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arXiv · 1907.04535

On general position sets in Cartesian products

Sandi Klavžar·Balázs Patkós·Gregor Rus·Ismael G. Yero
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Abstract

The general position number ${\rm gp}(G)$ 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 $P_r\,\square\, C_s$ is deduced. It is proved that ${\rm gp}(C_r\,\square\, C_s)\in \{6,7\}$ whenever $r\ge s \ge 3$, $s\ne 4$, and $r\ge 6$. 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 $P_r\,\square\, P_s$, where $r,s\ge 2$, is also determined.

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Sandi Klavžar, Balázs Patkós, Gregor Rus, Ismael G. Yero. 2019-07-10. On general position sets in Cartesian products. https://arxiv.org/abs/1907.04535

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