Edge coloring of graphs with small maximum degrees

By applying a discharging method, we give new lower bounds for the sizes of edge chromatic critical graphs for small maximum degrees. Furthermore, it is also proved that if GG is a graph embeddable in a surface SS with characteristic cS=−4cS=−4 or −5−5 or −6−6, then GG is class one if its maximum degree Δ≥10Δ≥10 or 1111 or 1212 respectively.