$χ(G) = 2$ if and only if $χ_f (G) = 2$.

Question

Can anyone help with this?

''Show that $χ(G)=2$ if and only if $χ_f(G)=2$.''

$χ(G)$ is a chromatic number and $χ_f(G)$ is a fractional chromatic number.

---

I tried to proove:

($=>$)

$χ(G)=2$ $⟺$ G is bipartite and $E(G)≠∅$ $=>$ $ω(G)=2$ $=>$ $χ_f(G)=2$

($<=$)

Suppose, that $χ_f(G)=2$.
Then $ω(G)≤2$.

(We can suppose that $ω(G)=2$, because if $ω(G)=1$, then $χ(G)=1$ $=>$ $χ_f(G)=1$ )

Answer 1

Hint: if $G$ is a connected graph and we give each vertex a subset of size $k$ of $\{1,...,2k\}$ such that adjacent vertices get disjoint sets, show that there is some subset $S$ such that all vertices get either $S$ or $S^c$.