141 research outputs found
Combinatorics
This is the report on the Oberwolfach workshop on Combinatorics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable structures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics
Error Graphs and the Reconstruction of Elements in Groups
Packing and covering problems for metric spaces, and graphs in particular,
are of essential interest in combinatorics and coding theory. They are
formulated in terms of metric balls of vertices. We consider a new problem in
graph theory which is also based on the consideration of metric balls of
vertices, but which is distinct from the traditional packing and covering
problems. This problem is motivated by applications in information transmission
when redundancy of messages is not sufficient for their exact reconstruction,
and applications in computational biology when one wishes to restore an
evolutionary process. It can be defined as the reconstruction, or
identification, of an unknown vertex in a given graph from a minimal number of
vertices (erroneous or distorted patterns) in a metric ball of a given radius r
around the unknown vertex. For this problem it is required to find minimum
restrictions for such a reconstruction to be possible and also to find
efficient reconstruction algorithms under such minimal restrictions.
In this paper we define error graphs and investigate their basic properties.
A particular class of error graphs occurs when the vertices of the graph are
the elements of a group, and when the path metric is determined by a suitable
set of group elements. These are the undirected Cayley graphs. Of particular
interest is the transposition Cayley graph on the symmetric group which occurs
in connection with the analysis of transpositional mutations in molecular
biology. We obtain a complete solution of the above problems for the
transposition Cayley graph on the symmetric group.Comment: Journal of Combinatorial Theory A 200
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