Improvement on Brook theorem for (3 Times K1)-free Graphs

TL;DR

This paper improves bounds on the chromatic number of (3 Times K1)-free graphs, confirming the Borodin & Kostochka Conjecture for this class by establishing new upper bounds based on maximum degree and clique number.

Contribution

It proves new upper bounds for the chromatic number of (3 Times K1)-free graphs, advancing understanding of their coloring properties and confirming the Borodin & Kostochka Conjecture in this case.

Findings

For maximum degree ≥ 8, χ ≤ max(Δ-1, ω).

If ω=4 and Δ ≥ 7, then χ ≤ Δ-1.

Borodin & Kostochka Conjecture holds for (3 Times K1)-free graphs.

Abstract

Problem of finding an optimal upper bound for the chromatic no. of a (3 Times K1)-free graph is still open and pretty hard. Here we prove that for a (3 Times K1)-free graph G with maximum degree greater than or equal to 8, {\chi} is less than or equal to max (maximum degree-1, {\omega}). We also prove that if G is (3 Times K1)-free, {\omega} is equal to 4 and maximum degree is greater than or equal to 7, then {\chi} is less than or equal to maximum degree-1. This implies that Borodin & Kostochka Conjecture is true for (3 Times K1)-free graphs as a corollary.