Abstract
Steinberg asked whether every planar graph without 4 and 5 cycles is 3-colorable. Borodin, and independently Sanders and Zhao, showed that every planar graph without any cycle of length between 4 and 9 is 3-colorable. We improve this result by showing that every planar graph without any cycle of length 4, 5, 6, or 9 is 3-choosable.
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