748 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
Continuum spin foam model for 3d gravity
An example illustrating a continuum spin foam framework is presented. This
covariant framework induces the kinematics of canonical loop quantization, and
its dynamics is generated by a {\em renormalized} sum over colored polyhedra.
Physically the example corresponds to 3d gravity with cosmological constant.
Starting from a kinematical structure that accommodates local degrees of
freedom and does not involve the choice of any background structure (e. g.
triangulation), the dynamics reduces the field theory to have only global
degrees of freedom. The result is {\em projectively} equivalent to the
Turaev-Viro model.Comment: 12 pages, 3 figure
Grad and classes with bounded expansion I. decompositions
We introduce classes of graphs with bounded expansion as a generalization of
both proper minor closed classes and degree bounded classes. Such classes are
based on a new invariant, the greatest reduced average density (grad) of G with
rank r, grad r(G). For these classes we prove the existence of several
partition results such as the existence of low tree-width and low tree-depth
colorings. This generalizes and simplifies several earlier results (obtained
for minor closed classes)
Monochromatic spanning trees and matchings in ordered complete graphs
Two independent edges in ordered graphs can be nested, crossing or separated.
These relations define six types of subgraphs, depending on which relations are
forbidden. We refine a remark by Erd\H{o}s and Rado that every 2-coloring of
the edges of a complete graph contains a monochromatic spanning tree. We show
that forbidding one relation we always have a monochromatic (non-nested,
non-crossing, non-separated) spanning tree in a 2-edge-colored ordered complete
graph. On the other hand, if two relations are forbidden, then it is possible
that we have monochromatic (nested, separated, crossing) subtrees of size only
half the number of vertices. The existence of a monochromatic non-nested
spanning tree in 2-colorings of ordered complete graphs verifies a more general
conjecture for twisted drawings. Our second subject is to refine the Ramsey
number of matchings for ordered complete graphs. Cockayne and Lorimer proved
that for given positive integers t, n, m=(t-1)(n-1)+2n is the smallest integer
with the following property: every t-coloring of the edges of a complete graph
Km contains a monochromatic matching with n edges. We conjecture a
strengthening: t-colored ordered complete graphs on m vertices contain
monochromatic non-nested and also non-separated matchings with n edges. We
prove this conjecture for some special cases. (i) Every t-colored ordered
complete graph on t+3 vertices contains a monochromatic non-nested matching of
size two. (ii) Every 2-colored ordered complete graph on 3n-1 vertices contains
a monochromatic non-separated matching of size n.
For nested, separated, and crossing matchings the situation is different. The
smallest m ensuring a monochromatic matching of size n in every t-coloring is
2(t(n-1))+1) in the first two cases and one less in the third case
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