On Finding Large Cliques when the Chromatic Number is close to the Maximum Degree
Abstract
We prove that every graph G with chromatic number χ(G) = ∆(G) − 1 and ∆(G) ≥ 66 contains a clique of size ∆(G) − 17. Our proof closely parallels a proof from Cranston and Rabern, who showed that graphs with χ = ∆ and ∆ ≥ 13 contain a clique of size ∆ − 3. Their result is the best currently known for general ∆ towards the Borodin-Kostochka conjecture, which posits that graphs with χ = ∆ and ∆ ≥ 9 contain a clique of size ∆. We also outline some related progress which has been made towards the conjecture.
Collections
Cite this version of the work
Colter MacDonald
(2021).
On Finding Large Cliques when the Chromatic Number is close to the Maximum Degree. UWSpace.
http://hdl.handle.net/10012/17820
Other formats