A Pósa rotation is an operation on a graph path used in the study of Hamiltonian paths and Hamiltonian cycles. Let
 |
(1)
|
be a path in a graph 
.
If the endpoint 
is adjacent to 
for some 
,
then replacing the edge 
by 
and reversing the terminal subpath
gives the new path
 |
(2)
|
The vertex 
is called the pivot of the rotation, the endpoint 
remains fixed, and the other endpoint changes from 
to 
.
Repeated Pósa rotations preserve the vertex set of the path while producing many possible endpoints. Together with steps that extend a path to a new vertex or close it into a cycle, these rotations form the Pósa rotation-extension technique, a standard tool in Hamiltonicity arguments, especially for random graphs.
For a longest path 
in 
, let 
be the set of endpoints obtainable from 
by a finite sequence of Pósa rotations with one endpoint
fixed. Pósa's lemma states that the external neighborhood 
, consisting of the vertices outside 
adjacent to a vertex of 
, satisfies
 |
(3)
|
(Pósa 1976).
