Geodesic
Acyclic 4-coloring of plane graphs without cycles of length~4 and~6
Diskretnyj analiz i issledovanie operacij, Tome 16 (2009) no. 6, pp. 3-11

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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. Bibl. 17.
Keywords: planar graph, acyclically coloring, acyclic choosability. 
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     title = {Acyclic 4-coloring of plane graphs without cycles of length~4 and~6},
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O. V. Borodin. Acyclic 4-coloring of plane graphs without cycles of length~4 and~6. Diskretnyj analiz i issledovanie operacij, Tome 16 (2009) no. 6, pp. 3-11. http://geodesic.mathdoc.fr/item/DA_2009_16_6_a0/