Tight Chromatic Number for a class of Graphs with two forbidden subgraphs

TL;DR

This paper establishes that for certain graphs with two forbidden subgraphs and bounded maximum degree, the chromatic number equals the size of the largest clique, providing a precise bound under these conditions.

Contribution

It proves a specific chromatic number bound for graphs with two forbidden subgraphs and maximum degree constraints, including necessary conditions and illustrative examples.

Findings

Chromatic number equals maximum clique size under given conditions

Maximum degree bound of 2ω - 3 is necessary

Provides examples demonstrating the necessity of the conditions

Abstract

Although the chromatic number of a graph is not known in general, attempts have been made to find good bounds for the number. Here we prove that for a graph G with two forbidden subgraphs and maximum degree less than or equal to 2{\omega} - 3, {\chi} equals its maximum clique size. We also give examples to show that the condition is necessary.