The average degree of an edge-chromatic critical graph

A graph G   with maximum degree ΔΔ and edge chromatic number χ′(G)>Δχ′(G)>Δ is edge  -ΔΔ-critical   if χ′(G-e)=Δχ′(G-e)=Δ for every edge e of G  . New lower bounds are given for the average degree of an edge-ΔΔ-critical graph, which improve on the best bounds previously known for most values of ΔΔ. Examples of edge-ΔΔ-critical graphs are also given. In almost all cases, there remains a large gap between the best lower bound known and the smallest average degree of any known edge-ΔΔ-critical graph.