On graphs with chromatic number and maximum degree both equal to nine

An equivalent version of the Borodin-Kostochka Conjecture, due to Cranston and Rabern, says that any graph with χ = Δ = 9 contains K₃ E₆ as a subgraph. Here we prove several results in support of this conjecture, where vertex-criticality and forbidden substructure conditions get us either close or all the way to containing K₃ E₆.