Open Access

On the classification problem for split graphs

  • Sheila Morais de Almeida1, 2Email author,
  • Célia Picinin de Mello1 and
  • Aurora Morgana3
Journal of the Brazilian Computer Society201118:46

https://doi.org/10.1007/s13173-011-0046-2

Received: 5 October 2011

Accepted: 20 October 2011

Published: 9 November 2011

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.

Keywords

Edge-coloring Overfull graph Split graph Classification problem