Geodesic
Acyclic 4-colorability of planar graphs without 4- and 5-cycles
Diskretnyj analiz i issledovanie operacij, Tome 17 (2010) no. 2, pp. 20-38

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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.
Keywords: planar graphs, acyclic coloring, forbidden cycles. 
@article{DA_2010_17_2_a1,
     author = {O. V. Borodin},
     title = {Acyclic 4-colorability of planar graphs without 4- and 5-cycles},
     journal = {Diskretnyj analiz i issledovanie operacij},
     pages = {20--38},
     publisher = {mathdoc},
     volume = {17},
     number = {2},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DA_2010_17_2_a1/}
}
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O. V. Borodin. Acyclic 4-colorability of planar graphs without 4- and 5-cycles. Diskretnyj analiz i issledovanie operacij, Tome 17 (2010) no. 2, pp. 20-38. http://geodesic.mathdoc.fr/item/DA_2010_17_2_a1/