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Chromatic index of graphs with no cycle with a unique chord

Machado, RCS, de Figueiredo, CMH and Vuskovic, K (2010) Chromatic index of graphs with no cycle with a unique chord. Theoretical Computer Science, 411 (7-9). 1221 - 1234 . ISSN 0304-3975

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Abstract

The class C of graphs that do not contain a cycle with a unique chord was recently studied by Trotignon and Vušković (in press) [23], who proved for these graphs strong structure results which led to solving the recognition and vertex-colouring problems in polynomial time. In the present paper, we investigate how these structure results can be applied to solve the edge-colouring problem in the class. We give computational complexity results for the edge-colouring problem restricted to C and to the subclass C′ composed of the graphs of C that do not have a 4-hole. We show that it is NP-complete to determine whether the chromatic index of a graph is equal to its maximum degree when the input is restricted to regular graphs of C with fixed degree Δ≥3. For the subclass C′, we establish a dichotomy: if the maximum degree is Δ=3, the edge-colouring problem is NP-complete, whereas, if Δ≠3, the only graphs for which the chromatic index exceeds the maximum degree are the odd holes and the odd order complete graphs, a characterization that solves edge-colouring problem in polynomial time. We determine two subclasses of graphs in C′ of maximum degree 3 for which edge-colouring is polynomial. Finally, we remark that a consequence of one of our proofs is that edge-colouring in NP-complete for r-regular tripartite graphs of degree Δ≥3, for r≥3.

Item Type: Article
Copyright, Publisher and Additional Information: ©2010, Elsevier. This is an author produced version of a paper published in Theoretical Computer Science. Uploaded in accordance with the publisher's self-archiving policy.
Keywords: Cycle with a unique chord, decomposition, recognition, Petersen graph, Heawood graph, edge-colouring, NP-Completeness, indifference graphs, regular graphs, edge
Academic Units: The University of Leeds > Faculty of Engineering (Leeds) > School of Computing (Leeds)
Depositing User: Symplectic Publications
Date Deposited: 22 Jun 2012 10:31
Last Modified: 08 Feb 2013 17:39
Published Version: http://dx.doi.org/10.1016/j.tcs.2009.12.018
Status: Published
Publisher: Elsevier
Identification Number: 10.1016/j.tcs.2009.12.018
URI: http://eprints.whiterose.ac.uk/id/eprint/74348

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