On (3,1)∗-choosability of planar graphs without adjacent short cycles

There exist planar graphs containing 4-cycles that are not (3,1)∗-choosable (Cowen et al., 1986  [1]). This partly explains the fact that in all above known sufficient conditions for the (3,1)∗-choosability of planar graphs the 4-cycles are completely forbidden. In this paper we allow 4-cycles nonadjacent to relatively short cycles. More precisely, we prove that every planar graph without 4-cycles adjacent to 3- and 4-cycles is (3,1)∗-choosable. This is a common strengthening of all above mentioned results. Moreover as a consequence we give a partial answer to a question of Xu and Zhang (2007)  [11] and show that every planar graph without 4-cycles is (3,1)∗-choosable.