tailieunhanh - Báo cáo toán học: "On a conjecture concerning dyadic oriented matroids"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: On a conjecture concerning dyadic oriented matroids. | On a conjecture concerning dyadic oriented matroids Matt Scobee Department of Mathematics University of Louisville. scobee@ Submitted January 7 1999 Accepted March 30 1999 Abstract A rational matrix is totally dyadic if all of its nonzero subdeterminants are in 2k k 2 Zg. 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 Al and A2 2 k det A1 det A2 2k 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 Lee 1990 . A rational matrix is totally dyadic if all of its nonzero subdeterminants are in 2k k 2 Zg. 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 det A1 2k 2 k det A2 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 Lee 1990 . 1 THE ELECTRONIC .JOURNAL OF COmBINATORICS 6 1999 R23 2 1 Background We assume some familiarity with matroid theory see Oxley 1992 . The ground set of a matroid M is denoted by E M . For a matrix A over a held F let M A denote the matroid represented by A. Let C M resp. C M denote the set of circuits cocircuits of a matroid M. Orientations of a matroid M arise by partitioning or signing each circuit X resp. cocircuit Y as X X- Y Y- so that X n Y u X- n Y X n Y- u X- n Y holds for all X 2 C M Y 2 C M . We note that negating any