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Keywords:

  • acyclic coloring;
  • planar graph;
  • choosability;
  • forbidden cycle

Abstract

Every planar graph is known to be acyclically 7-choosable and is conjectured to be acyclically 5-choosable (O. V. Borodin, D. G. Fon-Der-Flaass, A. V. Kostochka, E. Sopena, J Graph Theory 40 (2002), 83–90). This conjecture if proved would imply both Borodin's (Discrete Math 25 (1979), 211–236) acyclic 5-color theorem and Thomassen's (J Combin Theory Ser B 62 (1994), 180–181) 5-choosability theorem. However, as yet it has been verified only for several restricted classes of graphs. Some sufficient conditions are also obtained for a planar graph to be acyclically 4- and 3-choosable. In particular, the acyclic 4-choosability was proved for the following planar graphs: without 3-, 4-, and 5-cycles (M. Montassier, P. Ochem, and A. Raspaud, J Graph Theory 51 (2006), 281–300), without 4-, 5-, and 6-cycles, or without 4-, 5-, and 7-cycles, or without 4-, 5-, and intersecting 3-cycles (M. Montassier, A. Raspaud, W. Wang, Topics Discrete Math (2006), 473–491), and neither 4- and 5-cycles nor 8-cycles having a triangular chord (M. Chen and A. Raspaud, Discrete Math. 310(15–16) (2010), 2113–2118). The purpose of this paper is to strengthen these results by proving that each planar graph without 4- and 5-cycles is acyclically 4-choosable.