108 research outputs found
On Choosability with Separation of Planar Graphs with Forbidden Cycles
We study choosability with separation which is a constrained version of list
coloring of graphs. A (k,d)-list assignment L on a graph G is a function that
assigns to each vertex v a list L(v) of at least k colors and for any adjacent
pair xy, the lists L(x) and L(y) share at most d colors. A graph G is
(k,d)-choosable if there exists an L-coloring of G for every (k,d)-list
assignment L. This concept is also known as choosability with separation. We
prove that planar graphs without 4-cycles are (3,1)-choosable and that planar
graphs without 5-cycles and 6-cycles are (3,1)-choosable. In addition, we give
an alternative and slightly stronger proof that triangle-free planar graphs are
-choosable.Comment: 27 pages, 10 figure
Complexity of choosability with a small palette of colors
A graph is -choosable if, for any choice of lists of colors for
each vertex, there is a list coloring, which is a coloring where each vertex
receives a color from its list. We study complexity issues of choosability of
graphs when the number of colors is limited. We get results which differ
surprisingly from the usual case where is implicit and which extend known
results for the usual case. We also exhibit some classes of graphs (defined by
structural properties of their blocks) which are choosable.Comment: 31 pages, 11 figure
DP-3-coloring of some planar graphs
In this article, we use a unified approach to prove several classes of planar
graphs are DP--colorable, which extend the corresponding results on
-choosability.Comment: 15 pages, five figures. We revised the previous version based on
referees' repor
List Colouring Big Graphs On-Line
In this paper, we investigate the problem of graph list colouring in the
on-line setting. We provide several results on paintability of graphs in the
model introduced by Schauz [13] and Zhu [20]. We prove that the on-line version
of Ohba's conjecture is true in the class of planar graphs. We also consider
several alternate on-line list colouring models
On choosability with separation of planar graphs with lists of different sizes
A (k,d)(k,d)-list assignment LL of a graph GG is a mapping that assigns to each vertex vv a list L(v)L(v) of at least kk colors and for any adjacent pair xyxy, the lists L(x)L(x) and L(y)L(y) share at most dd colors. A graph GG is (k,d)(k,d)-choosable if there exists an LL-coloring of GG for every (k,d)(k,d)-list assignment LL. This concept is also known as choosability with separation. It is known that planar graphs are (4, 1)-choosable but it is not known if planar graphs are (3, 1)-choosable. We strengthen the result that planar graphs are (4, 1)-choosable by allowing an independent set of vertices to have lists of size 3 instead of 4. Our strengthening is motivated by the observation that in (4, 1)-list assignment, vertices of an edge have together at least 7 colors, while in (3, 1)-list assignment, they have only at least 5. Our setting gives at least 6 colors
On group choosability of total graphs
In this paper, we study the group and list group colorings of total graphs
and we give two group versions of the total and list total colorings
conjectures. We establish the group version of the total coloring conjecture
for the following classes of graphs: graphs with small maximum degree,
two-degenerate graphs, planner graphs with maximum degree at least 11, planner
graphs without certain small cycles, outerplanar and near-outerplanar graphs.
In addition, the group version of the list total coloring conjecture is
established for forests, outerplanar graphs and graphs with maximum degree at
most two
Defective and Clustered Choosability of Sparse Graphs
An (improper) graph colouring has "defect" if each monochromatic subgraph
has maximum degree at most , and has "clustering" if each monochromatic
component has at most vertices. This paper studies defective and clustered
list-colourings for graphs with given maximum average degree. We prove that
every graph with maximum average degree less than is
-choosable with defect . This improves upon a similar result by Havet and
Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with
maximum average degree , no bound on the number of colours
was previously known. The above result with solves this problem. It
implies that every graph with maximum average degree is
-choosable with clustering 2. This extends a
result of Kopreski and Yu [Discrete Math., 2017] to the setting of
choosability. We then prove two results about clustered choosability that
explore the trade-off between the number of colours and the clustering. In
particular, we prove that every graph with maximum average degree is
-choosable with clustering , and is
-choosable with clustering . As an
example, the later result implies that every biplanar graph is 8-choosable with
bounded clustering. This is the best known result for the clustered version of
the earth-moon problem. The results extend to the setting where we only
consider the maximum average degree of subgraphs with at least some number of
vertices. Several applications are presented
3 List Coloring Graphs of Girth at least Five on Surfaces
Grotzsch proved that every triangle-free planar graph is 3-colorable.
Thomassen proved that every planar graph of girth at least five is 3-choosable.
As for other surfaces, Thomassen proved that there are only finitely many
4-critical graphs of girth at least five embeddable in any fixed surface. This
implies a linear-time algorithm for deciding 3-colorablity for graphs of girth
at least five on any fixed surface. Dvorak, Kral and Thomas strengthened
Thomassen's result by proving that the number of vertices in a 4-critical graph
of girth at least five is linear in its genus. They used this result to prove
Havel's conjecture that a planar graph whose triangles are pairwise far enough
apart is 3-colorable. As for list-coloring, Dvorak proved that a planar graph
whose cycles of size at most four are pairwise far enough part is 3-choosable.
In this article, we generalize these results. First we prove a linear
isoperimetric bound for 3-list-coloring graphs of girth at least five. Many new
results then follow from the theory of hyperbolic families of graphs developed
by Postle and Thomas. In particular, it follows that there are only finitely
many 4-list-critical graphs of girth at least five on any fixed surface, and
that in fact the number of vertices of a 4-list-critical graph is linear in its
genus. This provides independent proofs of the above results while generalizing
Dvorak's result to graphs on surfaces that have large edge-width and yields a
similar result showing that a graph of girth at least five with crossings
pairwise far apart is 3-choosable. Finally, we generalize to surfaces
Thomassen's result that every planar graph of girth at least five has
exponentially many distinct 3-list-colorings. Specifically, we show that every
graph of girth at least five that has a 3-list-coloring has
distinct 3-list-colorings.Comment: 33 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
On the choosability with separation of planar graphs and its correspondence colouring analogue
A list assignment for a graph is an -list assignment if
for each and for each
. We say is -choosable if it admits an -colouring
for every -list assignment . We prove that if is a planar
graph with -list assignment and for every triangle
we have that , then is -colourable.
In fact, we prove a slightly stronger result: if contains a clique such
that for every triangle with
, then is -colourable. Additionally, we
give a counterexample to the correspondence colouring analogue of
-choosability for planar graphs.Comment: 8 pages, 1 figure. Removed a theorem that was already implied by an
existing resul
- …