An upper bound on the number of edges of edge-coloring critical graphs with high maximum degree

In this paper, we prove that any edge-coloring critical graph G with maximum degree Δ⩾(11+49−24ε)/2, where ε⩽1, has the size at least 3(|V(G)|−ε)+1 if δ⩽7 or if δ⩾8 and |V(G)|>2Δ−δ−4−(Δ+6)/(δ−6), where δ is the minimum degree of G. It generalizes a result of Sanders and Zhao.