Abstract
In this note, we prove that a planar graph is 3-choosable if it contains neither cycles of length 4, 7, and 9 nor 6-cycle with one chord. In particular, every planar graph without cycles of length 4, 6, 7, or 9 is 3-choosable. Together with other known parallel results, this completes a theorem on 3-choosability of planar graphs: planar graphs without cycles of length 4, i, j, 9 with i < j and i, j ∈ { 5 , 6 , 7 , 8 } are 3-choosable. Moreover, some further problems on this direction are proposed.
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