A bi-Cayley graph over a group 
is a graph admitting a semiregular
group action of 
by graph automorphisms with exactly two group orbits on its vertex
set. Equivalently, for suitable subsets 
, a bi-Cayley graph is a graph 
that has vertices
 |
with right edges 
whenever 
,
left edges 
whenever 
,
and spoke edges 
whenever 
.
For a simple graph, the subsets 
and 
are inverse-closed (closed under taking inverse
elements) and do not contain the identity element.
The term semi-Cayley graph is sometimes used for the same notion.
When 
is a cyclic group, bi-Cayley graphs are bicirculant
graphs. In particular, the bi-Cayley graphs over cyclic
groups with both orbits independent are the Haar graphs.
