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.
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.
| 👁 (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 |
Rather surprisingly, the 👁 n×(n+1)
black bishop graph is isomorphic to the 👁 n
-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, ..., 👁 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, Triangular Honeycomb Bishop Graph, White Bishop GraphThis entry contributed by Stan Wagon
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References
Wagon, S. "Graph Theory Problems from Hexagonal and Traditional Chess." College Math. J. 45, 278-287, 2014.Referenced on Wolfram|Alpha
Black Bishop GraphCite this as:
Wagon, Stan. "Black Bishop Graph." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BlackBishopGraph.html