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Black Bishop Graph


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 (m,n)-black bishop graph is therefore a connected component of the general (m,n)-bishop graph. It is isomorphic to the (m,n)-white bishop graph unless both m and n are odd.

Special cases are summarized in the following table.

(m,n)graph
(1,n)empty graph K^__(|_n/2_|)
(2,n)path graph P_n
(3,3)butterfly graph
(n,n+1)n-triangular honeycomb bishop graph

Stan Wagon (pers. comm., Dec. 5, 2018) considered the set of graphs with vertices corresponding to all subsets of the integers 1, ..., n-k of size n-1 and with edges between vertices that agree as vectors in exactly one position. Wagon noted that the graphs with n=3 correspond to the (k+2,k+3)-black bishop graphs.


See also

Bishop Graph, King Graph, Knight Graph, Rook Graph, White Bishop Graph

This entry contributed by Stan Wagon

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Cite this as:

Wagon, Stan. "Black Bishop Graph." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BlackBishopGraph.html

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