A bishop graph is a graph formed from possible moves of a bishop chess piece, which may make diagonal moves of any length on a chessboard (or any other board). To form the graph, each chessboard square is considered a vertex, and vertices connected by allowable bishop moves are considered edges.
Because bishops starting on squares of one color and moving diagonally always remain on the same color squares, all bishop graphs are disconnected (except for the trivial
singleton graph on a 
board which is trivially connected).
Special cases are summarized in the following table.
 -bishop graph | graph |
 |  -empty
graph  |
 | 2  -path
graphs  |
The connected components of an 
-bishop graph corresponding to bishops moving on white
squares and black squares (i.e., the white bishop
graph and black bishop graph, respectively),
illustrated above for small square chessbaords, are isomorphic iff 
and 
are not both odd. Note that here, "white" and "black"
refer to the color of the squares a given bishop moves on irrespective of the color
of the bishop piece itself.
Closed formulas for the numbers 
of 
-graph cycles of 
are given by
 |  |  |
(1)
|
 |  |  |
(2)
|
and
 |
(3)
|
for 
, 7, ..., the last of which is due
to Perepechko and Voropaev.
S. Wagon (pers. comm., Aug. 17, 2012) showed that the 
-white bishop graph B(m,n) is Hamiltonian
for 
and when 
and 
, and nonhamiltonian for 
and the trivial cases 
or 1.
All bishop graphs are perfect.
