
On orthogonal symmetric chain decompositions
The ncube is the poset obtained by ordering all subsets of {1,...,n} by...
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On the central levels problem
The central levels problem asserts that the subgraph of the (2m+1)dimen...
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Graph Sparsification, Spectral Sketches, and Faster Resistance Computation, via Short Cycle Decompositions
We develop a framework for graph sparsification and sketching, based on ...
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A short proof of the middle levels theorem
Consider the graph that has as vertices all bitstrings of length 2n+1 wi...
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Further Results on Circuit Codes
We present a new characterization of circuit codes of spread k based on ...
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Property analysis of symmetric travelling salesman problem instances acquired through evolution
We show how an evolutionary algorithm can successfully be used to evolve...
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A ChainDetection Algorithm for TwoDimensional Grids
We describe a general method of detecting valid chains or links of piece...
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Gray codes and symmetric chains
We consider the problem of constructing a cyclic listing of all bitstrings of length 2n+1 with Hamming weights in the interval [n+1ℓ,n+ℓ], where 1≤ℓ≤ n+1, by flipping a single bit in each step. This is a farranging generalization of the wellknown middle two levels problem (the case ℓ=1). We provide a solution for the case ℓ=2 and solve a relaxed version of the problem for general values of ℓ, by constructing cycle factors for those instances. Our proof uses symmetric chain decompositions of the hypercube, a concept known from the theory of posets, and we present several new constructions of such decompositions. In particular, we construct four pairwise edgedisjoint symmetric chain decompositions of the ndimensional hypercube for any n≥ 12.
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