Coloring a graph with Δ-1 colors: Conjectures equivalent to the Borodin-Kostochka conjecture that appear weaker

Borodin and Kostochka conjectured that every graph G with maximum degree ∆ ≥ 9 satisfies χ ≤ max {ω,∆− 1}. We carry out an in-depth study of minimum counterexamples to the Borodin-Kostochka conjecture. Our main tool is the classification of graph joins A ∗B with |A| ≥ 2, |B| ≥ 2 which are f -choosable, where f(v) := d(v)− 1 for each vertex v. Since such a join cannot be an induced subgraph of a vertex critical graph with χ = ∆, we have a wealth of structural information about minimum counterexamples to the Borodin-Kostochka conjecture. Our main result proves that certain conjectures that are prima facie weaker than the Borodin-Kostochka conjecture are in fact equivalent to it. One such equivalent conjecture is the following: Any graph with χ ≥ ∆ = 9 contains K3 ∗E6 as a subgraph.