TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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Hall's marriage theorem for a finite bipartite graph

Statement

For a finite bipartite graph with parts $(X,Y)$ , there is a matching saturating $X$ if and only if $|N(S)|\ge |S|$ for every $S\subseteq X$ .

Facts & Assumptions

Given: A finite bipartite graph with specified parts $(X,Y)$ .

[L1]

The strict and tight Hall subinstances in the induction have Hall's condition (Hall's condition passes to the strict and tight induction subinstances).

[L2]

The induction principle proves a statement for every natural number from its base case and successor step (The principle of mathematical induction).

Proof

technique · induction on $|X|$

1.1

For $|X|=0$ the empty matching saturates $X$ ; for $|X|=1$ , Hall gives a neighbour and its incident edge saturates $X$ .

L2base

1.2

Any matching that saturates $X$ assigns distinct neighbours to each $S\subseteq X$ , hence has $|N(S)|\ge|S|$ .

given

1.3

In the strict case with $|X|\ge2$ , choose any edge $xy$ ; [L1] gives Hall after deleting $x,y$ , so induction supplies a matching there saturating $X\setminus\{x\}$ , and adjoining $xy$ saturates $X$ .

L1ih

1.4

In the tight case, [L1] gives Hall on the two smaller left parts $S$ and $X\setminus S$ ; induction gives saturating matchings in each, and their disjoint vertex sets let their union saturate $X$ .

L1ih

2.1

The base cases and the two alternatives prove Hall's sufficient direction by [L2], and step 1.2 proves its necessary direction.

L2step 1.2step 1.3step 1.4discharge-induction∎

Remarks

This is the finite theorem only. No infinite-family or choice-principle claim is being made here.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 25 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

F. Gotti, Matching and Hall's Theorem (standard reference, not scraped)