We are indebted to Weifan Wang , and independently to Manu Basavaraju and L. Sunil Chandran  for pointing out that Theorem 1.1 (ii) is wrong in our paper , as there exist infinite outerplanar graphs G with and containing no subgraph isomorphic to Q that have acyclic edge chromatic number five. The proof is incorrect. In the line 36th on page 33 (below Case 3), we let . This is not always true, as may be an edge of G.
The problem that determining the acyclic edge chromatic number of outerplanar graphs G with remains open and seems difficult. The following result give infinite counterexample of Theorem 1.1 (ii).
Theorem 1. Let G be an outerplanar graph with and , and be an outer edge of G with and . If H is an outerplanar graph by replacing the edge with the graph , where and , then .
Proof. Otherwise, let ϕ be an acyclic edge 4-coloring of H with , , and . By Lemma 4.2, , , and (See Fig. 1). Let and for the other edges of G. It is easy to see that is an acyclic edge 4-coloring of G, a contradiction.
The rest of the paper is not affected by this error.