867 research outputs found
Universal targets for homomorphisms of edge-colored graphs
A -edge-colored graph is a finite, simple graph with edges labeled by
numbers . A function from the vertex set of one -edge-colored
graph to another is a homomorphism if the endpoints of any edge are mapped to
two different vertices connected by an edge of the same color. Given a class
of graphs, a -edge-colored graph (not necessarily
with the underlying graph in ) is -universal for
when any -edge-colored graph with the underlying graph in
admits a homomorphism to . We characterize graph classes that admit
-universal graphs. For such classes, we establish asymptotically almost
tight bounds on the size of the smallest universal graph.
For a nonempty graph , the density of is the maximum ratio of the
number of edges to the number of vertices ranging over all nonempty subgraphs
of . For a nonempty class of graphs, denotes
the density of , that is the supremum of densities of graphs in
.
The main results are the following. The class admits
-universal graphs for if and only if there is an absolute constant
that bounds the acyclic chromatic number of any graph in . For any
such class, there exists a constant , such that for any , the size
of the smallest -universal graph is between and
.
A connection between the acyclic coloring and the existence of universal
graphs was first observed by Alon and Marshall (Journal of Algebraic
Combinatorics, 8(1):5-13, 1998). One of their results is that for planar
graphs, the size of the smallest -universal graph is between and
. Our results yield that there exists a constant such that for all
, this size is bounded from above by
Acyclic Subgraphs of Planar Digraphs
An acyclic set in a digraph is a set of vertices that induces an acyclic
subgraph. In 2011, Harutyunyan conjectured that every planar digraph on
vertices without directed 2-cycles possesses an acyclic set of size at least
. We prove this conjecture for digraphs where every directed cycle has
length at least 8. More generally, if is the length of the shortest
directed cycle, we show that there exists an acyclic set of size at least .Comment: 9 page
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
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