Acyclic 4-colorability of planar graphs without 4- and 5-cycles

Every planar graph is known to be acyclically 5-colorable (Borodin, 1976). Some sufficient conditions are also obtained for a planar graph to be acyclically 4- and 3-colorable. In particular, the acyclic 4-colorability was proved for the following planar graphs: without 3-, and 4-cycles (Borodin, Kostochka, Woodall, 1999), without 4-, 5- and 6-cycles, without 4-, 5- and 7-cycles, and with neither 4- or 5-cycles nor intersecting 3-cycles (Montassier, Raspaud and Wang, 2006), and also without cycles of length 4, 5 and 8 (Chen, Raspaud, 2009).
In this paper it is proved that each planar graph without 4-cycles and 5-cycles is acyclically 4-colorable. Bibl. 23.