A Sufficient Condition of the Four Color Theorem

Abstract

Grotzsch conjectured that if $G$ is planar, bridgeless with $\mathrm{\Delta }=3$ and $n_2\ge 2$, 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 $\mathrm{\Delta }=3$ and $n_2=1$. Then we prove that the conjecture implies the Four Color Theorem.