In mathematical graph theory, a chordal graph is a closed-loop having 4 or more vertices such that an edge can be drawn from one vertex to another vertex is present in it. In other words, a chordal graph is a graph of length 4 or more containing at least one chord. A chord is nothing but an edge connecting two vertices of the graph and is drawn inside the graph.
Important Points to Remember:
- A chordal graph is a subset of the complete graph.
- A graph containing no chord is called a hole or chordless cycle.
- A complete graph is a chordal graph itself.
1) Consider the below graph ABCD c. An edge CB is drawn from vertex C to vertex B which acts as a chord in this graph. Hence the below graph is said to be a chordal graph.
Chordal Graph
2) Consider the below example of the complete graph. A graph consisting of n*(n-1)/2 edges is said to be a complete graph, where n is the number of vertices. In the complete graph, the edges connecting all vertices are already present and act as a chord also. Hence, a complete graph is a superset of a chordal graph.
Complete graph
Minimum-coloring Algorithm :
In this article, we will discuss coloring the chordal graphs by using the greedy coloring algorithm.
Greedy Coloring Algorithm:
- First, we need to find the perfect elimination ordering (PEO) of the given chordal graph.
- Then, traverse the vertices in the reverse order of that PEO that we have already found.
- Gives the smallest coloring to each vertex such that the current vertex color is not used in its neighboring vertices.
- If the color of the current vertex is already given to its neighbor then increment the color and assign it to the current vertex.
- Repeat steps 3 and 4 until all the vertices are colored.
Example: Consider the below chordal graph ABCD. Let the perfect elimination ordering of this graph be [D, C, B, A]. We will traverse the graph PEO in the reverse direction. Hence, [A, B, C, D] is the reverse of PEO.
1. First, we will color the vertex A with 1 as we need to color with the smallest present color.
2. Now for vertex B, we will check whether its neighbors contain color 1 or not. As vertex ‘A’ contains color 1, therefore we will increment the color by 1 to color vertex B.
3. Similarly, for vertex C, its neighbor A contains color 1, and B contains color 2 so we will color vertex C to 3.
4. Now, for vertex D, as its neighbors do not contain the color 1 so we will color 1 to vertex D.