3-choosable planar graphs with some precolored vertices and no $5^{-}$-cycles normally adjacent to $8^{-}$-cycles
Fangyao Lu, Qianqian Wang, Tao Wang · arXiv (Cornell University) · 2019
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.