79 research outputs found
Planar graphs without normally adjacent short cycles
Let be the class of plane graphs without triangles normally
adjacent to -cycles, without -cycles normally adjacent to
-cycles, and without normally adjacent -cycles. In this paper, it is
showed that every graph in is -choosable. Instead of proving
this result, we directly prove a stronger result in the form of "weakly"
DP--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 -, -, -cycles is -choosable, and every
planar graph without -, -, -, -cycles is -choosable. In the
third section, it is proved that the vertex set of every graph in
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
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DP-4-colorability of two classes of planar graphs
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 without -cycles adjacent to
-cycles is DP--colorable for and . As a consequence, we obtain
two new classes of -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
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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