  The direct product $G\times H$ of graphs $G$ and $H$ is defined by:
\[V(G\times H)=V(G)\times V(H)\] and \[E(G\times
H)=\left\{[(u_1,v_1),(u_2,v_2)]: (u_1,u_2)\in E(G) \mbox{\ and\ } (v_1,v_2)\in
E(H)\right\}.\]
  In this paper, we will prove that the equality $$\alpha(G\times
H)=\max\{\alpha(G)|H|, \alpha(H)|G|\}$$ holds for all vertex-transitive graphs
$G$ and $H$, which provides an affirmative answer to a problem posed by Tardif
(Discrete Math. 185 (1998) 193-200). Furthermore, the structure of all maximum
independent sets of $G\times H$ are determined.
