See , a perfect matching, every vertex of the graph is incident to exactly one edge of the matching. A perfect matching is therefore a matching of a graph containing 
edges, the largest possible, meaning perfect matchings are only possible on graphs with an even number of vertices. A perfect matching is sometimes called a complete matching or 1-factor.
The nine perfect matchings of the cubical graph are illustrated above.
Note that rather confusingly, the class of graphs known as perfect graphs are distinct from the class of graphs with perfect matchings.
A graph 
has a perfect matching iff its matching number 
satisfies
where 
is the vertex count of 
.
