Let $k\geq2$ be an integer, a graph $G=(V,E)$ is called $k$-partite if $V$ admits a partition into $k$ classes such that every edge of $G$ has its ends in different classes: vertices in the same class must not be adjacent.

Now let $G=(V,E)$ be $k$-partite and $V_1,V_2,\cdots,V_k$ be the $k$ classes into which $V$ Is partitioned such that:

$(1)\Delta (G)=2;$

$(2)|V_i|=3,i=1,2,\cdots,k;$

$(3)$For any $1\leq i\neq j\leq k$, there does not exist $u\in V_i$ and $v,w\in V_j$ such that $u$ are adjacent with both $u$ and $w$.

I want to ask if I can choose $v_i$ from $V_i, i=1,2,\cdots,k$ such that $\{v_1,v_2,\cdots,v_k\}$ is an independent set in $G$.