3-choosable planar graphs with some precolored vertices and no 5^--cycles normally adjacent to 8^--cycles

08/14/2019

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by   Fangyao Lu, et al.

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DP-coloring was introduced by Dvořák and Postle [J. Combin. Theory Ser. B 129 (2018) 38–54] as a generalization of list coloring. They used a "weak" version of DP-coloring to solve a longstanding conjecture by Borodin, stating that every planar graph without cycles of length 4 to 8 is 3-choosable. Liu and Li improved the result by showing that every planar graph without adjacent cycles of length at most 8 is 3-choosable. In this paper, it is showed that every planar graph without 5^--cycles normally adjacent to 8^--cycles is 3-choosable. Actually, all these three papers give more stronger results by stating them in the form of "weakly" DP-3-coloring and color extension.