Further split graphs known to be Class 1 and a characterization of subgraph-overfull split graphs

Abstract

The chromatic index, χ′(G), is the smallest integer k for which a graph G has a proper edge coloring. The Classification Problem involves determining whether a graph G is Class 1 (χ′(G)=Δ(G)) or Class 2 (χ′(G)=Δ(G)+1). It is known that subgraph-overfull graphs must be Class 2. In this paper we are concerned with the Classification Problem for split graphs G[Q∪S] where Q is a clique and S is an independent set. When Δ(G) is odd, G is known to be Class 1. The original proof presented by Chen et al. (1995) has a minor flaw which we detail in this paper while also clarifying that it does not compromise their result. We prove that their technique can be adapted in a non-trivial way to show that some split graphs with even Δ(G) are also Class 1. We show that to solve the Classification Problem for split graphs it suffices to consider that all vertices in Q have degree Δ(G). Considering the subset X of S of the vertices of degree at most Δ(G)/2, we show that if the neighborhood of X has at least ⌊|Q|/2⌋ vertices, then G is Class 1; in the remaining cases we characterize the subgraph-overfull split graphs.