The middle layer graph of order 
is the graph whose vertex set consists of all bitstrings of
length 
that have exactly 
or 
entries equal to 1, with an edge between any two vertices for which the corresponding
bitstrings differ in exactly one bit (Mütze 2016). The conjecture (now proved)
that the middle layer graph has a Hamilton cycle
for every 
is known as the middle levels conjecture.
Since this conjecture has been proved, middle layer graphs are therefore uniquely
Hamiltonian.
The middle layer graph corresponds to the bipartite Kneser graph 
.
It is the sparsest bipartite Kneser graph
and so is in some sense is the hardest obstacle in proving Hamiltonicity for this
family of graphs (Mütze 2016). Special cases are summarized in the table below.
 |  -middle layer graph |
| 1 | cycle graph  |
| 2 | Desargues graph  |
| 3 | Danzer graph |
The middle layer graph is bipartite, connected, and has 
vertices (Mütze 2016).
The middle layer graph of order 
is implemented in the Wolfram
Language as GraphData[
"MiddleLayer", n
].
