On General Position Sets in Cartesian Products
Sandi Klavžar, Balázs Patkós, Gregor Rus, Ismael G. Yero · Results in Mathematics · 2021
Abstract The general position number $$\mathrm{gp}(G)$$ gp(G) of a connected graphGis the cardinality of a largest setSof vertices such that no three distinct vertices fromSlie on a common geodesic; such sets are refereed to as gp-sets ofG. The general position number of cylinders $$P_r\,\square \,C_s$$ Pr□Cs is deduced. It is proved that $$\mathrm{gp}(C_r\,\square \,C_s)\in \{6,7\}$$ gp(Cr□Cs)∈{6,7} whenever $$r\ge s \ge 3$$ r≥s≥3 , $$s e 4$$ 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 $$P_r\,\square \,P_s$$ Pr□Ps , where $$r,s\ge 2$$ r,s≥2 , is also determined.