A black 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), when starting from a black square on the board. To form the graph, each chessboard square is considered a vertex, and vertices connected by allowable bishop moves are considered edges.
The 
-black
bishop graph is therefore a connected component
of the general 
-bishop graph. It is isomorphic to the 
-white bishop graph
unless both 
and 
are 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.
Special cases are summarized in the following table.
-
Rather surprisingly, the 
black bishop graph is isomorphic to the 
-triangular
honeycomb bishop graph (Wagon 2014).
Stan Wagon (pers. comm., Dec. 5, 2018) considered the set of graphs with vertices corresponding to all subsets of the integers 1, ..., 
of size 
and with edges between vertices that agree as vectors in
exactly one position. Wagon noted that the graphs with 
correspond to the 
-black bishop graphs.
