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

TL;DR

This paper investigates graphs with chromatic number and maximum degree nine, providing partial results supporting a conjecture that such graphs contain a specific subgraph, under certain structural conditions.

Contribution

It offers new partial proofs and conditions that bring us closer to confirming the Borodin-Kostochka Conjecture for graphs with  =  = 9.

Findings

Graphs with  =  = 9 contain the subgraph K_3  E_6 under certain conditions.

Vertex-criticality and forbidden substructures help identify subgraph containment.

Results support the conjecture in specific graph classes.

Abstract

An equivalent version of the Borodin-Kostochka Conjecture, due to Cranston and Rabern, says that any graph with $\chi = \Delta = 9$ contains $K_3 \lor E_6$ 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 $K_3 \lor E_6$.