34 research outputs found
Coloring directed cycles
Sopena in his survey [E. Sopena, The oriented chromatic number of graphs: A
short survey, preprint 2013] writes, without any proof, that an oriented cycle
can be colored with three colors if and only if ,
where is the number of forward arcs minus the number of
backward arcs in . This is not true. In this paper we show that can be colored with three colors if and only if
or does not contain three consecutive arcs going in the same
direction
The Oriented Chromatic Number of the Hexagonal Grid is 6
The oriented chromatic number of a directed graph is the minimum order of
an oriented graph to which has a homomorphism. The oriented chromatic
number of a graph family is the maximum oriented
chromatic number over any orientation of any graph in . For the
family of hexagonal grids , Bielak (2006) proved that . Here we close the gap by showing that .Comment: 8 pages, 5 figure
Upper oriented chromatic number of undirected graphs and oriented colorings of product graphs
The oriented chromatic number of an oriented graph is the minimum
order of an oriented graph \vev H such that admits a homomorphism to
\vev H. The oriented chromatic number of an undirected graph is then the
greatest oriented chromatic number of its orientations. In this paper, we
introduce the new notion of the upper oriented chromatic number of an
undirected graph , defined as the minimum order of an oriented graph \vev
U such that every orientation of admits a homomorphism to . We give some properties of this parameter, derive some general upper bounds
on the ordinary and upper oriented chromatic numbers of Cartesian, strong,
direct and lexicographic products of graphs, and consider the particular case
of products of paths.Comment: 14 page
Acyclic edge coloring of subcubic graphs
AbstractAn acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and it is denoted by a′(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors
Colourings of -coloured mixed graphs
A mixed graph is, informally, an object obtained from a simple undirected
graph by choosing an orientation for a subset of its edges. A mixed graph is
-coloured if each edge is assigned one of colours, and each
arc is assigned one of colours. Oriented graphs are -coloured mixed graphs, and 2-edge-coloured graphs are -coloured
mixed graphs. We show that results of Sopena for vertex colourings of oriented
graphs, and of Kostochka, Sopena and Zhu for vertex colourings oriented graphs
and 2-edge-coloured graphs, are special cases of results about vertex
colourings of -coloured mixed graphs. Both of these can be regarded as
a version of Brooks' Theorem.Comment: 7 pages, no figure