Maximum Size of a Maximal Clique in a Bipartite Graph

In graph theory, a bipartite graph is a special type of graph where the set of vertices can be divided into two disjoint sets such that no two vertices within the same set are adjacent. This property sets bipartite graphs apart from other graph types, particularly in determining the size of a maximal clique. In this article, we delve into the specific characteristics that define the maximum size of a maximal clique in a bipartite graph.

Definition and Properties of Bipartite Graphs

First, it is crucial to understand the definition and properties of a bipartite graph. A bipartite graph G (V, E) is a graph where the set of vertices V can be divided into two disjoint sets, X and Y, such that every edge in the set of edges E connects a vertex in X to a vertex in Y. These sets are referred to as the parts of the graph.

This unique property has significant implications for the structure of the graph, particularly in the context of cliques. A clique is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. In other words, a clique is a complete subgraph. Since edges are only allowed to connect vertices between X and Y in a bipartite graph, the only possible maximal cliques consist of pairs of vertices, one from X and one from Y.

Maximal Clique in Bipartite Graphs

The key insight here is that any attempt to form a clique of size greater than two would violate the bipartite property. Let's consider a clique of size three pictorially as a "triangle". In such a triangle, at least two vertices must belong to the same part, but since edges connect vertices from different parts, this creates a contradiction. Therefore, a bipartite graph cannot contain a triangle, and consequently, cannot contain a clique of size three or larger.

This reasoning extends to any clique of size k geq 3. Any subset of three vertices in a clique of size k would form a triangle. Hence, the maximal size of any clique in a bipartite graph is at most two. Mathematically, this conclusion can be succinctly stated as follows:

The maximal size of a maximal clique in a bipartite graph is 2.

Summary of Key Concepts

1. **Bipartite Graph**: A graph in which the vertices can be partitioned into two disjoint sets such that no two vertices in the same set are adjacent.

2. **Clique**: A complete subgraph where every two distinct vertices are adjacent.

3. **Maximal Clique**: The largest possible clique within a graph.

4. **Bipartite Property**: Edges in a bipartite graph can only connect vertices from different parts of the graph.

These concepts together ensure that the maximal clique size in a bipartite graph is strictly limited to two vertices, as exemplified by the reasoning provided.