2,778 research outputs found
Nombre chromatique fractionnaire, degré maximum et maille
We prove new lower bounds on the independence ratio of graphs of maximum degree ∆ ∈ {3,4,5} and girth g ∈ {6,…,12}, notably 1/3 when (∆,g)=(4,10) and 2/7 when (∆,g)=(5,8). We establish a general upper bound on the fractional chromatic number of triangle-free graphs, which implies that deduced from the fractional version of Reed's bound for triangle-free graphs and improves it as soon as ∆ ≥ 17, matching the best asymptotic upper bound known for off-diagonal Ramsey numbers. In particular, the fractional chromatic number of a triangle-free graph of maximum degree ∆ is less than 9.916 if ∆=17, less than 22.17 if ∆=50 and less than 249.06 if ∆=1000. Focusing on smaller values of ∆, we also demonstrate that every graph of girth at least 7 and maximum degree ∆ has fractional chromatic number at most min (2∆ + 2^{k-3}+k)/k pour k ∈ ℕ. In particular, the fractional chromatic number of a graph of girth 7 and maximum degree ∆ is at most (2∆+9)/5 when ∆ ∈ [3,8], at most (∆+7)/3 when ∆ ∈ [8,20], at most (2∆+23)/7 when ∆ ∈ [20,48], and at most ∆/4+5 when ∆ ∈ [48,112]
Eigenvalue interlacing and weight parameters of graphs
Eigenvalue interlacing is a versatile technique for deriving results in
algebraic combinatorics. In particular, it has been successfully used for
proving a number of results about the relation between the (adjacency matrix or
Laplacian) spectrum of a graph and some of its properties. For instance, some
characterizations of regular partitions, and bounds for some parameters, such
as the independence and chromatic numbers, the diameter, the bandwidth, etc.,
have been obtained. For each parameter of a graph involving the cardinality of
some vertex sets, we can define its corresponding weight parameter by giving
some "weights" (that is, the entries of the positive eigenvector) to the
vertices and replacing cardinalities by square norms. The key point is that
such weights "regularize" the graph, and hence allow us to define a kind of
regular partition, called "pseudo-regular," intended for general graphs. Here
we show how to use interlacing for proving results about some weight parameters
and pseudo-regular partitions of a graph. For instance, generalizing a
well-known result of Lov\'asz, it is shown that the weight Shannon capacity
of a connected graph \G, with vertices and (adjacency matrix)
eigenvalues , satisfies \Theta\le
\Theta^* \le \frac{\|\vecnu\|^2}{1-\frac{\lambda_1}{\lambda_n}} where
is the (standard) Shannon capacity and \vecnu is the positive
eigenvector normalized to have smallest entry 1. In the special case of regular
graphs, the results obtained have some interesting corollaries, such as an
upper bound for some of the multiplicities of the eigenvalues of a
distance-regular graph. Finally, some results involving the Laplacian spectrum
are derived. spectrum are derived
Degree Sequence Index Strategy
We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by
which to bound graph invariants by certain indices in the ordered degree
sequence. As an illustration of the DSI strategy, we show how it can be used to
give new upper and lower bounds on the -independence and the -domination
numbers. These include, among other things, a double generalization of the
annihilation number, a recently introduced upper bound on the independence
number. Next, we use the DSI strategy in conjunction with planarity, to
generalize some results of Caro and Roddity about independence number in planar
graphs. Lastly, for claw-free and -free graphs, we use DSI to
generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester
Towards on-line Ohba's conjecture
The on-line choice number of a graph is a variation of the choice number
defined through a two person game. It is at least as large as the choice number
for all graphs and is strictly larger for some graphs. In particular, there are
graphs with whose on-line choice numbers are larger
than their chromatic numbers, in contrast to a recently confirmed conjecture of
Ohba that every graph with has its choice number
equal its chromatic number. Nevertheless, an on-line version of Ohba conjecture
was proposed in [P. Huang, T. Wong and X. Zhu, Application of polynomial method
to on-line colouring of graphs, European J. Combin., 2011]: Every graph
with has its on-line choice number equal its chromatic
number. This paper confirms the on-line version of Ohba conjecture for graphs
with independence number at most 3. We also study list colouring of
complete multipartite graphs with all parts of size 3. We prove
that the on-line choice number of is at most , and
present an alternate proof of Kierstead's result that its choice number is
. For general graphs , we prove that if then its on-line choice number equals chromatic number.Comment: new abstract and introductio
Some results on chromatic number as a function of triangle count
A variety of powerful extremal results have been shown for the chromatic
number of triangle-free graphs. Three noteworthy bounds are in terms of the
number of vertices, edges, and maximum degree given by Poljak \& Tuza (1994),
and Johansson. There have been comparatively fewer works extending these types
of bounds to graphs with a small number of triangles. One noteworthy exception
is a result of Alon et. al (1999) bounding the chromatic number for graphs with
low degree and few triangles per vertex; this bound is nearly the same as for
triangle-free graphs. This type of parametrization is much less rigid, and has
appeared in dozens of combinatorial constructions.
In this paper, we show a similar type of result for as a function
of the number of vertices , the number of edges , as well as the triangle
count (both local and global measures). Our results smoothly interpolate
between the generic bounds true for all graphs and bounds for triangle-free
graphs. Our results are tight for most of these cases; we show how an open
problem regarding fractional chromatic number and degeneracy in triangle-free
graphs can resolve the small remaining gap in our bounds
Local Graph Coloring and Index Coding
We present a novel upper bound for the optimal index coding rate. Our bound
uses a graph theoretic quantity called the local chromatic number. We show how
a good local coloring can be used to create a good index code. The local
coloring is used as an alignment guide to assign index coding vectors from a
general position MDS code. We further show that a natural LP relaxation yields
an even stronger index code. Our bounds provably outperform the state of the
art on index coding but at most by a constant factor.Comment: 14 Pages, 3 Figures; A conference version submitted to ISIT 2013;
typos correcte
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