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8 papers
Research papers
Acyclic $3$-choosability of planar graphs with no cycles of length from $4$ to $11$
O. V. Borodinab,
A. O. Ivanovac a Sobolev Institute of Mathematics, Novosibirsk, Russia
b Novosibirsk State University
c Institute of Mathematics at Yakutsk State University
Abstract:
Every planar graph is known to be acyclically
$7$-choosable and is conjectured to be acyclically
$5$-choosable (Borodin et al., 2002). This conjecture if proved would imply both Borodin's acyclic
$5$-color theorem (1979) and Thomassen's
$5$-choosability theorem (1994). However, as yet it has been verified only for several restricted classes of graphs. Some sufficient conditions are also obtained for a planar graph to be acyclically
$4$- and
$3$-choosable.
In particular, a planar graph of girth at least
$7$ is acyclically
$3$-colorable (Borodin, Kostochka and Woodall, 1999) and acyclically
$3$-choosable (Borodin et al., 2010). A natural measure of sparseness, introduced by Erdős and Steinberg, is the absence of
$k$-cycles, where
$4\le k\le C$. Here, we prove that every planar graph with no cycles of length from
$4$ to
$11$ is acyclically
$3$-choosable.
Keywords:
acyclic coloring, planar graph, forbidden cycles.
UDC:
519.172
MSC: 05C15 Received August 9, 2010, published
September 17, 2010
Language: English