837 research outputs found
An update on the middle levels problem
The middle levels problem is to find a Hamilton cycle in the middle levels,
M_{2k+1}, of the Hasse diagram of B_{2k+1} (the partially ordered set of
subsets of a 2k+1-element set ordered by inclusion). Previously, the best
result was that M_{2k+1} is Hamiltonian for all positive k through k=15. In
this note we announce that M_{33} and M_{35} have Hamilton cycles. The result
was achieved by an algorithmic improvement that made it possible to find a
Hamilton path in a reduced graph of complementary necklace pairs having
129,644,790 vertices, using a 64-bit personal computer.Comment: 11 pages, 5 figure
Gray code order for Lyndon words
International audienceAt the 4th Conference on Combinatorics on Words, Christophe Reutenauer posed the question of whether the dual reflected order yields a Gray code on the Lyndon family. In this paper we give a positive answer. More precisely, we present an O(1)-average-time algorithm for generating length n binary pre-necklaces, necklaces and Lyndon words in Gray code order
Partitioning de Bruijn Graphs into Fixed-Length Cycles for Robot Identification and Tracking
We propose a new camera-based method of robot identification, tracking and
orientation estimation. The system utilises coloured lights mounted in a circle
around each robot to create unique colour sequences that are observed by a
camera. The number of robots that can be uniquely identified is limited by the
number of colours available, , the number of lights on each robot, , and
the number of consecutive lights the camera can see, . For a given set of
parameters, we would like to maximise the number of robots that we can use. We
model this as a combinatorial problem and show that it is equivalent to finding
the maximum number of disjoint -cycles in the de Bruijn graph
.
We provide several existence results that give the maximum number of cycles
in in various cases. For example, we give an optimal
solution when . Another construction yields many cycles in larger
de Bruijn graphs using cycles from smaller de Bruijn graphs: if
can be partitioned into -cycles, then
can be partitioned into -cycles for any divisor of
. The methods used are based on finite field algebra and the combinatorics
of words.Comment: 16 pages, 4 figures. Accepted for publication in Discrete Applied
Mathematic
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