ACYCLIC 3-CHOOSABILITY OF PLANAR GRAPHS WITH NO CYCLES OF LENGTH FROM 4 TO 11

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 Erdos and Steinberg, is the absence of k-cycles, where 4 <= k <= C. Here, we prove that every planar graph with no cycles of length from 4 to 11 is acyclically 3-choosable.