Abstract :
Due to Grötzsch and Grünbaum, the following configurations are known to be 3-color reducible: (a) a quadrangle, (b) a pentagon incident with four 3-vertices, and (c) a triangle incident with a 3-vertex. Planar graphs are constructed with arbitrarily large minimal distance d between triangles, having no (b) or (c), and such that ‘clenching’ any quadrangle decreases d.
