Tight Chromatic Upper Bound for {3 Times K1, {2 Times K1 + (K2 UNION K1)}-free Graphs

TL;DR

This paper improves the upper bound on the chromatic number for certain 3K1-free graphs, showing it is at most 1.5 times the clique number for most cases, with specific bounds for the case when the clique size is five.

Contribution

It establishes a tighter upper bound on the chromatic number for (3K1, (2K1 + (K2 ∪ K1)))-free graphs, advancing previous results.

Findings

Upper bound of 3ω/2 for chromatic number when ω ≠ 5

Chromatic number ≤ 8 when ω = 5

Examples of extremal graphs provided

Abstract

Problem of finding an optimal upper bound for {\chi} of (3 Times K1)-free graphs is still open and pretty hard. It was proved by Choudum et al that upper bound on the {\chi} of {3 Times K1, {2 Times K1 + (K2 UNION K1)}-free graphs is 2{\omega}. We improve this by proving that if G is {3 Times K1, {2 Times K1 + (K2 UNION K1)}-free, then {\chi} less than or equal to 3{\omega} divided by 2 for {\omega} not equal to 5, and {\chi} less than or equal to 8 for {\omega} = 5 where {\omega} is the size of a maximum clique in G. We also give examples of extremal graphs.