A Proof of Albertson's Conjecture

Abstract

Albertson conjectured that every graph of chromatic number r has crossing number at least that of K_r. We prove the conjecture for every r. After the known case r <= 18, an r-critical counterexample is reduced to two order ranges. Near r, we use Gallai's decomposition, completion, and a reserved weak-immersion routing argument. In the remaining middle range, we compress repeated independent-triple reductions into an exact terminal edge bound and combine it with sampled crossing-number inequalities. The remaining finite and interval inequalities are verified by exact certificates.

24 Pages. This version extends and supersedes the previous v1 (r<=26 result). Exact verification certificates are archived on GitHub and Zenodo