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

AbstractIn 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 G is 3-choosable if it is satisfied one of the following conditions: (1)G has no cycles of length 4 or 9 and no 6-cycle is adjacent to a 3-cycle. Moreover, for each vertex x, there exists an integer ix∈{5,7,8} such that x is not incident to cycles of length ix.(2)G has no cycles of length 4, 7, or 9, and for each vertex x, there exists an integer ix∈{5,6,8} such that x is not incident to cycles of length ix. 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])