Abstract We study reduced density matrices of the integrable critical restricted solid-on-solid (RSOS) model in a particular topological sector containing ground state. Similar as spin- 1 / 2 Heisenberg it has been obse...

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...