Planar graphs without pairwise adjacent 3-, 4-, 5-, and 6-cycle are 4-choosable

  title={Planar graphs without pairwise adjacent 3-, 4-, 5-, and 6-cycle are 4-choosable},
  author={Pongpat Sittitrai and Kittikorn Nakprasit},
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. 

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Planar graphs without intersecting 5-cycles are 4-choosable
Planar Graphs Without Cycles of Specific Lengths
It is shown that planar graphs without 3-cycles are 3-degenerate, and more surprisingly, that the same holds for planar graph without 6-cycles.
The 4-Choosability of Plane Graphs without 4-Cycles'
In this paper, it is shown that each plane graph without 4-cycles is 4-choosable.
Choosability and Edge Choosability of Planar Graphs without Intersecting Triangles
It is proved that G is 4-choosable and G is edge-$(\Delta(G)+1)$-Choosable when its maximum degree $\Delta (G)\ne 5$ is 5.
Choosability and edge choosability of planar graphs without five cycles
The complexity of planar graph choosability
List colourings of planar graphs
Every Planar Graph Is 5-Choosable
We prove the statement of the title, which was conjectured in 1975 by V. G. Vizing and, independently, in 1979 by P. Erdos, A. L. Rubin, and H Taylor.