On the Tight Chromatic Bounds for a Class of Graphs without Three Induced Subgraphs

TL;DR

This paper proves that graphs excluding certain three induced subgraphs have chromatic numbers at most one more than their maximum clique size, establishing tight bounds and demonstrating the necessity of each forbidden subgraph.

Contribution

The authors establish a tight upper bound on the chromatic number for graphs excluding three specific induced subgraphs, with examples confirming the bounds are optimal.

Findings

Chromatic number is at most maximum clique size plus one for these graphs.

Bounds are tight, with examples showing necessity of each forbidden subgraph.

Provides a characterization of graph classes with bounded chromatic number based on forbidden subgraphs.

Abstract

Here we prove that a graph without some three induced subgraphs has chromatic number at the most equal to its maximum clique size plus one. Further we show that the bounds are tight and give examples to show that each of the three forbidden subgraphs is necessary in the hypothesis.