Albertson's Conjecture Holds for r at Most 26

Abstract

Albertson conjectured that every graph with chromatic number r has crossing number at least cr(K_r). The conjecture was verified for r <= 12 by Albertson, Cranston and Fox, for r <= 16 by Bar&#39;at and T&#39;oth, for r <= 18 by Ackerman, and recently for r <= 24 by Cranston, who reduced the remaining cases r in {25, 26} to three orders. We settle those three orders, so that Albertson&#39;s Conjecture holds for all r <= 26. Only published results are used, and an appendix reproves the range 19 <= r <= 24 so that the case r <= 26 does not rest on unpublished work. We also show that if chi(G) = 27 and cr(G) < cr(K_27), then G has a 27-critical subgraph of order 53 or 54 whose complement is connected.