# Lecture 9 – Chromatic Number vs. Clique Number & Girth

You may want to download the lecture slides that were used for these videos (PDF).

## 1. Chromatic Number, Maximum Clique Size, & Why the Inequality is not Tight

We recall the definitions of chromatic number and maximum clique size that we introduced in previous lectures. Then, we state the theorem that there exists a graph G with maximum clique size 2 and chromatic number t for t arbitrarily large.  (4:46)

## 2. A Construction Using the Pigeonhole Principle

This video provides a complete proof that triangle-free graphs with arbitrarily large chromatic number exist, by using the Pigeonhole principle. (16:54)

## 3. The Mycielski Construction

This video provides a complete proof that uses something called the Mycielski Construction. (17:04)

## 4. Shift Graphs

This video introduces shift graphs, and introduces a theorem that we will later prove: the chromatic number of a shift graph is the least positive integer t so that 2t ≥ n.  The video also discusses why shift graphs are triangle-free.  (3:44)

## 5. Proof that the Chromatic Number is at Least t

We want to show that the chromatic number of a shift graph is at least t for some t with 2t ≥ n. (10:03)

## 6. Coloring Our Graph with t Colors

We want to show that if t satisfies our inequality, then we can color our graph with t colors. (9:25)

## 7. Blanche Descartes

Before moving on, a brief historical anecdote. (0:58)

## 8. The Girth of a Graph

This video introduces the concept of the girth of a graph, along with forests. (2:39)

## 9. Chromatic Number & Girth

For every pair (g,t) of positive integers with g,t ≥ 3 there is a graph G with girth g and chromatic number t.  Erdős proved this using the probabilistic method.  (10:55)