# Usage of the term “perfect matching” for bipartite graphs

A textbook written by my discrete mathematics teacher defines a "perfect matching" in a bipartite graph as a matching that covers at least one side of the graph (i.e. for $G = (V_1, V_2, E)$ with $V_1$ and $V_2$ as the two sets of vertices and $E$ as the set of edges, the number of edges in the "perfect matching" should be equal to the number of vertices in $V_1$ or $V_2$).

However, other sources (including Wikipedia) define a "perfect matching" as containing all vertices of a graph.

Do these definitions of the term conflict? If yes, which is the correct one?

A perfect matching has a $1$ to $1$ correspondence between both sides of the bipartite graph.
• @Mints97: The term is usually defined as in Wikipedia, so that a bipartite graph cannot have a perfect matching unless $|V_1|=|V_2|$, but I have seen the more general definition as well, requiring that only one part be completely matched. – Brian M. Scott Feb 16 '15 at 10:40
• @Mints97 I agree with Brian, I've seen both in use. The terminology I've seen that is the most clear is to define a bipartite matching as perfect only if $|V_1| = |V_2|$, and say there is a matching saturating $V_1$ when a matching meets all vertices of $V_1$, where $|V_1| \leq |V_2|$. – Perry Elliott-Iverson Feb 17 '15 at 17:49