A Proof of Albertson's Conjecture

@misc{sadhu2026proof,
  title = {{A Proof of Albertson's Conjecture}},
  author = {Ankan Sadhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.01682},
  eprint = {2609.01682},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01682},
}

TY  - UNPB
TI  - A Proof of Albertson's Conjecture
AU  - Ankan Sadhu
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.01682
UR  - https://arxiv.org/abs/2609.01682
AB  - Albertson's conjecture states that every graph of chromatic number r has crossing number at least that of the complete graph K_r. We prove the conjecture for all r. The proof combines new clique-sensitive edge bounds for critical graphs, maximum-clique sampling, and a reserved immersion construction. For graphs of order close to r, we construct edge-disjoint routes while preserving a complete bipartite subgraph whose forced crossings compensate for those introduced when the immersion is smoothed into a drawing. For larger orders, we derive explicit edge bounds using independent-triple deletion and structural properties of critical graphs, then combine them with sampling inequalities that account for crossings within a maximum clique. The remaining numerical inequalities are certified using exact rational arithmetic, with a reproducible certificate and standalone Python verifier.
ER  -