# Graph Coloring With Example

Hence the chromatic number of the graph is 3. Following is an example of a graph that can be coloured with 3 different colours.

### V – S such that c u c v whenever there exists an edge uv in G.

**Graph coloring with example**. Mathmatically a proper vertex coloring of an undirected graph G VE is a map c. However non-vertex coloring problems are often stated and studied as is. Look at example and say that since it has a 6-coloring therefore it is.

4-color theorem finally proved in 1977 Appel HakenFirst major computer-based proof Optimal 4 coloring example. If any of the permutations is valid for the given graph and colors we output the result otherwise not. The other graph coloring problems like Edge Coloring No vertex is incident to two edges of same color and Face Coloring Geographical Map Coloring can be transformed into vertex coloring.

The least possible value of m required to color the graph successfully is known as the chromatic number of the given graph. Let G G G have a graph coloring with colors 1 χ G 1 dots chiG 1 χ G and G G G have a graph coloring with colors χ G 1 χ G χ G chiG 1 dots chiG chiG χ G 1 χ G χ G. In this approach using the brute force method we find all permutations of color combinations that can color the graph.

A k-coloring of G is an assignment of k colors to the vertices of G in such a way that adjacent vertices are assigned different colors. N1 n2 n2 Consider this example with K 4. Some applications of graph coloring.

In the above figure at first vertex a is colored red. For example an edge coloring of a graph is just a vertex coloring of its line graph and a face coloring of a plane graph is just a vertex coloring of its dual. Coloring of a graph is an assignment of colors either to the edges of the graph G or to vertices or to maps in such a way that adjacent edgesverticesmaps are colored differently.

Scheduling via Graph Coloring. Here the colors would be schedule times such as 8MWF 9MWF 11TTh etc. Applications of Graph Coloring.

Applications of Graph Coloring. Graph Coloring Solution Using Naive Algorithm. If G has a k-coloring then G is said to be k-coloring then G is said to be k-colorableThe chromatic number of G denoted by XG is the smallest number k for which is k-colorable.

For example an edge coloring of a graph is just a vertex coloring of its line graph and a face coloring of a plane graph is just a vertex coloring of its dual. The chromatic number of Kn is. As the adjacent vertices of vertex a are again adjacent vertex b and vertex d are colored with different color green and blue respectively.

For example consider the following graph. Graph Coloring Applications- Some important applications of graph coloring are as follows-Map Coloring. Look at previous example.

1007 3137 3157 3203 3261 4115 4118 4156 Suppose also that there are no students in common taking the following pairs of. Then vertex c is colored as red as no adjacent vertex of c is colored red. This graph is coloured as follows.

Hence each vertex requires a new color. In the complete graph each vertex is adjacent to remaining n 1 vertices. Simply state that the since we found the chromatic number to be N the graph is N-chromatic.

Graph Coloring Example- The following graph is an example of a properly colored graph- In this graph No two adjacent vertices are colored with the same color. Francis Guthrie 1852 The four color map theorem states that given any separation of a plane into contiguous regions producing a figure called a map no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. It has many failed proofs.

Four color theorem. Generate all possible configurations of colors. We strongly recommend that you click here and practice it before moving on to the solution.

For example the following can be colored minimum 2 colors. Such a graph is called as a Properly colored graph. Please note that we cant color the above graph using two colors.

Since each node can be coloured using any of the m available colours the total number of colour configurations possible are mV. Final Exam Example Suppose want to schedule some inal exams for CS courses with following course numbers. Hence we could color the graph by 3 colors.

A coloring using at most k colors is called a proper kcoloring and a graph that can be assigned a proper kcoloring is kcolorable. Hence the chromatic number of K n n. Example 582 If the vertices of a graph represent academic classes and two vertices are adjacent if the corresponding classes have people in common then a coloring of the vertices can be used to schedule class meetings.

We can color it in many ways by using the minimum of 3 colors. The elements of set S are called the available colors. Before discussing the greedy algorithm to color graphs lets talk about basic graph coloring terminology.

Let G be a graph with no loops. However non-vertex coloring problems are often stated and studied as is. Look at previous example.

The smallest number of colors needed to color a graph G is called its chromatic number. Therefore it is a properly colored graph.

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