Graph-theory open problems

Crossing numbers and coloring

High ★★★ Graph Theory » Topological Graph Theory » Crossing numbers
Author: Albertson, Michael O.— posted by mdevos on 2009-09-04

Status partial high confidence

Albertson's conjecture remains open in general, but substantial progress has been made since it was posted in 2009. Ackerman (2019) extended the verified range from $t \le 16$ to $t \le 18$; Cranston (2025) further pushed this to $t \le 24$ and greatly restricts possible counterexamples for $t \in \{25, 26\}$; Fox, Pach, and Suk (SoCG 2025) proved the conjecture for all graphs with chromatic number $k$ having at most $1.4(k-1)$ vertices (for sufficiently large $k$), via the Lescure–Meyniel weak immersion conjecture.

Cited literature (3)

  • partial On topological graphs with at most four crossings per edge (2019)
    Eyal Ackerman · Computational Geometry · arXiv:1509.01932

    As a corollary of results on topological graphs with at most four crossings per edge, proves Albertson's conjecture for $r \le 18$, extending the prior bound of $r \le 16$.

  • partial Immersions and Albertson's Conjecture (2025)
    Jacob Fox, János Pach, Andrew Suk · 41st International Symposium on Computational Geometry (SoCG 2025) · arXiv:2510.05893 · doi:10.4230/LIPIcs.SoCG.2025.50

    Proves Albertson's conjecture for all graphs with chromatic number $k$ having at most $1.4(k-1)$ vertices (for sufficiently large $k$), via a proof of the Lescure–Meyniel weak immersion conjecture in that range.

  • partial Progress on Albertson's Conjecture (2025)
    Daniel W. Cranston · arXiv preprint · arXiv:2512.08020

    Extends verification of Albertson's conjecture to $r \le 24$ (from $r \le 18$) and greatly restricts the possible counterexamples for $r \in \{25, 26\}$, also improving bounds on the order of a minimum counterexample.

Reviewer notes. The Ackerman paper (arXiv:1509.01932) was submitted in 2015 and revised/published in 2019; Wikipedia identifies the venue as Computational Geometry but I did not independently verify the journal DOI. The Fox–Pach–Suk arXiv preprint (2510.05893) appeared October 2025 after the SoCG 2025 conference; the published DROPS version cites the 1.4(k-1) vertex bound. The Cranston 2025 paper (arXiv:2512.08020) is a preprint as of December 2025; journal publication status unknown. The conjecture remains wide open for general graphs.

Auto-reviewed 2026-05-08 with claude-sonnet-4-6 (worker 06) (web search enabled).

Conjecture. Every graph $ G $ with $ \chi(G) \ge t $ satisfies $ cr(G) \ge cr(K_t) $ .
Keywords: coloring · complete graph · crossing number

Discussion

This conjecture is an interesting weakening of the disproved Hajos Conjecture which asserted that $ \chi(G) \ge t $ implies that $ G $ contains a subdivision of $ K_t $ . A minimal counterexample to Albertson's conjecture is critical, with minimum degree $ \ge t $ . Using this and the crossing lemma, Albertson, Cranston and Fox showed that a minimum counterexample has at most $ 4t $ vertices. They then analyzed small cases to show that the conjecture holds for $ t \le 12 $ . More recently, Barat and Toth [BT] sharpened these arguments to show that the conjecture holds for $ t \le 16 $ .

Bibliography