Average Degrees of Critical Graphs

Abstract

Let \(G\) be a simple graph with the average degree \(d_{ave}\) and the maximum degree \(\Delta\). It is proved, in this paper, that \(G\) is not critical if \(d_{ave} \leq \frac{103}{12}\) and \(\Delta \geq 12\). It also improves the current result by L.Y. Miao and J.L. Wu [7] on the number of edges of critical graphs for \(\Delta \geq 12\).