Validity of Borodin & Kostochka Conjecture for a Class of Graphs

TL;DR

This paper proves Borodin & Kostochka's conjecture for specific classes of graphs, namely {P3 UNION K1}-free and {K2 UNION complement of K2}-free graphs, expanding the conjecture's verified scope.

Contribution

The paper establishes the validity of Borodin & Kostochka's conjecture for two new classes of graphs, which was previously unverified.

Findings

Conjecture holds for {P3 UNION K1}-free graphs.

Conjecture holds for {K2 UNION complement of K2}-free graphs.

Expands the classes of graphs for which the conjecture is proven.

Abstract

Borodin & Kostochka conjectured that if maximum degree of a graph is greater than or equal to 9, then the chromatic number of the graph is less than or equal to maximum of {\omega} and maximum degree minus 1. Here we prove that this Conjecture is true for {P3 UNION K1}-free graphs and {K2 UNION complement of K2}-free graphs.