291 research outputs found
Track Layouts of Graphs
A \emph{-track layout} of a graph consists of a (proper) vertex
-colouring of , a total order of each vertex colour class, and a
(non-proper) edge -colouring such that between each pair of colour classes
no two monochromatic edges cross. This structure has recently arisen in the
study of three-dimensional graph drawings. This paper presents the beginnings
of a theory of track layouts. First we determine the maximum number of edges in
a -track layout, and show how to colour the edges given fixed linear
orderings of the vertex colour classes. We then describe methods for the
manipulation of track layouts. For example, we show how to decrease the number
of edge colours in a track layout at the expense of increasing the number of
tracks, and vice versa. We then study the relationship between track layouts
and other models of graph layout, namely stack and queue layouts, and geometric
thickness. One of our principle results is that the queue-number and
track-number of a graph are tied, in the sense that one is bounded by a
function of the other. As corollaries we prove that acyclic chromatic number is
bounded by both queue-number and stack-number. Finally we consider track
layouts of planar graphs. While it is an open problem whether planar graphs
have bounded track-number, we prove bounds on the track-number of outerplanar
graphs, and give the best known lower bound on the track-number of planar
graphs.Comment: The paper is submitted for publication. Preliminary draft appeared as
Technical Report TR-2003-07, School of Computer Science, Carleton University,
Ottawa, Canad
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
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