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 $\chi =\mathrm{\Delta }=9$ contains $K_3\vee E_6$ 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 $K_3\vee E_6$.