Tight Chromatic Upper Bound for {3K1, C5}-free Graphs

TL;DR

This paper establishes a precise upper bound on the chromatic number for graphs that are free of both 3K1 and C5 subgraphs, relating it to the size of their maximum clique.

Contribution

The paper introduces a tight upper bound for the chromatic number of {3K1, C5}-free graphs, linking it to the maximum clique size and proving its optimality.

Findings

The chromatic number of {3K1, C5}-free graphs is at most (3ω - 1)/2.

The bound is proven to be tight with specific examples.

Provides a new characterization for coloring such graphs.

Abstract

Problem of finding an optimal upper bound for a chromatic no. of 3K1-free graphs is still open and pretty hard. Here we find a tight chromatic upper bound for {3K1, C5}-free graphs. We prove that if G is {3K1, C5}-free, then the chromatic no. <= (3{\omega}-1)/2 where {\omega} is the size of a maximum clique in G and show with examples that the bound is tight.