A density bound for triangle‐free 4‐critical graphs

We prove that every triangle-free 4-critical graph G $G$ satisfies e ( ) ≥ 5 v + 2 3 $e(G)\ge \frac{5v(G)+2}{3}$ . This result gives a unified proof planar graphs are 3-colourable, as well of girth at least five which embed in either the projective plane, torus or Klein Bottle results Grötzsch, Thomassen, and Thomas Walls. Our is nearly best possible, Davies has constructed such = 4 $e(G)=\frac{5v(G)+4}{3}$ To this result, we more general characterizing sparse with few vertex-disjoint triangles.