Towards The Albertson Conjecture

TL;DR

This paper extends the proof of Albertson's conjecture, which relates the chromatic number of a graph to its crossing number, up to chromatic number 16, building on previous results for smaller values.

Contribution

The paper proves Albertson's conjecture for graphs with chromatic number up to 16, advancing the known cases from 12 to 16.

Findings

Confirmed the conjecture for r ≤ 16

Extended previous proofs from r ≤ 12

Strengthened the relationship between chromatic number and crossing number

Abstract

Albertson conjectured that if a graph $G$ has chromatic number $r$ then its crossing number is at least as much as the crossing number of $K_r$. Albertson, Cranston, and Fox verified the conjecture for $r\le 12$. We prove the statement for $r\le 16$.