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Volume 18 Supplement 2

GraphCliques

On the classification problem for split graphs

Abstract

The Classification Problem is the problem of deciding whether a simple graph has chromatic index equal to Δ or Δ+1. In the first case, the graphs are called Class 1, otherwise, they are Class 2. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. Split graphs are a subclass of chordal graphs. Figueiredo at al. (J. Combin. Math. Combin. Comput. 32:79–91, 2000) state that a chordal graph is Class 2 if and only if it is neighborhood-overfull. In this paper, we give a characterization of neighborhood-overfull split graphs and we show that the above conjecture is true for some split graphs.

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Correspondence to Sheila Morais de Almeida.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Morais de Almeida, S., Picinin de Mello, C. & Morgana, A. On the classification problem for split graphs. J Braz Comput Soc 18, 95–101 (2012). https://doi.org/10.1007/s13173-011-0046-2

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Keywords

  • Edge-coloring
  • Overfull graph
  • Split graph
  • Classification problem