A Delta-subgraph condition for a graph to be class 1

The Δ-subgraph GΔ of a simple graph G is the subgraph of G induced by the vertices of maximum degree Δ = Δ(G). We prove a number of results to the effect that if Δ(G) is large and the minimum degree of GΔ is at most one, then G satisfies χ′(G) = Δ(G), where χ′(G) is the edge-chromatic number of G.