TITLE: Is the Cheeger constant of an induced subgraph of a cube at most 1?
QUESTION [8 upvotes]: It is known that the
Cheeger constant
of a
hypercube graph $Q_n$
is exactly $1$, regardless of its dimension $n$.  Is $1$ also an upper bound
on the Cheeger constant of nontrivial induced connected subgraphs of $Q_n$?
The Cheeger constant $h(G)$ is also known as the
edge expansion
or the isoperimetic number.
To indirectly address a comment,
here are the Cheeger constants of more graphs:
$h($$Q_n$$) = 1$ 
$h($$P_n$$) = 1 / \lfloor n / 2 \rfloor$ 
$h($$C_n$$) = 2 / \lfloor n / 2 \rfloor$ 
$h($$K_n$$) = \lceil n / 2 \rceil$

REPLY [6 votes]: Your conjecture is true, every subgraph of the cube has expansion constant at most $1$.
Proof: Suppose we are given a subgraph $G\subset Q_n$, $n>1$ and cut the cube into $2$ $(n-1)$-dimensional subcubes $A_1,A_2$. (So that $A_1\cup A_2=Q_n$)  The key is to notice that each vertex in $A_1$ is connected to one and only one vertex in $A_2$.  Then split $G$ into two parts, $G_1=G\cap A_1$ and $G_2=G\cap A_2$. One of these will have size $\leq \frac{|G|}{2}$, suppose it is $G_1$.  Because $G_1$ is in $A_1$, it is only connected to vertices in $A_2$, and we have $\partial G_1 \leq |G_1|$.   Since the expansion constant is the minimum, we conclude that $$h(G)\leq 1.$$