Rainbow Independent Sets in Cycles

TL;DR

This paper investigates the minimal number of independent sets needed to guarantee a rainbow independent subset in odd cycle graphs, confirming a specific case of a broader conjecture.

Contribution

It proves that for odd cycle graphs, the minimal number equals the size of the independent sets, advancing understanding of rainbow independent sets in cycle graphs.

Findings

f_{C_{2s+1}}(s, s) = s for odd cycle graphs

Supports a conjecture by Aharoni et al.

Advances combinatorial understanding of rainbow independent sets

Abstract

For a given class ${\cal C}$ of graphs and given integers $m \le n$, let $f_{\cal C}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in any graph belonging to ${\cal C}$ have a (possibly partial) rainbow independent $m$-set. In this paper, we consider the case ${\cal C}=\{C_{2s+1}\}$ and show that $f_{C_{2s+1}}(s, s) = s$. Our result is a special case of the conjecture (Conjecture 2.9) proposed by Aharoni et al in \cite{Aharoni}.