The Chromatic Number of a Graph

The chromatic number of a graph is the minimum number of colors needed to color the vertices of the graph in such a way that no two adjacent vertices have the same color. In other words, it is the minimum number of colors needed to color the graph such that no two adjacent vertices have the same color.

Graphs Without Odd Length Cycles

An odd length cycle in a graph is a cycle (a path that starts and ends at the same vertex) with an odd number of edges. A graph that does not contain any odd length cycle is called bipartite. A bipartite graph is a graph whose vertices can be divided into two disjoint sets such that no two vertices within the same set are adjacent.

Chromatic Number of a Graph Without Odd Length Cycles

For a graph without any odd length cycle, it is always possible to color the vertices using only two colors. This is because we can divide the vertices into two sets, such that no two vertices within the same set are adjacent. Therefore, the chromatic number of the graph is 2.

Explanation

To understand why the chromatic number of a graph without any odd length cycle is 2, let's consider the following steps:

1. Assume that the graph has n vertices.
2. Since the graph is simple and connected, it must be a tree or a forest (a collection of disjoint trees).
3. If the graph is a tree, it is trivially bipartite, as we can color the vertices alternatively using two colors.
4. If the graph is a forest, we can still color the vertices using two colors. We can start with any vertex and assign it color 1. Then, for each adjacent vertex, assign it the opposite color. Continue this process until all vertices are colored.
5. Since the graph does not contain any odd length cycle, there will be no adjacent vertices with the same color, as the colors are assigned alternately.
6. Therefore, we have successfully colored the graph using only two colors.
7. Hence, the chromatic number of the graph is 2.

Conclusion

The chromatic number of an n-vertex simple connected graph which does not contain any odd length cycle is 2. This means that the graph can be colored using only two colors such that no two adjacent vertices have the same color.