55 research outputs found
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
DP-4-coloring of planar graphs with some restrictions on cycles
DP-coloring was introduced by Dvo\v{r}\'{a}k and Postle as a generalization
of list coloring. It was originally used to solve a longstanding conjecture by
Borodin, stating that every planar graph without cycles of lengths 4 to 8 is
3-choosable. In this paper, we give three sufficient conditions for a planar
graph is DP-4-colorable. Actually all the results (Theorem 1.3, 1.4 and 1.7)
are stated in the "color extendability" form, and uniformly proved by vertex
identification and discharging method.Comment: 13 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1908.0490
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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Ramified rectilinear polygons: coordinatization by dendrons
Simple rectilinear polygons (i.e. rectilinear polygons without holes or
cutpoints) can be regarded as finite rectangular cell complexes coordinatized
by two finite dendrons. The intrinsic -metric is thus inherited from the
product of the two finite dendrons via an isometric embedding. The rectangular
cell complexes that share this same embedding property are called ramified
rectilinear polygons. The links of vertices in these cell complexes may be
arbitrary bipartite graphs, in contrast to simple rectilinear polygons where
the links of points are either 4-cycles or paths of length at most 3. Ramified
rectilinear polygons are particular instances of rectangular complexes obtained
from cube-free median graphs, or equivalently simply connected rectangular
complexes with triangle-free links. The underlying graphs of finite ramified
rectilinear polygons can be recognized among graphs in linear time by a
Lexicographic Breadth-First-Search. Whereas the symmetry of a simple
rectilinear polygon is very restricted (with automorphism group being a
subgroup of the dihedral group ), ramified rectilinear polygons are
universal: every finite group is the automorphism group of some ramified
rectilinear polygon.Comment: 27 pages, 6 figure
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