RETRACTED ARTICLE: On 3-choosability of planar graphs with neither adjacent triangles nor 5-, 6- and 9-cycles

Zhang, Haihui

doi:10.1016/j.ipl.2010.09.006

Cited by 2 publications

(1 citation statement)

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“…Every planar graph without 3-, 7-and 8-cycles is DP-3-colorable Zhang and Wu [34] showed that every planar graph without 4-, 5-, 6-and 9-cycles is 3-choosable. Zhang [29] generalized this result by showing that every planar graph with neither adjacent triangles nor 5-, 6and 9-cycles is 3-choosable. Liu et al [17] showed that every planar graph without 4-, 5-, 6-and 9-cycles is DP-3-colorable.…”

Section: Introductionmentioning

confidence: 99%

DP-3-coloring of planar graphs without certain cycles

Rao,

Wang

2020

Preprint

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DP-coloring is a generalization of list coloring, which was introduced by Dvořák and Postle [J. Combin. Theory Ser. B 129 (2018) 38-54]. Zhang [Inform. Process. Lett. 113 (9) (2013) 354-356] showed that every planar graph with neither adjacent triangles nor 5-, 6-, 9-cycles is 3-choosable. Liu et al. [Discrete Math. 342 (2019) 178-189] showed that every planar graph without 4-, 5-, 6-and 9-cycles is DP-3colorable. In this paper, we show that every planar graph with neither adjacent triangles nor 5-, 6-, 9-cycles is DP-3-colorable, which generalizes these results. Yu et al. gave three Bordeaux-type results by showing that (i) every planar graph with the distance of triangles at least three and no 4-, 5-cycles is DP-3-colorable; (ii) every planar graph with the distance of triangles at least two and no 4-, 5-, 6-cycles is DP-3-colorable; (iii) every planar graph with the distance of triangles at least two and no 5-, 6-, 7-cycles is DP-3-colorable. We also give two Bordeaux-type results in the last section: (i) every plane graph with neither 5-, 6-, 8-cycles nor triangles at distance less than two is DP-3-colorable; (ii) every plane graph with neither 4-, 5-, 7-cycles nor triangles at distance less than two is DP-3-colorable.

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Section: Introductionmentioning

confidence: 99%