212 research outputs found
Largest reduced neighborhood clique cover number revisited
Let be a graph and . The largest reduced neighborhood clique
cover number of , denoted by , is the largest, overall
-shallow minors of , of the smallest number of cliques that can cover
any closed neighborhood of a vertex in . It is known that
, where is an incomparability graph and is
the number of leaves in a largest shallow minor which is isomorphic to an
induced star on leaves. In this paper we give an overview of the
properties of including the connections to the greatest
reduced average density of , or , introduce the class
of graphs with bounded neighborhood clique cover number, and derive a simple
lower and an upper bound for this important graph parameter. We announce two
conjectures, one for the value of , and another for a
separator theorem (with respect to a certain measure) for an interesting class
of graphs, namely the class of incomparability graphs which we suspect to have
a polynomial bounded neighborhood clique cover number, when the size of a
largest induced star is bounded.Comment: The results in this paper were presented in 48th Southeastern
Conference in Combinatorics, Graph Theory and Computing, Florida Atlantic
University, Boca Raton, March 201
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
On the Generalised Colouring Numbers of Graphs that Exclude a Fixed Minor
The generalised colouring numbers and
were introduced by Kierstead and Yang as a generalisation
of the usual colouring number, and have since then found important theoretical
and algorithmic applications. In this paper, we dramatically improve upon the
known upper bounds for generalised colouring numbers for graphs excluding a
fixed minor, from the exponential bounds of Grohe et al. to a linear bound for
the -colouring number and a polynomial bound for the weak
-colouring number . In particular, we show that if
excludes as a minor, for some fixed , then
and
.
In the case of graphs of bounded genus , we improve the bounds to
(and even if
, i.e. if is planar) and
.Comment: 21 pages, to appear in European Journal of Combinatoric
A note on circular chromatic number of graphs with large girth and similar problems
In this short note, we extend the result of Galluccio, Goddyn, and Hell,
which states that graphs of large girth excluding a minor are nearly bipartite.
We also prove a similar result for the oriented chromatic number, from which
follows in particular that graphs of large girth excluding a minor have
oriented chromatic number at most , and for the th chromatic number
, from which follows in particular that graphs of large girth
excluding a minor have
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