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

Fangyao Lu, Qianqian Wang, Tao Wang

Published 2019-08-14Version 1

DP-coloring was introduced by Dvo\v{r}\'{a}k 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.