21 research outputs found
A connection between circular colorings and periodic schedules
AbstractWe show that there is a curious connection between circular colorings of edge-weighted digraphs and periodic schedules of timed marked graphs. Circular coloring of an edge-weighted digraph was introduced by Mohar [B. Mohar, Circular colorings of edge-weighted graphs, J. Graph Theory 43 (2003) 107–116]. This kind of coloring is a very natural generalization of several well-known graph coloring problems including the usual circular coloring [X. Zhu, Circular chromatic number: A survey, Discrete Math. 229 (2001) 371–410] and the circular coloring of vertex-weighted graphs [W. Deuber, X. Zhu, Circular coloring of weighted graphs, J. Graph Theory 23 (1996) 365–376]. Timed marked graphs G→ [R.M. Karp, R.E. Miller, Properties of a model for parallel computations: Determinancy, termination, queuing, SIAM J. Appl. Math. 14 (1966) 1390–1411] are used, in computer science, to model the data movement in parallel computations, where a vertex represents a task, an arc uv with weight cuv represents a data channel with communication cost, and tokens on arc uv represent the input data of task vertex v. Dynamically, if vertex u operates at time t, then u removes one token from each of its in-arc; if uv is an out-arc of u, then at time t+cuv vertex u places one token on arc uv. Computer scientists are interested in designing, for each vertex u, a sequence of time instants {fu(1),fu(2),fu(3),…} such that vertex u starts its kth operation at time fu(k) and each in-arc of u contains at least one token at that time. The set of functions {fu:u∈V(G→)} is called a schedule of G→. Computer scientists are particularly interested in periodic schedules. Given a timed marked graph G→, they ask if there exist a period p>0 and real numbers xu such that G→ has a periodic schedule of the form fu(k)=xu+p(k−1) for each vertex u and any positive integer k. In this note we demonstrate an unexpected connection between circular colorings and periodic schedules. The aim of this note is to provide a possibility of translating problems and methods from one area of graph coloring to another area of computer science
Two results on the digraph chromatic number
It is known (Bollob\'{a}s (1978); Kostochka and Mazurova (1977)) that there
exist graphs of maximum degree and of arbitrarily large girth whose
chromatic number is at least . We show an analogous
result for digraphs where the chromatic number of a digraph is defined as
the minimum integer so that can be partitioned into acyclic
sets, and the girth is the length of the shortest cycle in the corresponding
undirected graph. It is also shown, in the same vein as an old result of Erdos
(1962), that there are digraphs with arbitrarily large chromatic number where
every large subset of vertices is 2-colorable
New Bounds for the Dichromatic Number of a Digraph
The chromatic number of a graph , denoted by , is the minimum
such that admits a -coloring of its vertex set in such a way that each
color class is an independent set (a set of pairwise non-adjacent vertices).
The dichromatic number of a digraph , denoted by , is the minimum
such that admits a -coloring of its vertex set in such a way that
each color class is acyclic.
In 1976, Bondy proved that the chromatic number of a digraph is at most
its circumference, the length of a longest cycle.
Given a digraph , we will construct three different graphs whose chromatic
numbers bound .
Moreover, we prove: i) for integers , and with and for each , that if all
cycles in have length modulo for some ,
then ; ii) if has girth and there are integers
and , with such that contains no cycle of length
modulo for each , then ;
iii) if has girth , the length of a shortest cycle, and circumference
, then , which improves,
substantially, the bound proposed by Bondy. Our results show that if we have
more information about the lengths of cycles in a digraph, then we can improve
the bounds for the dichromatic number known until now.Comment: 14 page
Uniquely D-colourable digraphs with large girth
Let C and D be digraphs. A mapping is a C-colouring if for
every arc of D, either is an arc of C or , and the
preimage of every vertex of C induces an acyclic subdigraph in D. We say that D
is C-colourable if it admits a C-colouring and that D is uniquely C-colourable
if it is surjectively C-colourable and any two C-colourings of D differ by an
automorphism of C. We prove that if a digraph D is not C-colourable, then there
exist digraphs of arbitrarily large girth that are D-colourable but not
C-colourable. Moreover, for every digraph D that is uniquely D-colourable,
there exists a uniquely D-colourable digraph of arbitrarily large girth. In
particular, this implies that for every rational number , there are
uniquely circularly r-colourable digraphs with arbitrarily large girth.Comment: 21 pages, 0 figures To be published in Canadian Journal of
Mathematic
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