Volume 16, No 5, 2009, P. 26-33
O. V. Borodin
Acyclic 3-choosability of plane graphs without cycles of length from 4 to 12
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., 2009).
A natural measure of sparseness, introduced by Erdős and Steinberg, is the absence of $k$-cycles, where $4\le k\le S$. Here, we prove that every planar graph with no cycles with length from 4 to 12 is acyclically 3-choosable.
Keywords: planar graph, acyclic coloring, acyclic choosability.
Borodin Oleg Veniaminovich 1
1. S. L. Sobolev Institute of Mathematics, SB RAS,
4 Acad. Koptyug Ave., 630090 Novosibirsk, Russia