22 research outputs found
The localization number and metric dimension of graphs of diameter 2
We consider the localization number and metric dimension of certain graphs of diameter , focusing on families of Kneser graphs and graphs without 4-cycles. For the Kneser graphs with a diameter of , we find upper and lower bounds for the localization number and metric dimension, and in many cases these parameters differ only by an additive constant. Our results on the metric dimension of Kneser graphs improve on earlier ones, yielding exact values in infinitely many cases. We determine bounds on the localization number and metric dimension of Moore graphs of diameter and polarity graphs
Rainbow Thresholds
We extend a recent breakthrough result relating expectation thresholds and
actual thresholds to include rainbow versions
The -visibility Localization Game
We study a variant of the Localization game in which the cops have limited
visibility, along with the corresponding optimization parameter, the
-visibility localization number , where is a non-negative
integer. We give bounds on -visibility localization numbers related to
domination, maximum degree, and isoperimetric inequalities. For all , we
give a family of trees with unbounded values. Extending results known
for the localization number, we show that for , every tree contains a
subdivision with . For many , we give the exact value of
for the Cartesian grid graphs, with the remaining cases
being one of two values as long as is sufficiently large. These examples
also illustrate that for all distinct choices of and
$j.