EN|RU Volume 16, No 5, 2009, P. 26-33 UDC 519.172.2 O. V. Borodin Acyclic 3-choosability of plane graphs without cycles of length from 4 to 12 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., 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. Bibl. 18. 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 e-mail: brdnoleg@math.nsc.ru © Sobolev Institute of Mathematics, 2015