The Chromatic Index of Proper Circular-arc Graphs of Odd Maximum Degree which are Chordal

Abstract

The complexity of the edge-coloring problem when restricted to chordal graphs, listed in the famous D. Johnson's NP-completeness column of 1985, is still undetermined. A conjecture of Figueiredo, Meidanis, and Mello, open since the late 1990s, states that all chordal graphs of odd maximum degree Δ have chromatic index equal to Δ. This conjecture has already been proved for proper interval graphs (a subclass of proper circular-arc ∩ chordal graphs) of odd Δ by a technique called pullback. Using a new technique called multi-pullback, we show that this conjecture holds for all proper circular-arc ∩ chordal graphs of odd Δ. We also believe that this technique can be used for further results on edge-coloring other graph classes.