12 research outputs found
On DP-Coloring of Digraphs
DP-coloring is a relatively new coloring concept by Dvo\v{r}\'ak and Postle
and was introduced as an extension of list-colorings of (undirected) graphs. It
transforms the problem of finding a list-coloring of a given graph with a
list-assignment to finding an independent transversal in an auxiliary graph
with vertex set . In this paper, we
extend the definition of DP-colorings to digraphs using the approach from
Neumann-Lara where a coloring of a digraph is a coloring of the vertices such
that the digraph does not contain any monochromatic directed cycle.
Furthermore, we prove a Brooks' type theorem regarding the DP-chromatic number,
which extends various results on the (list-)chromatic number of digraphs.Comment: 23 pages, 6 figure
Colouring Complete Multipartite and Kneser-type Digraphs
The dichromatic number of a digraph is the smallest such that can
be partitioned into acyclic subdigraphs, and the dichromatic number of an
undirected graph is the maximum dichromatic number over all its orientations.
Extending a well-known result of Lov\'{a}sz, we show that the dichromatic
number of the Kneser graph is and that the
dichromatic number of the Borsuk graph is if is large
enough. We then study the list version of the dichromatic number. We show that,
for any and , the list
dichromatic number of is . This extends a recent
result of Bulankina and Kupavskii on the list chromatic number of ,
where the same behaviour was observed. We also show that for any ,
and , the list dichromatic number of the complete
-partite graph with vertices in each part is , extending
a classical result of Alon. Finally, we give a directed analogue of Sabidussi's
theorem on the chromatic number of graph products.Comment: 15 page
Digraph Coloring and Distance to Acyclicity
In -Digraph Coloring we are given a digraph and are asked to partition its
vertices into at most sets, so that each set induces a DAG. This well-known
problem is NP-hard, as it generalizes (undirected) -Coloring, but becomes
trivial if the input digraph is acyclic. This poses the natural parameterized
complexity question what happens when the input is "almost" acyclic. In this
paper we study this question using parameters that measure the input's distance
to acyclicity in either the directed or the undirected sense.
It is already known that, for all , -Digraph Coloring is NP-hard
on digraphs of DFVS at most . We strengthen this result to show that, for
all , -Digraph Coloring is NP-hard for DFVS . Refining our
reduction we obtain two further consequences: (i) for all , -Digraph
Coloring is NP-hard for graphs of feedback arc set (FAS) at most ;
interestingly, this leads to a dichotomy, as we show that the problem is FPT by
if FAS is at most ; (ii) -Digraph Coloring is NP-hard for graphs
of DFVS , even if the maximum degree is at most ; we show
that this is also almost tight, as the problem becomes FPT for DFVS and
.
We then consider parameters that measure the distance from acyclicity of the
underlying graph. We show that -Digraph Coloring admits an FPT algorithm
parameterized by treewidth, whose parameter dependence is . Then,
we pose the question of whether the factor can be eliminated. Our main
contribution in this part is to settle this question in the negative and show
that our algorithm is essentially optimal, even for the much more restricted
parameter treedepth and for . Specifically, we show that an FPT algorithm
solving -Digraph Coloring with dependence would contradict the
ETH
Extension of Gyárfás-Sumner conjecture to digraphs
The dichromatic number of a digraph D is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has been a recent center of study. In this work we look at possible extensions of Gyárfás-Sumner conjecture. More precisely, we propose as a conjecture a simple characterization of finite sets F of digraphs such that every oriented graph with sufficiently large dichromatic number must contain a member of F as an induce subdigraph. Among notable results, we prove that oriented triangle-free graphs without a directed path of length 3 are 2-colorable. If condition of "triangle-free" is replaced with "K 4-free", then we have an upper bound of 414. We also show that an orientation of complete multipartite graph with no directed triangle is 2-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest
Digraph Colouring and Arc-Connectivity
The dichromatic number of a digraph is the minimum size of
a partition of its vertices into acyclic induced subgraphs. We denote by
the maximum local edge connectivity of a digraph . Neumann-Lara
proved that for every digraph , . In this
paper, we characterize the digraphs for which . This generalizes an analogue result for undirected graph proved by Stiebitz
and Toft as well as the directed version of Brooks' Theorem proved by Mohar.
Along the way, we introduce a generalization of Haj\'os join that gives a new
way to construct families of dicritical digraphs that is of independent
interest.Comment: 34 pages, 11 figure