A \(\Delta\)-subgraph condition for a graph to be class 1 (Q1087551)

A \(\Delta\)-subgraph \(G_{\Delta}\) of a simple graph G is the subgraph of G induced by the vertices of maximum degree \(\Delta =\Delta (G)\). We prove a number of results to the effect that, if \(\Delta\) (G) is large and the minimum degree of \(G_{\Delta}\) is at most one, then G satisfies \(\chi '(G)=\Delta (G)\), where \(\chi\) '(G) is the edge-chromatic number of G.