Some structural properties of planar graphs and their applications to 3-choosability

In this article, we consider planar graphs in which each vertex is not incident to some cycles of given lengths, but all vertices can have different restrictions. This generalizes the approach based on forbidden cycles which corresponds to the case where all vertices have the same restrictions on the incident cycles. We prove that a planar graph GG is 3-choosable if it is satisfied one of the following conditions: (1)GG has no cycles of length 4 or 9 and no 6-cycle is adjacent to a 3-cycle. Moreover, for each vertex xx, there exists an integer ix∈{5,7,8}ix∈{5,7,8} such that xx is not incident to cycles of length ixix.(2)GG has no cycles of length 4, 7, or 9, and for each vertex xx, there exists an integer ix∈{5,6,8}ix∈{5,6,8} such that xx is not incident to cycles of length ixix. This result generalizes several previously published results (Zhang and Wu, 2005 [12], Chen et al., 2008 [3], Shen and Wang, 2007 [6], Zhang and Wu, 2004 [13], Shen et al., 2011 [7]).