Tight Chromatic Upper Bound for {3K1, K1+C4}-free Graphs

TL;DR

This paper establishes a tighter upper bound on the chromatic number for {3K1, K1+C4}-free graphs, improving previous bounds and demonstrating the bound's tightness with examples.

Contribution

It proves a new upper bound of 3ω/2 for the chromatic number of {3K1, K1+C4}-free graphs, refining earlier results.

Findings

New upper bound of 3ω/2 for chromatic number.

Examples demonstrating the bound's tightness.

Abstract

Problem of finding an optimal upper bound for the chromatic no. of 3K1-free graphs is still open and pretty hard. It was proved by Choudum et al that an upper bound on the chromatic no. of {3K1, K1+C4}-free graphs, is 2{\omega}. We improve this by proving that if G is {3K1, K1+C4}-free, then its chromatic no. is less than or equal to 3{\omega} divided by 2, where {\omega} is the size of a maximum clique in G. Also we give examples to show that this bound is tight.