Chromatic Number of a graph in Graph Theory

The process of assigning colors to the vertices of a graph in such a way that no two adjacent vertices have the same color, while minimizing the total number of colors used.
Find the Chromatic number of the K3,3 and K2,3. Applications of chromatic number of a graph: Scheduling
Map coloring
Traffic light systems
Cellular networks
Resource allocation

1. Contents

2. Introduction to Graph Theory 1 Vertices and Edges Graphs are mathematical structures consisting of vertices (nodes) and edges (connections) between them. 2 Adjacency Two vertices are considered adjacent if they are connected by an edge. 3 Graph Representations Graphs can be represented visually or using adjacency matrices and lists.

3. Graph Coloring The process of assigning colors to the vertices of a graph in such a way that no two adjacent vertices have the same color, while minimizing the total number of colors used.

4. Chromatic Number of a Graph Formal Definition The minimal number of colours needed to colour the vertices in such a way that no two adjacent vertices have the same colour. Representation The Chromatic number of a graph is represented by X(G). Importance Understanding the chromatic number helps in solving various optimization problems related to graph coloring.

5. X(G)

6. Find the chromatic number of the K 3,3 and K 2,3 1 2 3 4 5 6 (i) K 3,3 X(G)=2

7. 1 2 3 4 5 (ii) K 2,3 X(G)=2

8. Applications of chromatic number of a graph Scheduling Map coloring Traffic light systems Cellular networks Resource allocation

9. Conclusion and Key Takeaways Chromatic Number The chromatic number is a fundamental concept in graph theory, representing the minimum number of colors needed to color the vertices of a graph. Bipartite Graphs Bipartite graphs have a chromatic number of 2, as their vertices can be divided into two disjoint sets with no edges within the same set. Applications Chromatic number and bipartite graphs have numerous applications in various fields, such as scheduling, resource allocation, and social network modeling.

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