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Erratum to “Acyclic Edge Chromatic Number of Outerplanar Graphs”
Author(s) -
Hou JianFeng,
Wu JianLiang
Publication year - 2013
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.21702
Subject(s) - mathematics , combinatorics , outerplanar graph , discrete mathematics , counterexample , lemma (botany) , 1 planar graph , graph , pathwidth , chordal graph , line graph , ecology , poaceae , biology
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.