62 research outputs found
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
Digraphs and homomorphisms: Cores, colorings, and constructions
A natural digraph analogue of the graph-theoretic concept of an `independent set\u27 is that of an acyclic set, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets
D-colorable digraphs with large girth
In 1959 Paul Erdos (Graph theory and probability, Canad. J. Math. 11 (1959), 34-38) famously proved, nonconstructively, that there exist graphs that have both arbitrarily large girth and arbitrarily large chromatic number. This result, along with its proof, has had a number of descendants (D. Bokal, G. Fijavz, M. Juvan, P.M. Kayll and B. Mohar, The circular chromatic number of a digraph, J. Graph Theory 46 (2004), 227-240; B. Bollobas and N. Sauer, Uniquely colourable graphs with large girth, Canad. J. Math. 28 (1976), 1340-1344; J. Nesetril and X. Zhu, On sparse graphs with given colorings and homomorphisms, J. Combin. Theory Ser. B 90 (2004), 161-172; X. Zhu, Uniquely H-colorable graphs with large girth, J. Graph Theory 23 (1996), 33-41) that have extended and generalized the result while strengthening the techniques used to achieve it. We follow the lead of Xuding Zhu (op. cit.) who proved that, for a suitable graph H, there exist graphs of arbitrarily large girth that are uniquely H-colorable. We establish an analogue of Zhu\u27s results in a digraph setting.
Let C and D be digraphs. A mapping f:V(D)&rarr V(C) is a C-coloring if for every arc uv of D, either f(u)f(v) is an arc of C or f(u)=f(v), and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colorable if it admits a C-coloring and that D is uniquely C-colorable if it is surjectively C-colorable and any two C-colorings of D differ by an automorphism of C. We prove that if D is a digraph that is not C-colorable, then there exist graphs of arbitrarily large girth that are D-colorable but not C-colorable. Moreover, for every digraph D that is uniquely D-colorable, there exists a uniquely D-colorable digraph of arbitrarily large girth
On the Complexity of Digraph Colourings and Vertex Arboricity
It has been shown by Bokal et al. that deciding 2-colourability of digraphs
is an NP-complete problem. This result was later on extended by Feder et al. to
prove that deciding whether a digraph has a circular -colouring is
NP-complete for all rational . In this paper, we consider the complexity
of corresponding decision problems for related notions of fractional colourings
for digraphs and graphs, including the star dichromatic number, the fractional
dichromatic number and the circular vertex arboricity. We prove the following
results:
Deciding if the star dichromatic number of a digraph is at most is
NP-complete for every rational .
Deciding if the fractional dichromatic number of a digraph is at most is
NP-complete for every .
Deciding if the circular vertex arboricity of a graph is at most is
NP-complete for every rational .
To show these results, different techniques are required in each case. In
order to prove the first result, we relate the star dichromatic number to a new
notion of homomorphisms between digraphs, called circular homomorphisms, which
might be of independent interest. We provide a classification of the
computational complexities of the corresponding homomorphism colouring problems
similar to the one derived by Feder et al. for acyclic homomorphisms.Comment: 21 pages, 1 figur
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
A short construction of highly chromatic digraphs without short cycles
A natural digraph analogue of the graph-theoretic concept of an `independent set' is that of an `acyclic set', namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets. In the spirit of a famous theorem of P. Erd\H{o}s [Graph theory and probability, Canad. J. Math. {\bf11} (1959), 34--38], it was shown probabilistically in [D. Bokal et al., The circular chromatic number of a digraph, J. Graph Theory {\bf46} (2004), no. 3, 227--240] that there exist digraphs with arbitrarily large girth and chromatic number. Here we give a construction of such digraphs.
Homomorphism complexes, reconfiguration, and homotopy for directed graphs
The neighborhood complex of a graph was introduced by Lov\'asz to provide
topological lower bounds on chromatic number. More general homomorphism
complexes of graphs were further studied by Babson and Kozlov. Such `Hom
complexes' are also related to mixings of graph colorings and other
reconfiguration problems, as well as a notion of discrete homotopy for graphs.
Here we initiate the detailed study of Hom complexes for directed graphs
(digraphs). For any pair of digraphs graphs and , we consider the
polyhedral complex that parametrizes the directed graph
homomorphisms . Hom complexes of digraphs have applications
in the study of chains in graded posets and cellular resolutions of monomial
ideals. We study examples of directed Hom complexes and relate their
topological properties to certain graph operations including products,
adjunctions, and foldings. We introduce a notion of a neighborhood complex for
a digraph and prove that its homotopy type is recovered as the Hom complex of
homomorphisms from a directed edge. We establish a number of results regarding
the topology of directed neighborhood complexes, including the dependence on
directed bipartite subgraphs, a digraph version of the Mycielski construction,
as well as vanishing theorems for higher homology. The Hom complexes of
digraphs provide a natural framework for reconfiguration of homomorphisms of
digraphs. Inspired by notions of directed graph colorings we study the
connectivity of for a tournament. Finally, we use
paths in the internal hom objects of digraphs to define various notions of
homotopy, and discuss connections to the topology of Hom complexes.Comment: 34 pages, 10 figures; V2: some changes in notation, clarified
statements and proofs, other corrections and minor revisions incorporating
comments from referee
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