A 3-acyclic edge-coloring is also known as an acyclic edge-coloring. The best already known upper bound for a′3(G) is due to Ndreca et al. [4], they proved that a′3(G) ≤ 9.62(G) for any graph G. In view of the results mentioned above, we can see that a′r(G) = O((G)) for r ≤ 3.