# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1702.08151