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 G is a graph embeddable in a surface S with characteristic c S = − 4 or − 5 or − 6 , then G is class one if its maximum degree Δ ≥ 10 or 11 or 12 respectively.