 Hall's marriage theorem

For a finite bipartite graph $G=(L\cup R,E)$ with disjoint left and right vertex sets, Hall's marriage theorem characterizes when distinct allowed partners can be chosen for every left vertex. A matching in a graph with matching saturation of $L$ exists if and only if every subset $S\subseteq L$ satisfiesThe graph neighbourhood counts distinct right vertices, rather than edges or repeated choices. This is the Hall condition. Its failure supplies a concrete Hall obstruction. Necessity follows because the chosen partners of $S$ are distinct members of $N_G(S)$; sufficiency is the surprising assertion that these collective counts rule out every obstacle.^[ref]

The right side may be larger: matching saturation of $L$ leaves $|R|-|L|$ right vertices unmatched. Only when $|L|=|R|$ does a matching in a graph saturating $L$ automatically become a perfect matching. For a specified subset $X\subseteq L$, apply the same criterion only to $S\subseteq X$, using the induced subgraph on $X\cup R$. The remaining left vertices need not be matched. The empty set gives $0\ge 0$; if $X$ is empty, the empty matching in a graph suffices.

Consider a scheduling model with three unit-duration jobs $a,b,c$ and four available slots $1,2,3,4$ on one machine. Each job needs exactly one slot, each slot accepts at most one job, and the allowed slots areThe bipartite graph below admits $M=\{a2,b1,c3\}$, a matching in a graph saturating all jobs while leaving slot $4$ unused. It is not a perfect matching of the seven-vertex graph. A greedy choice $a1$ initially blocks $b$, but the augmenting path in a matching $b,1,a,2$ replaces $a1$ by $b1,a2$; subsequent assignment of $c3$ succeeds. A bad partial choice is not an impossibility certificate.

There are eight subsets of $L=\{a,b,c\}$. Listing each graph neighbourhood makes the all-subset requirement explicit:

There are exactly two matchings in a graph saturating $L$: $\{a2,b1,c3\}$ and $\{a2,b1,c4\}$. Job $b$ forces slot $1$, then job $a$ forces slot $2$, leaving two choices for $c$.

Hall's Marriage Theorem is a result in combinatorial mathematics, specifically in the area of graph theory and bipartite matching. It provides a necessary and sufficient condition for the existence of a perfect matching in a bipartite graph.