A New Lower Bound on Critical Graphs with Maximum Degree of $8$ and $9$

Abstract

In this article, we give new lower bounds for the size of edge chromatic critical graphs with maximum degrees of $8$ and $9$, respectively. Furthermore, it implies that if $G$ is a graph embeddable in a surface $S$ with characteristics $c(S)=-1$ or $-2$, then $G$ is class one if maximum degree $\mathrm{\Delta }\ge 8$ or $9$, respectively.