8 research outputs found
Euler tours in hypergraphs
We show that a quasirandom -uniform hypergraph has a tight Euler tour
subject to the necessary condition that divides all vertex degrees. The
case when is complete confirms a conjecture of Chung, Diaconis and Graham
from 1989 on the existence of universal cycles for the -subsets of an
-set.Comment: version accepted for publication in Combinatoric
Universal and Near-Universal Cycles of Set Partitions
We study universal cycles of the set of -partitions of the
set and prove that the transition digraph associated
with is Eulerian. But this does not imply that universal cycles
(or ucycles) exist, since vertices represent equivalence classes of partitions!
We use this result to prove, however, that ucycles of exist for
all when . We reprove that they exist for odd when and that they do not exist for even when . An infinite family
of for which ucycles do not exist is shown to be those pairs for which
is odd (). We also show that there exist
universal cycles of partitions of into subsets of distinct sizes when
is sufficiently smaller than , and therefore that there exist universal
packings of the partitions in . An analogous result for
coverings completes the investigation.Comment: 22 page
On Universal Cycles for new Classes of Combinatorial Structures
A universal cycle (u-cycle) is a compact listing of a collection of
combinatorial objects. In this paper, we use natural encodings of these objects
to show the existence of u-cycles for collections of subsets, matroids,
restricted multisets, chains of subsets, multichains, and lattice paths. For
subsets, we show that a u-cycle exists for the -subsets of an -set if we
let vary in a non zero length interval. We use this result to construct a
"covering" of length for all subsets of of size
exactly with a specific formula for the term. We also show that
u-cycles exist for all -length words over some alphabet which
contain all characters from Using this result we provide
u-cycles for encodings of Sperner families of size 2 and proper chains of
subsets
The existence of k-radius sequences
Let and be positive integers, and let be an alphabet of size .
A sequence over of length is a \emph{-radius sequence} if any two
distinct elements of occur within distance of each other somewhere in
the sequence. These sequences were introduced by Jaromczyk and Lonc in 2004, in
order to produce an efficient caching strategy when computing certain functions
on large data sets such as medical images.
Let be the length of the shortest -ary -radius sequence. The
paper shows, using a probabilistic argument, that whenever is fixed and
The paper observes that the same argument generalises to the situation when
we require the following stronger property for some integer such that
: any distinct elements of must simultaneously occur
within a distance of each other somewhere in the sequence.Comment: 8 pages. More papers cited, and a minor reorganisation of the last
section, since last version. Typo corrected in the statement of Theorem
Recommended from our members
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem