The middle layer graph of order 
is the vertex-transitive
graph whose vertex set consists of all bitstrings
of length 
that have exactly 
or 
entries equal to 1, with a graph 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 Hamiltonian
cycle for every 
is known as the middle levels conjecture.
The middle layer graph corresponds to the bipartite Kneser graph 
.
It is the sparsest bipartite Kneser graph
and so is in some sense 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
].
