8,190 research outputs found
Rainbow Connection Number and Connected Dominating Sets
Rainbow connection number rc(G) of a connected graph G is the minimum number
of colours needed to colour the edges of G, so that every pair of vertices is
connected by at least one path in which no two edges are coloured the same. In
this paper we show that for every connected graph G, with minimum degree at
least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2,
where {\gamma}_c(G) is the connected domination number of G. Bounds of the form
diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special
graph classes follow as easy corollaries from this result. This includes
interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and
chain graphs all with minimum degree at least 2 and connected. We also show
that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of
these cases, we also demonstrate the tightness of the bounds. An extension of
this idea to two-step dominating sets is used to show that for every connected
graph on n vertices with minimum degree {\delta}, the rainbow connection number
is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of
Schiermeyer (2009), improving the previously best known bound of 20n/{\delta}
by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up
to additive factors by a construction of Caro et al. (2008).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
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
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