Validity of Borodin and Kostochka Conjecture for {4 Times K1}-free Graphs

TL;DR

This paper proves the Borodin and Kostochka Conjecture for 4K1-free graphs, establishing an upper bound on the chromatic number based on maximum degree and clique number for graphs with degree at least 9.

Contribution

It extends the validity of the Borodin and Kostochka Conjecture to 4K1-free graphs, a previously unresolved class of graphs.

Findings

Proved the conjecture for 4K1-free graphs with maximum degree ≥ 9.

Established that the chromatic number is bounded by the maximum of clique number and degree minus one.

Confirmed the conjecture's applicability to a new class of graphs.

Abstract

Problem of finding an optimal upper bound for the chromatic no. of even 3K1-free graphs is still open and pretty hard. Here we prove Borodin & Kostochka Conjecture for 4K1-free graphs G i.e. If maximum degree of a {4 Times K1}-free 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 {\delta-1}.