On the classification problem for split graphs

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.