The graph coloring represents the painting of the graph or we can say that the labeling of the graph with different types of colors. This technique is very much helpful from the Point of view of differentiating the vertexes from the edges. As in the graph coloring process the adjacent vertexes and the adjacent edges are colored with different colors, the coloring of vertices is called vertex coloring and the coloring of edges is called the edge coloring. Where as the faces of the graph those are sharing the same boundary are colored differently and this is called face coloring.
Graph Coloring AlgorithmBack to Top
Where ‘G’ denotes the graph,
‘V’ denotes the vertices of a graph,
And ‘E’ denotes the edge of a graph is known as bipartite graph. The vertices (V) in the graph are located into two Set X and Y such that each edges is incident to a vertex in X and ‘Y’ vertex, which means there is no two vertices in ‘X’ and ‘Y’ are adjacent. Bipartite graph does not have any odd length.
Now let’s switch towards the Graph Coloring Algorithm. We know that, the matching problems are related to the bipartite graphs. To find a maximal bipartite matching in a bipartite graph a simple form is define which is given as:
G = (v = (S, T), E),
Each path of this algorithm is created in O (E) time, and the running time is O (VE). This algorithm gives the complexity of O (V2.376). Let’s have a graph which is denoted by ‘G’ and G = (V, E), then a matching M is a Perfect Matching Bipartite Graph if it matches all the vertices of a graph. Every vertex in the graph is matched with one of the edges of a graph. If it matches with more than one edge then the graph is not perfect matching graph or it s an unmatched graph. Each edge in a weighted graph has an associative value.
In the graph a vertex is matched if endpoint of one of the edges in the graph is present in the matching. If it not so then the graph vertex is said to be unmatched. A matching M in the graph G is said to be maximal matching if any edge in the graph is not present in M is added to M. In a matching if large Numbers of edges are present then it is known as maximal matching. These all are the Graph Coloring Algorithms through which we can find any of the graphs easily.
Graph Coloring ApplicationsBack to Top
1. Vertex coloring :- When we are coloring vertices we have to remember one thing that colour of vertices which share same edge should be different .
2. Edge coloring :- When we are coloring edges then we have to remember that color of two edges which share same vertex must be different.
Graph can be colored in number of ways, so chromatic polynomial number counts all possible ways by which we can color the graph.
Suppose we have a graph :
We have four colors to fill this graph, so according to chromatic polynomial.
We can say that there are 12 possible ways to color this graph. We need minimum of 3 colors to fill this graph. If we are using four colors then there are 4! = 24 possible ways of coloring the graph. We can’t fill this with 2 colors.
This is one of the most important applications of graph coloring.