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

TL;DR

This paper investigates the Borodin-Kostochka conjecture on graph coloring, analyzing minimal counterexamples and establishing equivalences with weaker conjectures through structural graph properties.

Contribution

It identifies structural conditions and graph joins that characterize minimal counterexamples, proving certain weaker conjectures are actually equivalent to the Borodin-Kostochka conjecture.

Findings

Certain weaker conjectures are equivalent to the Borodin-Kostochka conjecture.

Structural properties of minimal counterexamples are characterized.

Identified specific subgraph conditions related to chromatic number and maximum degree.

Abstract

Borodin and Kostochka conjectured that every graph $G$ with maximum degree $\Delta \ge 9$ satisfies $\chi \le \max\{\omega, \Delta-1\}$. We carry out an in-depth study of minimum counterexamples to the Borodin-Kostochka conjecture. Our main tool is the identification of graph joins that 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 $\chi = \Delta$, 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 $\chi \ge \Delta = 9$ contains $K_3 * \bar{K_6}$ as a subgraph.