The general position problem on Kneser graphs and on some graph operations

TL;DR

This paper investigates the general position number in Kneser graphs and various graph operations, providing exact values and bounds for these complex graph families.

Contribution

It determines the gp-number for specific Kneser graphs and establishes bounds for Cartesian products, joins, coronas, and line graphs of complete graphs.

Findings

Exact gp-number for K(n,2) and K(n,3)

Sharp lower bounds for Cartesian products

gp-number for joins, coronas, and line graphs

Abstract

A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number (gp-number) ${\rm gp}(G)$ of $G$. The gp-number is determined for some families of Kneser graphs, in particular for $K(n,2)$ and $K(n,3)$. A sharp lower bound on the gp-number is proved for Cartesian products of graphs. The gp-number is also determined for joins of graphs, coronas over graphs, and line graphs of complete graphs.