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.

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.

Karavaev, A. M. "FlowProblem: Statistics of Simple Cycles." http://flowproblem.ru/paths/statistics-of-simple-cycles.Perepechko,
S. N. and Voropaev, A. N. "The Number of Fixed Length Cycles in an
Undirected Graph. Explicit Formulae in Case of Small Lengths."