Rainbow independent sets in graphs with maximum degree two

Given a graph $G$, let $f_{G}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in $G$ have a rainbow $m$-set. Let $mathcal{D}(2)$ be the family of all graphs with maximum degree at most two. Aharoni et al. (2019) conjectured that (i) $f_G(n,n-1)=n-1$ for all graphs $Ginmathcal{D}(2)$ and (ii) $f_{C_t}(n,n)=n$ for $tge 2n+1$. Lv and Lu (2020) showed that the conjecture (ii) holds when $t=2n+1$. In this article, we show that the conjecture (ii) holds for $tgefrac{1}{3}n^2+frac{44}{9}n$. Let $C_t$ be a cycle of length $t$ with vertices being arranged in a clockwise order. An ordered set $I=(a_1,a_2,ldots,a_n)$ on $C_t$ is called a $2$-jump independent $n$-set of $C_t$ if $a_{i+1}-a_i=2pmod{t}$ for any $1le ile n-1$. We also show that a collection of 2-jump independent $n$-sets $mathcal{F}$ of $C_t$ with $|mathcal{F}|=n$ admits a rainbow independent $n$-set, i.e. (ii) holds if we restrict $mathcal{F}$ on the family of 2-jump independent $n$-sets. Moreover, we prove that if the conjecture (ii) holds, then (i) holds for all graphs $Ginmathcal{D}(2)$ with $c_e(G)le 4$, where $c_e(G)$ is the number of components of $G$ isomorphic to cycles of even lengths.