1,063 research outputs found
The chromatic numbers of double coverings of a graph
If we fix a spanning subgraph of a graph , we can define a chromatic
number of with respect to and we show that it coincides with the
chromatic number of a double covering of with co-support . We also find
a few estimations for the chromatic numbers of with respect to .Comment: 10 page
Partitions and Coverings of Trees by Bounded-Degree Subtrees
This paper addresses the following questions for a given tree and integer
: (1) What is the minimum number of degree- subtrees that partition
? (2) What is the minimum number of degree- subtrees that cover
? We answer the first question by providing an explicit formula for the
minimum number of subtrees, and we describe a linear time algorithm that finds
the corresponding partition. For the second question, we present a polynomial
time algorithm that computes a minimum covering. We then establish a tight
bound on the number of subtrees in coverings of trees with given maximum degree
and pathwidth. Our results show that pathwidth is the right parameter to
consider when studying coverings of trees by degree-3 subtrees. We briefly
consider coverings of general graphs by connected subgraphs of bounded degree
Decompositions into subgraphs of small diameter
We investigate decompositions of a graph into a small number of low diameter
subgraphs. Let P(n,\epsilon,d) be the smallest k such that every graph G=(V,E)
on n vertices has an edge partition E=E_0 \cup E_1 \cup ... \cup E_k such that
|E_0| \leq \epsilon n^2 and for all 1 \leq i \leq k the diameter of the
subgraph spanned by E_i is at most d. Using Szemer\'edi's regularity lemma,
Polcyn and Ruci\'nski showed that P(n,\epsilon,4) is bounded above by a
constant depending only \epsilon. This shows that every dense graph can be
partitioned into a small number of ``small worlds'' provided that few edges can
be ignored. Improving on their result, we determine P(n,\epsilon,d) within an
absolute constant factor, showing that P(n,\epsilon,2) = \Theta(n) is unbounded
for \epsilon
n^{-1/2} and P(n,\epsilon,4) = \Theta(1/\epsilon) for \epsilon > n^{-1}. We
also prove that if G has large minimum degree, all the edges of G can be
covered by a small number of low diameter subgraphs. Finally, we extend some of
these results to hypergraphs, improving earlier work of Polcyn, R\"odl,
Ruci\'nski, and Szemer\'edi.Comment: 18 page
Partitioning edge-coloured complete graphs into monochromatic cycles and paths
A conjecture of Erd\H{o}s, Gy\'arf\'as, and Pyber says that in any
edge-colouring of a complete graph with r colours, it is possible to cover all
the vertices with r vertex-disjoint monochromatic cycles. So far, this
conjecture has been proven only for r = 2. In this paper we show that in fact
this conjecture is false for all r > 2. In contrast to this, we show that in
any edge-colouring of a complete graph with three colours, it is possible to
cover all the vertices with three vertex-disjoint monochromatic paths, proving
a particular case of a conjecture due to Gy\'arf\'as. As an intermediate result
we show that in any edge-colouring of the complete graph with the colours red
and blue, it is possible to cover all the vertices with a red path, and a
disjoint blue balanced complete bipartite graph.Comment: 25 pages, 3 figure
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
Tur\'an and Ramsey Properties of Subcube Intersection Graphs
The discrete cube is a fundamental combinatorial structure. A
subcube of is a subset of of its points formed by fixing
coordinates and allowing the remaining to vary freely. The subcube
structure of the discrete cube is surprisingly complicated and there are many
open questions relating to it.
This paper is concerned with patterns of intersections among subcubes of the
discrete cube. Two sample questions along these lines are as follows: given a
family of subcubes in which no of them have non-empty intersection, how
many pairwise intersections can we have? How many subcubes can we have if among
them there are no which have non-empty intersection and no which are
pairwise disjoint? These questions are naturally expressed as Tur\'an and
Ramsey type questions in intersection graphs of subcubes where the intersection
graph of a family of sets has one vertex for each set in the family with two
vertices being adjacent if the corresponding subsets intersect.
Tur\'an and Ramsey type problems are at the heart of extremal combinatorics
and so these problems are mathematically natural. However, a second motivation
is a connection with some questions in social choice theory arising from a
simple model of agreement in a society. Specifically, if we have to make a
binary choice on each of separate issues then it is reasonable to assume
that the set of choices which are acceptable to an individual will be
represented by a subcube. Consequently, the pattern of intersections within a
family of subcubes will have implications for the level of agreement within a
society.
We pose a number of questions and conjectures relating directly to the
Tur\'an and Ramsey problems as well as raising some further directions for
study of subcube intersection graphs.Comment: 18 page
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