# Equality cases for a bound on the chromatic number

## TL;DR

This paper characterizes the graphs that achieve equality in a known bound relating the chromatic number, vertices, and edges, showing they are composed of complete graphs or odd cycles with attached trees.

## Contribution

It provides a complete description of all graphs attaining equality in the bound, extending understanding of the structure of such graphs.

## Key findings

- Equality holds for graphs made of complete graphs or odd cycles with attached trees.
- The characterization includes all such graphs with these specific structures.
- The result clarifies the extremal cases for the chromatic number bound.

## Abstract

It is known that the inequality $$ \frac{\chi(G)(\chi(G)-1)}{2} + |V| - \chi(G) \leq |E|$$ holds for all connected graphs, where $\chi(G)$ denotes the chromatic number of $G$. We prove that equality holds whenever the graph consists of a complete graph or an odd cycle, together with finitely many trees attached to its vertices.