Day 07Part 05- Discrete mathematics Graph Theory-Concept of Complete bipartite graph

Day 07Part 05- Discrete mathematics Graph Theory-Concept of Complete bipartite graph

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Complete Bipartite Graph in Discrete Mathematics (Graph Theory – Day 07, Part 05)

 What is a Bipartite Graph?

A bipartite graph is a type of graph G = (V, E) where:

• The set of vertices V is divided into two disjoint sets (U and V).
• Every edge (u, v) ∈ E connects a vertex from U to V.
• There are no edges within the same set (i.e., no edges connecting vertices within U or within V).

 What is a Complete Bipartite Graph?

A complete bipartite graph, denoted as Kₘ,ₙ, is a special type of bipartite graph where:
 Every vertex in set U (m vertices) is connected to every vertex in set V (n vertices).
 There are m × n edges (each vertex in U connects to every vertex in V).
 It is maximally connected for a bipartite structure.

 Example of a Complete Bipartite Graph (K₃,₄)

Consider K₃,₄, where:

• Set U has 3 vertices → {A, B, C}
• Set V has 4 vertices → {1, 2, 3, 4}
• Every vertex in U is connected to every vertex in V.

Graph Representation:

Total edges = 3 × 4 = 12

 Properties of a Complete Bipartite Graph

No Odd-Length Cycles – A bipartite graph cannot have odd-length cycles.
Chromatic Number (χ) = 2 – Since it’s bipartite, it only requires two colors for proper coloring.
Degree of Each Vertex

• In Kₘ,ₙ, each vertex in U has a degree of n.
• Each vertex in V has a degree of m.
  Planarity
• K₂,₃ is planar, but K₃,₃ and beyond are non-planar (by Kuratowski’s Theorem).

 Real-World Applications of Complete Bipartite Graphs

Job Assignment Problem – Connecting workers (U) to tasks (V).
Network Design – Connecting data centers and servers efficiently.
Social Network Analysis – Representing relationships between two distinct groups (e.g., authors and research papers).
Computer Science & AI – Used in matching algorithms and recommendation systems.

 Conclusion

A complete bipartite graph Kₘ,ₙ is a fundamental structure in graph theory, representing maximum connections between two independent sets. Its applications in real-world problems make it an essential concept in discrete mathematics, networking, and optimization.

Would you like an example problem or visualization for better understanding?

Day 07Part 05- Discrete mathematics Graph Theory-Concept of Complete bipartite graph

GRAPH THEORY WITH APPLICATIONS

Bipartite Graphs Discrete Mathematics

Lecture 17: Graph Theory Bipartite Graphs