Description
In 2007, Mike Albertson made the following conjecture: If the chromatic number of a graph is at least k, then its crossing number is at least as large as the crossing number of the complete graph of k vertices. Albertson, Cranston, and Fox
verified this conjecture for graphs with at least 3k vertices, while Barat and G. Toth proved that it also holds for graphs with at most k+2 vertices. We show that Albertson's conjecture is true for all graphs with at most 1.5k vertices. Joint work with Jacob Fox and Andrew Suk.