A Sufficient Condition of the Four Color Theorem

Grotzsch conjectured that if \(G\) is planar, bridgeless with \(\Delta = 3\) and \(n_2 \geq 2\), then \(G\) is of Class one. We prove that when \(n_2 = 2\) the conjecture is equivalent to the statement: \(G\) is \(3\)-critical if \(G\) is planar, bridgeless with \(\Delta = 3\) and \(n_2 = 1\). Then we prove that the conjecture implies the Four Color Theorem.