The strong perfect graph theorem

In 1960 Berge came up with the concept of perfect graphs, and in doing so, conjectured some characteristics about them. A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph [2]. Two conjectures are now known as the Perfect Graph Theorem and the Strong Perfect Graph Theorem. Both of these theorems make detemining if a graph is perfect much easier than using the standard definition. Simply looking at any graph which has not been colored or arranged in any way tells very little about the relationship of the vertices. For example, look at the graph below, we say to vertecies are adjacent if they have an edge between them, which vertex is adjacent to the most vertecies? how many vertecies could you take so that all the vertecies you select are adjacent to each other? If you wanted to color each vertex in the graph so that it was not adjacent to a vertex of the same color, how many colors would you need? Let’s start with the first question, which vertex has the most neighbors, vertecies adjacent to it? You might have guessed G, and you would be right, but a guess is not enough. So, count the vertecies it is adjacent to, and see it has six neighbors, which means it is connected to every vertex on the graph. Therefore, it is surly has the most neighbors, but we have to count the neighbors of the other vertecies to see that it is the only one with six neighbors. So, now that we know which vertex is adjacent to the most vertecies, we can use this to help answer our other questions. We want to find the maximum number of vertecies we could take so that all the vertecies selected are adjacent to one another. In our example we know