The Overfull Conjecture on split-comparability and split-interval graphs

Abstract

An edge-coloring of a graph G is an assignment of colors to the edges of G so that adjacent edges have distinct colors. The chromatic index χ′(G) is the smallest number of colors needed in an edge-coloring of G. Clearly, χ′(G)≥Δ(G) and the celebrated Vizing’s Theorem (1964) states that χ′(G)≤Δ(G)+1, for any simple graph G. A simple graph G is Class 1 if χ′(G)=Δ(G), and Class 2 otherwise. The Classification Problem is to determine whether a simple graph is Class 1. The Overfull Conjecture, proposed by Chetwynd and Hilton (1984), states that when Δ(G)>|V(G)|/3, the graph G is Class 1 if and only if it is not subgraph-overfull. Figueiredo et al. (2000) conjectured that any chordal graph is Class 1 if and only if it is not subgraph-overfull. Chordal graphs are a superclass of split and of interval graphs. In this paper we prove that both the Overfull and Figueiredo et al.’s Conjectures hold for all split-comparability and split-interval graphs. Our proofs lead to polynomial-time algorithms to solve the Classification Problem in these classes.