# (Open) packing number of some graph products

## TL;DR

This paper investigates the open packing number in various graph products, providing exact solutions for some cases and bounds for others, advancing understanding of neighborhood packings in complex graph structures.

## Contribution

It offers a complete characterization of the open packing number for lexicographic and rooted products, and establishes bounds for Cartesian, strong, and direct products.

## Key findings

- Exact solutions for lexicographic and rooted products.
- Bounds for Cartesian, strong, and direct products.
- Enhanced understanding of neighborhood packings in graph products.

## Abstract

The packing number of a graph $G$ is the maximum number of closed neighborhoods of vertices in $G$ with pairwise empty intersections. Similarly, the open packing number of $G$ is the maximum number of open neighborhoods in $G$ with pairwise empty intersections. We consider the packing and open packing numbers on graph products. In particular we give a complete solution with respect to some properties of factors in the case of lexicographic and rooted products. For Cartesian, strong and direct products, we present several lower and upper bounds on these parameters.