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 \(\Delta \geq 8\) or \(9\), respectively.