480 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
Generalizations of Bounds on the Index of Convergence to Weighted Digraphs
We study sequences of optimal walks of a growing length, in weighted
digraphs, or equivalently, sequences of entries of max-algebraic matrix powers
with growing exponents. It is known that these sequences are eventually
periodic when the digraphs are strongly connected. The transient of such
periodicity depends, in general, both on the size of digraph and on the
magnitude of the weights. In this paper, we show that some bounds on the
indices of periodicity of (unweighted) digraphs, such as the bounds of
Wielandt, Dulmage-Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman, apply
to the weights of optimal walks when one of their ends is a critical node.Comment: 17 pages, 3 figure
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
Short directed cycles in bipartite digraphs
The Caccetta-H\"aggkvist conjecture implies that for every integer ,
if is a bipartite digraph, with vertices in each part, and every vertex
has out-degree more than , then has a directed cycle of length at
most . If true this is best possible, and we prove this for
and all .
More generally, we conjecture that for every integer , and every pair
of reals with , if is a bipartite
digraph with bipartition , where every vertex in has out-degree at
least , and every vertex in has out-degree at least ,
then has a directed cycle of length at most . This implies the
Caccetta-H\"aggkvist conjecture (set and very small), and again is
best possible for infinitely many pairs . We prove this for , and prove a weaker statement (that suffices) for
A family of mixed graphs with large order and diameter 2
A mixed regular graph is a connected simple graph in which each vertex has both a fixed outdegree (the same indegree) and a fixed undirected degree. A mixed regular graphs is said to be optimal if there is not a mixed regular graph with the same parameters and bigger order.
We present a construction that provides mixed graphs of undirected degree qq, directed degree View the MathML sourceq-12 and order 2q22q2, for qq being an odd prime power. Since the Moore bound for a mixed graph with these parameters is equal to View the MathML source9q2-4q+34 the defect of these mixed graphs is View the MathML source(q-22)2-14.
In particular we obtain a known mixed Moore graph of order 1818, undirected degree 33 and directed degree 11 called Bosák’s graph and a new mixed graph of order 5050, undirected degree 55 and directed degree 22, which is proved to be optimal.Peer ReviewedPostprint (author's final draft
- …