475 research outputs found
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
adde
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
Defective Coloring on Classes of Perfect Graphs
In Defective Coloring we are given a graph and two integers ,
and are asked if we can -color so that the maximum
degree induced by any color class is at most . We show that this
natural generalization of Coloring is much harder on several basic graph
classes. In particular, we show that it is NP-hard on split graphs, even when
one of the two parameters , is set to the smallest possible
fixed value that does not trivialize the problem ( or ). Together with a simple treewidth-based DP algorithm this completely
determines the complexity of the problem also on chordal graphs. We then
consider the case of cographs and show that, somewhat surprisingly, Defective
Coloring turns out to be one of the few natural problems which are NP-hard on
this class. We complement this negative result by showing that Defective
Coloring is in P for cographs if either or is fixed; that
it is in P for trivially perfect graphs; and that it admits a sub-exponential
time algorithm for cographs when both and are unbounded
- β¦