16,858 research outputs found
Matchings and Hamilton Cycles with Constraints on Sets of Edges
The aim of this paper is to extend and generalise some work of Katona on the
existence of perfect matchings or Hamilton cycles in graphs subject to certain
constraints. The most general form of these constraints is that we are given a
family of sets of edges of our graph and are not allowed to use all the edges
of any member of this family. We consider two natural ways of expressing
constraints of this kind using graphs and using set systems.
For the first version we ask for conditions on regular bipartite graphs
and for there to exist a perfect matching in , no two edges of which
form a -cycle with two edges of .
In the second, we ask for conditions under which a Hamilton cycle in the
complete graph (or equivalently a cyclic permutation) exists, with the property
that it has no collection of intervals of prescribed lengths whose union is an
element of a given family of sets. For instance we prove that the smallest
family of -sets with the property that every cyclic permutation of an
-set contains two adjacent pairs of points has size between
and . We also give bounds on the general version of this problem
and on other natural special cases.
We finish by raising numerous open problems and directions for further study.Comment: 21 page
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
Mixing fuel particles for space combustion research using acoustics
Part of the microgravity science to be conducted aboard the Shuttle (STS) involves combustion using solids, particles, and liquid droplets. The central experimental facts needed for characterization of premixed quiescent particle cloud flames cannot be adequately established by normal gravity studies alone. The experimental results to date of acoustically mixing a prototypical particulate, lycopodium, in a 5 cm diameter by 75 cm long flame tube aboard a Learjet aircraft flying a 20 sec low gravity trajectory are described. Photographic and light detector instrumentation combine to measure and characterize particle cloud uniformity
Set Systems Containing Many Maximal Chains
The purpose of this short problem paper is to raise an extremal question on
set systems which seems to be natural and appealing. Our question is: which set
systems of a given size maximise the number of -element chains in the
power set ? We will show that for each fixed
there is a family of sets containing
such chains, and that this is asymptotically best possible. For smaller set
systems we are unable to answer the question. We conjecture that a `tower of
cubes' construction is extremal. We finish by mentioning briefly a connection
to an extremal problem on posets and a variant of our question for the grid
graph.Comment: 5 page
Disparities in Cause-Specific Cancer Survival by Census Tract Poverty Level in Idaho, U.S.
Objective. This population-based study compared cause-specific cancer survival by socioeconomic status using methods to more accurately assign cancer deaths to primary site. Methods. The current study analyzed Idaho data used in the Accuracy of Cancer Mortality Statistics Based on Death Certificates (ACM) study supplemented with additional information to measure cause-specific cancer survival by census tract poverty level. Results. The distribution of cases by primary site group differed significantly by poverty level (chi-square = 265.3, 100 df, p In the life table analyses, for 8 of 24 primary site groups investigated, and all sites combined, there was a significant gradient relating higher poverty with poorer survival. For all sites combined, the absolute difference in 5-year cause-specific survival rate was 13.6% between the lowest and highest poverty levels. Conclusions. This study shows striking disparities in cause-specific cancer survival related to the poverty level of the area a person resides in at the time of diagnosis
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