Finding the matching number of a graph

A matching in a graph is a set of edges where no two edges share a common vertex. The matching number of a graph is the maximum number of edges in any matching − in other words, the largest set of edges you can select such that no vertex appears more than once.

The matching number is denoted by β₁.

Upper Bound

For a graph with n vertices, the matching number has the following upper bound −

β₁ ≤ ⌊n / 2⌋

This is because each edge in a matching uses exactly 2 vertices, and no vertex can be reused. So the maximum possible matching size is ⌊n / 2⌋.

Problem Statement

What is the matching number for the following graph ?

Solution

The graph has 9 vertices. The theoretical upper bound is −

β? ≤ ⌊n / 2⌋
β? ≤ ⌊9 / 2⌋
β? ≤ 4

Since 9 is an odd number, at most 8 vertices can be matched (one vertex must remain unmatched). We can find a matching that uses 4 edges covering 8 of the 9 vertices, as shown below −

The highlighted edges form a maximum matching − each selected edge connects two vertices, no vertex is used more than once, and no larger matching is possible. Vertex 'e' remains unmatched.

The four matched edges are −

Therefore, the matching number is −

β₁ = 4

Conclusion

The matching number of a graph is the maximum number of edges that can be selected without any two edges sharing a vertex. For a graph with n vertices, it is always at most ⌊n / 2⌋. When n is odd, at least one vertex will remain unmatched in any maximum matching.