Chromatic index of graphs with no cycle with a unique chord

  • Authors:

  • R. C. S. Machado;C. M. H. de Figueiredo;K. Vukovi

  • Affiliations:

  • COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil and Instituto Nacional de Metrologia Normalização e Qualidade Industrial, Rio de Janeiro, RJ, Brazil;COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil;School of Computing, University of Leeds, Leeds LS2 9JT, United Kingdom

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2010

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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 Vuskovic (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 @D=3. For the subclass C^', we establish a dichotomy: if the maximum degree is @D=3, the edge-colouring problem is NP-complete, whereas, if @D3, 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 @D=3, for r=3.