Erratum to “Acyclic Edge Chromatic Number of Outerplanar Graphs”

Authors

Errata

This article corrects:

  1. Acyclic edge chromatic number of outerplanar graphs Volume 64, Issue 1, 22–36, Article first published online: 1 July 2009

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 inline image 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 inline image. This is not always true, as inline image may be an edge of G.

The problem that determining the acyclic edge chromatic number of outerplanar graphs G with inline image 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 inline image and inline image, and inline image be an outer edge of G with inline image and inline image. If H is an outerplanar graph by replacing the edge inline image with the graph inline image, where inline image and inline image, then inline image.

Proof. Otherwise, let ϕ be an acyclic edge 4-coloring of H with inline image, inline image, inline image and inline image. By Lemma 4.2, inline image, inline image, inline image and inline image (See Fig. 1). Let inline image and inline image for the other edges of G. It is easy to see that inline image is an acyclic edge 4-coloring of G, a contradiction. inline image

Figure 1.

The coloring.

The rest of the paper is not affected by this error.

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