Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 17G (Graph Theory)

The Hall marriage theorem states that a finite bipartite graph with vertex classes $U$ and $V$ has a matching in a graph saturating $U$ if and only ifNecessity is immediate: distinct vertices of $A$ must be matched to distinct vertices of $N(A)$.

For sufficiency, induct on $|U|$. If every nonempty proper $A\subset U$ satisfies the strict bound $|N(A)|\ge |A|+1$, choose any edge $uv$ and delete its endpoints. Every $A\subseteq U\setminus \{u\}$ loses at most the neighbour $v$, so Hall's condition remains true; induction supplies a matching of the smaller graph, to which $uv$ is added.

Otherwise there is a nonempty proper tight set $A\subset U$ with $|N(A)|=|A|$. Hall's condition holds on the induced graph with parts $A,N(A)$, so induction matches $A$. For $B\subseteq U\setminus A$, Hall's condition on $A\cup B$ givesand hence $|N(B)\setminus N(A)|\ge |B|$. Induction therefore matches $U\setminus A$ into $V\setminus N(A)$. The two matchings are disjoint and together saturate $U$, completing the proof through Hall induction through a tight set.

Identify a subset of $[n]$ with its binary characteristic vector. Then $Q$ is the hypercube graph, and every edge changes set size by exactly one. The induced subgraph on $X_{i-1}\cup X_i$ is therefore bipartite with those two levels as its vertex classes.

Let $\mathcal{A}\subseteq X_{i-1}$. Every member of $\mathcal{A}$ has $n-i+1$ neighbours in $X_i$, obtained by adding one element, while every member of $N(\mathcal{A})$ is incident with at most $i$ of these edges. Double-counting the edges between the two families givesSince $i\le n/2$, we have $n-i+1\ge i$, and hence $|N(\mathcal{A})|\ge |\mathcal{A}|$. Hall's theorem yields the requested adjacent-level matching in a Boolean lattice, saturating $X_{i-1}$.

Finally, every chain in a partially ordered set in the Boolean lattice contains at most one member of the middle level $X_{n/2}$. Any chain partition therefore uses at leastchains.

For every $i\le n/2$, choose one of the matchings just constructed and orient its edges upward. These edges partition all levels at or below $n/2$ into chains ending at distinct middle-level sets. Taking complements gives, for every $i\ge n/2$, a matching from $X_{i+1}$ into $X_i$; orienting its edges upward extends those chains through all upper levels. Every subset belongs to exactly one resulting chain and every chain meets $X_{n/2}$ exactly once. Thus this symmetric chain decomposition of a Boolean lattice has one chain per middle-level set, and the minimum is