ACYCLIC 4-COLORING OF PLANE GRAPHS WITHOUT CYCLES OF LENGTH 4 AND 6.
Every planar graph is known to be acyclically 5-colorable (Borodin, 1976), which bound is precise. Some sufficient conditions are also obtained for a planar graph to be acyclically 4-colorable. In particular, the acyclic 4-colorability was proved for the following planar graphs: without 3- and 4-cycles (Borodin, Kostochka and Woodall, 1999), without 4-, 5- and 6-cycles, (Montassier, Raspaud and Wang, 2006), without 4-, 6- and 7-cycles, and without 4-, 6- and 8-cycles (Chen, Raspaud, and Wang, 2009). In this paper it is proved that each planar graph without 4- and 6-cycles is acyclically 4-colorable.