11,626 research outputs found
Colouring exact distance graphs of chordal graphs
For a graph and positive integer , the exact distance- graph
is the graph with vertex set and with an edge between
vertices and if and only if and have distance . Recently,
there has been an effort to obtain bounds on the chromatic number
of exact distance- graphs for from certain
classes of graphs. In particular, if a graph has tree-width , it has
been shown that for odd ,
and for even . We
show that if is chordal and has tree-width , then for odd , and for even .
If we could show that for every graph of tree-width there is a
chordal graph of tree-width which contains as an isometric subgraph
(i.e., a distance preserving subgraph), then our results would extend to all
graphs of tree-width . While we cannot do this, we show that for every graph
of genus there is a graph which is a triangulation of genus and
contains as an isometric subgraph.Comment: 11 pages, 2 figures. Versions 2 and 3 include minor changes, which
arise from reviewers' comment
Reconstruction/Non-reconstruction Thresholds for Colourings of General Galton-Watson Trees
The broadcasting models on trees arise in many contexts such as discrete
mathematics, biology statistical physics and cs. In this work, we consider the
colouring model. A basic question here is whether the root's assignment affects
the distribution of the colourings at the vertices at distance h from the root.
This is the so-called "reconstruction problem". For a d-ary tree it is well
known that d/ln (d) is the reconstruction threshold. That is, for
k=(1+eps)d/ln(d) we have non-reconstruction while for k=(1-eps)d/ln(d) we have.
Here, we consider the largely unstudied case where the underlying tree is
chosen according to a predefined distribution. In particular, our focus is on
the well-known Galton-Watson trees. This model arises naturally in many
contexts, e.g. the theory of spin-glasses and its applications on random
Constraint Satisfaction Problems (rCSP). The aforementioned study focuses on
Galton-Watson trees with offspring distribution B(n,d/n), i.e. the binomial
with parameters n and d/n, where d is fixed. Here we consider a broader version
of the problem, as we assume general offspring distribution, which includes
B(n,d/n) as a special case.
Our approach relates the corresponding bounds for (non)reconstruction to
certain concentration properties of the offspring distribution. This allows to
derive reconstruction thresholds for a very wide family of offspring
distributions, which includes B(n,d/n). A very interesting corollary is that
for distributions with expected offspring d, we get reconstruction threshold
d/ln(d) under weaker concentration conditions than what we have in B(n,d/n).
Furthermore, our reconstruction threshold for the random colorings of
Galton-Watson with offspring B(n,d/n), implies the reconstruction threshold for
the random colourings of G(n,d/n)
Forwarding and optical indices of 4-regular circulant networks
An all-to-all routing in a graph is a set of oriented paths of , with
exactly one path for each ordered pair of vertices. The load of an edge under
an all-to-all routing is the number of times it is used (in either
direction) by paths of , and the maximum load of an edge is denoted by
. The edge-forwarding index is the minimum of
over all possible all-to-all routings , and the arc-forwarding index
is defined similarly by taking direction into
consideration, where an arc is an ordered pair of adjacent vertices. Denote by
the minimum number of colours required to colour the paths of such
that any two paths having an edge in common receive distinct colours. The
optical index is defined to be the minimum of over all possible
, and the directed optical index is defined
similarly by requiring that any two paths having an arc in common receive
distinct colours. In this paper we obtain lower and upper bounds on these four
invariants for -regular circulant graphs with connection set , . We give approximation algorithms with performance ratio a
small constant for the corresponding forwarding index and routing and
wavelength assignment problems for some families of -regular circulant
graphs.Comment: 19 pages, no figure in Journal of Discrete Algorithms 201
Directed Ramsey number for trees
In this paper, we study Ramsey-type problems for directed graphs. We first
consider the -colour oriented Ramsey number of , denoted by
, which is the least for which every
-edge-coloured tournament on vertices contains a monochromatic copy of
. We prove that for any oriented
tree . This is a generalisation of a similar result for directed paths by
Chv\'atal and by Gy\'arf\'as and Lehel, and answers a question of Yuster. In
general, it is tight up to a constant factor.
We also consider the -colour directed Ramsey number
of , which is defined as above, but, instead
of colouring tournaments, we colour the complete directed graph of order .
Here we show that for any
oriented tree , which is again tight up to a constant factor, and it
generalises a result by Williamson and by Gy\'arf\'as and Lehel who determined
the -colour directed Ramsey number of directed paths.Comment: 27 pages, 14 figure
Reconstruction of Random Colourings
Reconstruction problems have been studied in a number of contexts including
biology, information theory and and statistical physics. We consider the
reconstruction problem for random -colourings on the -ary tree for
large . Bhatnagar et. al. showed non-reconstruction when and reconstruction when . We tighten this result and show non-reconstruction when and reconstruction when .Comment: Added references, updated notatio
On the editing distance of graphs
An edge-operation on a graph is defined to be either the deletion of an
existing edge or the addition of a nonexisting edge. Given a family of graphs
, the editing distance from to is the smallest
number of edge-operations needed to modify into a graph from .
In this paper, we fix a graph and consider , the set of
all graphs on vertices that have no induced copy of . We provide bounds
for the maximum over all -vertex graphs of the editing distance from
to , using an invariant we call the {\it binary chromatic
number} of the graph . We give asymptotically tight bounds for that distance
when is self-complementary and exact results for several small graphs
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