Applications of the Baudet–Schur–Van der Waerden

At the end of Chap. 14 , I left you with the embedding in the plane of the 352,735-vertex Blanche Descartes graph by Paul O’Donnell. One may ask, would attaching longer k-cycles (k > 7) to the foundation vertices increase the graph’s girth while keeping the chromatic number at 4? The answer is no – not if k-cycles were attached to all k-element subsets of the foundation set – because some k-cycles would have two or more vertex intersection that could cut down the girth of the graph. We would get a chance to succeed at this construction if we were to dramatically limit the number of attached k-cycles, by, say, allowing at most a single point intersection for the k-subsets of the foundation, to which k-cycles are allowed to be attached. This is exactly what O’Donnell implemented.