On General Position Sets in Cartesian Products - Klavžar, Sandi; Patkós, Balázs; Rus, Gregor; Yero, Ismael G. (Resultate der Mathematik, 2021)
The general position number gp ( G ) of a connected graph G is the cardinality of a largest set S of vertices such that no three distinct vertices from S lie on a common geodesic; such sets are refereed to as gp-sets of G . The general position number of cylinders P r □ C s is deduced. It is proved that gp ( C r □ C s ) ∈ { 6 , 7 } whenever r ≥ s ≥ 3 , s ≠ 4 , and r ≥ 6 . A probabilistic lower bound on the general position number of Cartesian graph powers is achieved. Along the way a formula for the number of gp-sets in P r □ P s , where r , s ≥ 2 , is also determined.