11 research outputs found
On non-Hamiltonian circulant digraphs of outdegree three
We construct infinitely many connected, circulant digraphs of outdegree three
that have no hamiltonian circuit. All of our examples have an even number of
vertices, and our examples are of two types: either every vertex in the digraph
is adjacent to two diametrically opposite vertices, or every vertex is adjacent
to the vertex diametrically opposite to itself
The Manhattan product of digraphs
We give a formal definition of a new product of bipartite digraphs, the Manhattan product, and we study some of its main properties. It is shown that when all the factors of the above product are (directed) cycles, then the obtained digraph is the Manhattan street network. To this respect, it is proved that many properties of such
networks, such as high symmetries and the presence of Hamiltonian cycles, are shared by the Manhattan product of some digraphs
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
Hamiltonian cycles in Cayley graphs of imprimitive complex reflection groups
Generalizing a result of Conway, Sloane, and Wilkes for real reflection
groups, we show the Cayley graph of an imprimitive complex reflection group
with respect to standard generating reflections has a Hamiltonian cycle. This
is consistent with the long-standing conjecture that for every finite group, G,
and every set of generators, S, of G the undirected Cayley graph of G with
respect to S has a Hamiltonian cycle.Comment: 15 pages, 4 figures; minor revisions according to referee comments,
to appear in Discrete Mathematic
On Hamiltonicity of circulant digraphs of outdegree three
This paper deals with Hamiltonicity of connected loopless circulant digraphs of outdegree three with connection set of the form ▫▫, where ▫▫ is an integer. In particular, we prove that if ▫▫ or ▫▫ such a circulant digraph is Hamiltonian if and only if it is not isomorphic to the circulant digraph on 12 vertices with connection set ▫▫