Rainbow independent sets in certain classes of graphs

TL;DR

This paper investigates the minimal number of independent sets needed to guarantee a rainbow independent subset of a certain size within various graph classes, extending known results beyond line graphs.

Contribution

It introduces and analyzes the function $f_ ext{class}(n,m)$ for different graph classes, broadening understanding of rainbow independent sets in graph theory.

Findings

Determined bounds for $f_ ext{class}(n,m)$ in specific graph classes

Extended known results from line graphs to other classes

Provided conditions for the finiteness of $f_ ext{class}(n,m)$

Abstract

For a given class $\mathcal{C}$ of graphs and given integers $m \leq n$, let $f_\mathcal{C}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in any graph belonging to $\mathcal{C}$ have a (possibly partial) rainbow independent $m$-set. Motivated by known results on the finiteness and actual value of $f_\mathcal{C}(n,m)$ when $\mathcal{C}$ is the class of line graphs of graphs, we study this function for various other classes.