On General Position Sets in Cartesian Products

Abstract

AbstractThe general position number $$\mathrm{gp}(G)$$ 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 $$P_r\,\square \,C_s$$ P r □ C s is deduced. It is proved that $$\mathrm{gp}(C_r\,\square \,C_s)\in \{6,7\}$$ gp ( C r □ C s ) ∈ { 6 , 7 } whenever $$r\ge s \ge 3$$ r ≥ s ≥ 3 , $$s\ne 4$$ s ≠ 4 , and $$r\ge 6$$ 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 $$P_r\,\square \,P_s$$ P r □ P s , where $$r,s\ge 2$$ r , s ≥ 2 , is also determined.