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Abstract: We consider the following question by Balister, Győri and Schelp: given 2^{n-1} nonzero vectors in F_2^n with zero sum, is it always possible to partition the elements of F_2^n into pairs such that the difference between the two elements of the i-th pair is equal to the i-th given vector for every i? An analogous question in F_p, which is a case of the so-called "seating couples" problem, has been resolved by Preissmann and Mischler in 2009. In this paper, we prove the conjecture in F_2^n in the case when the number of distinct values among the given difference vectors is at most n-2log(n)-1, and also in the case when at least a fraction 1/2+ε of the given vectors are equal (for all ε>0 and n sufficiently large based on ε).
Keywords: binary vector spaces, seating couples, prescribed differences, perfect matching, functional batch code, graph colourings
Published in RUP: 21.12.2025; Views: 177; Downloads: 0
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Keywords: popolna koda, učinkovita dominacija, grafi z učinkovito dominantno množico, polinomski algoritmi, hereditarni grafovski razredi, perfect code, efficient domination, efficiently dominatable graphs, polynomial time algorithms, hereditary graph classes
Published in RUP: 15.10.2015; Views: 5858; Downloads: 340
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