Graph Theory - Complete Bipartite Graphs

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Complete Bipartite Graphs

A complete bipartite graph is a special type of bipartite graph where every vertex in one set is connected to every vertex in the other set. These graphs are denoted as K_m,n, where m and n represent the number of vertices in each of the two sets.

The following image shows an example of a complete bipartite graph −

Properties of Complete Bipartite Graphs

Complete bipartite graphs have several important properties, which include −

• Disjoint Sets: The vertices are divided into two disjoint sets, and no edges exist within the same set. All edges are between vertices of different sets.
• Complete Connectivity: Every vertex in the first set is connected to every vertex in the second set. This results in m n edges.
• Planarity: Complete bipartite graphs K_m,n are planar if and only if m 2 or n 2.
• Chromatic Number: The chromatic number of a complete bipartite graph is 2, since the vertices can be colored with two colors, one for each set.

The structure of a complete bipartite graph ensures that vertices within the same set are not connected to each other, providing a clear separation between the two sets of vertices.

Classification of Complete Bipartite Graphs

Complete bipartite graphs can be classified based on the sizes of the two sets of vertices −

• K_1,3: This is a star graph with one central vertex connected to three outer vertices.
• K_2,2: This is a cycle of length 4, where each set has 2 vertices.
• K_3,3: This is a non-planar graph known as the utility graph, often used to illustrate the three utilities problem.
• K_2,3: This graph consists of 2 vertices in one set and 3 vertices in the other set, with each vertex from the first set connected to every vertex in the second set.

K_1,3 - Star Graph

A K_1,3 graph is a star graph with one central vertex connected to three outer vertices. This graph is a specific case of a complete bipartite graph where one set contains just a single vertex (the center) and the other set contains three vertices (the leaves).

The central vertex is connected to all the other three vertices, and there are no edges between the three outer vertices.

The following image shows a K_1,3 star graph −

K_2,2 - Cycle of Length 4

A K_2,2 graph is a type of graph that can be divided into two sets of vertices, with each set containing two vertices, and every vertex from one set is connected to every vertex in the other set. In this particular case, the graph K_2,2 forms a cycle of length 4.

The following image shows a K_2,2 cycle graph where each set contains 2 vertices, forming a square −

K_3,3 - Utility Graph

A K_3,3 graph is a complete bipartite graph with 3 vertices in each set. Every vertex in one set is connected to all vertices in the other set, while there are no edges between vertices within the same set. This graph is non-planar.

This graph is also known as a utility graph due to its application in the classic three utilities problem.

The following image shows a K_3,3 utility graph −

In this diagram:

• u1, u2, u3 are the vertices in one set.
• v1, v2, v3 are the vertices in the other set.
• Every vertex in the first set is connected to every vertex in the second set, forming 9 edges in total.

K_2,3 - Complete Bipartite Graph

A K_2,3 graph consists of 2 vertices in one set and 3 vertices in the other set. Each vertex from the first set is connected to every vertex in the second set.

The following image shows a K_2,3 complete bipartite graph −

Applications of Complete Bipartite Graphs

Complete bipartite graphs are used in various practical applications, including −

• Network Design: Complete bipartite graphs are useful for designing networks with two distinct sets of nodes, ensuring complete connectivity between the sets.
• Matching Problems: These graphs are used in matching problems where elements from one set need to be paired with elements from another set, such as job assignments.
• Scheduling: In scheduling, complete bipartite graphs can represent tasks and resources, ensuring each task is connected to every available resource.
• Database Relationships: They are used to model relationships between two sets of entities, such as customers and products in a database.

Testing Bipartiteness in Graphs

We can use various methods to test whether a graph is bipartite, such as −

• Two-Coloring: Attempting to color the graph with two colors. If successful, the graph is bipartite.
• Breadth-First Search (BFS): Using BFS to check if the graph can be divided into two sets with no edges within the same set.

Various Types of Complete Bipartite Graphs

There are several special types of complete bipartite graphs with unique properties, such as −

Graph Type	Sets	Notes
General Bipartite	m, n	Any complete bipartite graph Km,n with m and n vertices in each set, respectively.
Equal Sets Km,m Bipartite Graph	m, m	A bipartite graph where both sets contain an equal number of vertices, and every vertex in one set is connected to every vertex in the other set.
Wheel Graph	1, n-1	A wheel graph consists of a cycle with one central vertex connected to all vertices in the cycle, forming a "wheel" structure.
Toric Grid Graph	m, n	A graph formed by connecting vertices arranged in a grid on a torus, used in physics and geometry.
Bipartite Multigraph	m, n	A complete bipartite graph where multiple edges can exist between pairs of vertices in different sets.