698 research outputs found
Directional discrepancy in two dimensions
In the present paper, we study the geometric discrepancy with respect to
families of rotated rectangles. The well-known extremal cases are the
axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in
all possible directions (polynomial discrepancy). We study several intermediate
situations: lacunary sequences of directions, lacunary sets of finite order,
and sets with small Minkowski dimension. In each of these cases, extensions of
a lemma due to Davenport allow us to construct appropriate rotations of the
integer lattice which yield small discrepancy
On the minimum degree of minimally -tough, claw-free graphs
A graph is minimally -tough if the toughness of is and
deletion of any edge from decreases its toughness. Katona et al.
conjectured that the minimum degree of any minimally -tough graph is and proved that the minimum degree of minimally -tough and -tough, claw-free graphs is 1 and 2, respectively. We have
show that every minimally -tough, claw-free graph has a vertex of degree
of . In this paper, we give an upper bound on the minimum degree of
minimally -tough, claw-free graphs for
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