minimum face cycle length g, g
5, can be colored using at least
(g+3)/2) in such a way that G does not contain a polychromatic face, i.e., a face whose all the vertices have mutually different colors. In particular, if the girth of an n-vertex plane graph is at least five, then there is a coloring using at least 
n/2
+1 colors. Author Keywords: Planar graphs; Planar hypergraphs; Coloring; Polychromatic
