Acyclic 4-choosability of planar graphs without adjacent short cycles

The acyclic 4-choosability was proved, in particular, for the following planar graphs: without 3- and 4-cycles (Montassier et al., 2006 [29]), without 4-, 5-, and 6-cycles (Montassier et al., 2006 [29]), either without 4-, 6-, and 7-cycles, or without 4-, 6-, and 8-cycles (Chen, Raspaud, and Wang, 2009), and with neither 4-cycles nor 6-cycles adjacent to a triangle (Borodin et al., 2010 [13]). exist planar acyclically non-4-colorable bipartite graphs (Kostochka and Mel’nikov, 1976 [25]). This partly explains the fact that in all previously known sufficient conditions for the acyclic 4-choosability of planar graphs the 4-cycles are completely forbidden. In this paper we allow 4-cycles nonadjacent to relatively short cycles; namely, it is proved that a planar graph is acyclically 4-choosable if it does not contain an i -cycle adjacent to a j -cycle, where 3 ≤ j ≤ 6 if i = 3 and 4 ≤ j ≤ 7 if i = 4 . In particular, this absorbs all the above-mentioned results.