On a Conjecture Concerning Dyadic Oriented Matroids
نویسنده
چکیده
A rational matrix is totally dyadic if all of its nonzero subdeterminants are in {±2 : k ∈ Z}. An oriented matriod is dyadic if it has a totally dyadic representation A. A dyadic oriented matriod is dyadic of order k if it has a totally dyadic representation A with full row rank and with the property that for each pair of adjacent bases A1 and A2 2 ≤ ∣∣∣∣det(A1) det(A2) ∣∣∣∣ ≤ 2. In this note we present a counterexample to a conjecture on the relationship between the order of a dyadic oriented matroid and the ratio of agreement to disagreement in sign of its signed circuits and cocircuits (Conjecture 5.2, Lee (1990)). A rational matrix is totally dyadic if all of its nonzero subdeterminants are in {±2 : k ∈ Z}. An oriented matriod is dyadic if it has a totally dyadic representation A. A dyadic oriented matriod is dyadic of order k if it has a totally dyadic representation A with full row rank and with the property that for each pair of adjacent bases A1 and A2 2 ≤ ∣∣∣∣det(A1) det(A2) ∣∣∣∣ ≤ 2. In (Lee (1990)) it is shown that the order of a dyadic oriented matroid provides a necessary condition on the ratio of agreement to disagreement in sign of its signed circuits and cocircuits. It is the point of this note to show that this necessary condition is not sufficient (Conjecture 5.2, Lee (1990)).
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عنوان ژورنال:
- Electr. J. Comb.
دوره 6 شماره
صفحات -
تاریخ انتشار 1999