Erratum to “Acyclic Edge Chromatic Number of Outerplanar Graphs”

We are indebted to Weifan Wang [3], and independently to Manu Basavaraju and L. Sunil Chandran [1] for pointing out that Theorem 1.1 (ii) is wrong in our paper [2], as there exist infinite outerplanar graphs G with (G) = 4 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 G′ = G \ {u, w, x} + vz. This is not always true, as vz may be an edge of G. The problem that determining the acyclic edge chromatic number of outerplanar graphs G with (G) = 4 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 (G) = 4 and χ ′ a(G) = 5, and xy be an outer edge of G with d(x) = 4 and d(y) = 2, 3. If H is an outerplanar graph by replacing the edge xy with the graph Q − uw, where u = x and w = y, then χ ′ a(H) = 5.