128 research outputs found
The number of point-splitting circles
Let S be a set of 2n+1 points in the plane such that no three are collinear
and no four are concyclic. A circle will be called point-splitting if it has 3
points of S on its circumference, n-1 points in its interior and n-1 in its
exterior. We show the surprising property that S always has exactly n^2 point-
splitting circles, and prove a more general result.Comment: 12 pages, 4 figure
The Bergman complex of a matroid and phylogenetic trees
We study the Bergman complex B(M) of a matroid M: a polyhedral complex which
arises in algebraic geometry, but which we describe purely combinatorially. We
prove that a natural subdivision of the Bergman complex of M is a geometric
realization of the order complex of its lattice of flats. In addition, we show
that the Bergman fan B'(K_n) of the graphical matroid of the complete graph K_n
is homeomorphic to the space of phylogenetic trees T_n.Comment: 15 pages, 6 figures. Reorganized paper and updated references. To
appear in J. Combin. Theory Ser.
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