49,658 research outputs found
Strong geodetic problem on Cartesian products of graphs
The strong geodetic problem is a recent variation of the geodetic problem.
For a graph , its strong geodetic number is the cardinality of
a smallest vertex subset , such that each vertex of lies on a fixed
shortest path between a pair of vertices from . In this paper, the strong
geodetic problem is studied on the Cartesian product of graphs. A general upper
bound for is determined, as well as exact values
for , , and certain prisms.
Connections between the strong geodetic number of a graph and its subgraphs are
also discussed.Comment: 18 pages, 9 figure
An Improved Point-Line Incidence Bound Over Arbitrary Fields
We prove a new upper bound for the number of incidences between points and
lines in a plane over an arbitrary field , a problem first
considered by Bourgain, Katz and Tao. Specifically, we show that points and
lines in , with , determine at most
incidences (where, if has positive
characteristic , we assume ). This improves on the
previous best known bound, due to Jones. To obtain our bound, we first prove an
optimal point-line incidence bound on Cartesian products, using a reduction to
a point-plane incidence bound of Rudnev. We then cover most of the point set
with Cartesian products, and we bound the incidences on each product
separately, using the bound just mentioned. We give several applications, to
sum-product-type problems, an expander problem of Bourgain, the distinct
distance problem and Beck's theorem.Comment: 18 pages. To appear in the Bulletin of the London Mathematical
Societ
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