# On the Crossing Number of the Cartesian Product of a Sunlet Graph and a Star Graph

## TL;DR

This paper determines the crossing numbers for the Cartesian product of sunlet and star graphs for specific cases and provides an upper bound for general cases, advancing understanding of graph crossing numbers.

## Contribution

It introduces the first analysis of the crossing number for the Cartesian product of sunlet and star graphs, providing exact values for certain cases and bounds for others.

## Key findings

- Crossing number of $ ext{Sunlet}_n imes K_{1,2}$ is $n$.
- Crossing number of $ ext{Sunlet}_n imes K_{1,3}$ is $3n$.
- An upper bound for $ ext{Sunlet}_n imes K_{1,m}$ is established.

## Abstract

The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the Sunlet graph, denoted $\mathcal{S}_n$, and the Star graph, denoted $K_{1,m}$, is considered for the first time. It is proved that the crossing number of $\mathcal{S}_n \Box K_{1,2}$ is $n$, and the crossing number of $\mathcal{S}_n \Box K_{1,3}$ is $3n$. An upper bound for the crossing number of $\mathcal{S}_n \Box K_{1,m}$ is also given.