30 research outputs found
Oriented Colouring Graphs of Bounded Degree and Degeneracy
This paper considers upper bounds on the oriented chromatic number, ,
of graphs in terms of their maximum degree and/or their degeneracy
. In particular we show that asymptotically,
where and . This improves a result of MacGillivray, Raspaud, and
Swartz of the form . The rest of the paper is
devoted to improving prior bounds for in terms of and by
refining the asymptotic arguments involved.Comment: 8 pages, 3 figure
A Study of -dipath Colourings of Oriented Graphs
We examine -colourings of oriented graphs in which, for a fixed integer , vertices joined by a directed path of length at most must be
assigned different colours. A homomorphism model that extends the ideas of
Sherk for the case is described. Dichotomy theorems for the complexity of
the problem of deciding, for fixed and , whether there exists such a
-colouring are proved.Comment: 14 page
On Colouring Oriented Graphs of Large Girth
We prove that for every oriented graph and every choice of positive integers and , there exists an oriented graph along with a surjective homomorphism such that: (i) girth; (ii) for every oriented graph with at most vertices, there exists a homomorphism from to if and only if there exists a homomorphism from to C; and (iii) for every -pointed oriented graph with at most vertices and for every homomorphism there exists a unique homomorphism such that . Determining the chromatic number of an oriented graph is equivalent to finding the smallest integer such that admits a homomorphism to an order- tournament, so our main theorem yields results on the girth and chromatic number of oriented graphs. While our main proof is probabilistic (hence nonconstructive), for any given and , we include a construction of an oriented graph with girth and chromatic number
On the oriented chromatic number of dense graphs
Let be a graph with vertices, edges, average degree , and maximum degree . The \emph{oriented chromatic number} of is the maximum, taken over all orientations of , of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which . We prove that every such graph has oriented chromatic number at least . In the case that , this lower bound is improved to . Through a simple connection with harmonious colourings, we prove a general upper bound of \Oh{\Delta\sqrt{n}} on the oriented chromatic number. Moreover this bound is best possible for certain graphs. These lower and upper bounds are particularly close when is ()-regular for some constant , in which case the oriented chromatic number is between and
Oriented coloring on recursively defined digraphs
Coloring is one of the most famous problems in graph theory. The coloring
problem on undirected graphs has been well studied, whereas there are very few
results for coloring problems on directed graphs. An oriented k-coloring of an
oriented graph G=(V,A) is a partition of the vertex set V into k independent
sets such that all the arcs linking two of these subsets have the same
direction. The oriented chromatic number of an oriented graph G is the smallest
k such that G allows an oriented k-coloring. Deciding whether an acyclic
digraph allows an oriented 4-coloring is NP-hard. It follows, that finding the
chromatic number of an oriented graph is an NP-hard problem. This motivates to
consider the problem on oriented co-graphs. After giving several
characterizations for this graph class, we show a linear time algorithm which
computes an optimal oriented coloring for an oriented co-graph. We further
prove how the oriented chromatic number can be computed for the disjoint union
and order composition from the oriented chromatic number of the involved
oriented co-graphs. It turns out that within oriented co-graphs the oriented
chromatic number is equal to the length of a longest oriented path plus one. We
also show that the graph isomorphism problem on oriented co-graphs can be
solved in linear time.Comment: 14 page
Oriented Colourings of Graphs with Maximum Degree Three and Four
We show that any orientation of a graph with maximum degree three has an
oriented 9-colouring, and that any orientation of a graph with maximum degree
four has an oriented 69-colouring. These results improve the best known upper
bounds of 11 and 80, respectively