Xu and Wu proved that if every $5$-cycle of a planar graph $G$ is not simultaneously adjacent to $3$-cycles and $4$-cycles, then $G$ is $4$-choosable. In this paper, we improve this result as follows. If $G$ is a planar graph without pairwise adjacent $3$-, $4$-, $5$-, and $6$-cycle, then $G$ is $4$-choosable.
This work has received scholarship under the Post-Doctoral
Training Program from Khon Kaen University, Thailand. Kittikorn Nakprasit is supported by
National Research Council of Thailand and Khon Kaen University under Mid-Career Research Grant
(years 2021–2024) in the project of Graph Structural Analysis for Solving Graph Coloring
The authors would like to thank the referees for their careful reading and valuable suggestions. The authors also would like to express our gratitude to Tao Wang for many important comments pointing out arguments that required some improvement.
"Planar Graphs Without Pairwise Adjacent $3$-, $4$-, $5$-, and $6$-cycle are $4$-choosable." Taiwanese J. Math. Advance Publication 1 - 23, 2021. https://doi.org/10.11650/tjm/210701