79 research outputs found

    Planar graphs without normally adjacent short cycles

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    Let G\mathscr{G} be the class of plane graphs without triangles normally adjacent to 8−8^{-}-cycles, without 44-cycles normally adjacent to 6−6^{-}-cycles, and without normally adjacent 55-cycles. In this paper, it is showed that every graph in G\mathscr{G} is 33-choosable. Instead of proving this result, we directly prove a stronger result in the form of "weakly" DP-33-coloring. The main theorem improves the results in [J. Combin. Theory Ser. B 129 (2018) 38--54; European J. Combin. 82 (2019) 102995]. Consequently, every planar graph without 44-, 66-, 88-cycles is 33-choosable, and every planar graph without 44-, 55-, 77-, 88-cycles is 33-choosable. In the third section, it is proved that the vertex set of every graph in G\mathscr{G} can be partitioned into an independent set and a set that induces a forest, which strengthens the result in [Discrete Appl. Math. 284 (2020) 626--630]. In the final section, tightness is considered.Comment: 19 pages, 3 figures. The result is strengthened, and a new result is adde

    DP-4-colorability of two classes of planar graphs

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    DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvo\v{r}\'ak and Postle (2017). In this paper, we prove that every planar graph GG without 44-cycles adjacent to kk-cycles is DP-44-colorable for k=5k=5 and 66. As a consequence, we obtain two new classes of 44-choosable planar graphs. We use identification of verticec in the proof, and actually prove stronger statements that every pre-coloring of some short cycles can be extended to the whole graph.Comment: 12 page

    DP-3-coloring of planar graphs without certain cycles

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    DP-coloring is a generalization of list coloring, which was introduced by Dvo\v{r}\'{a}k 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-3-colorable. 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.Comment: 16 pages, 4 figure
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