Triangular Honeycomb Bishop Graph
The π n
-triangular
honeycomb bishop graph π B_n
(DeMaio and Tran 2013), called the hex bishop graph and
denoted π HB_n
by Wagon (2014), is a graph consisting of vertices on a triangular
honeycomb board with π n
vertices along each side, where vertices are connected by
an edge if they lie on the same π +60 degrees
or π -60 degrees
diagonal line of the chessboard (DeMaio and Tran
2013, Wagon 2014). The graphs for π n=3
and 4 are illustrated above.
Note the moves considered in this definition differ from those allowed by the bishop piece in GliΕski's hexagonal chess (GliΕski 1973).
Rather surprisingly, the π n
-triangular honeycomb bishop graph is isomorphic to the π nΓ(n+1)
black bishop graph (Wagon 2014). Other special
cases are summarized in the following table.
| π n | isomorphic graph |
| 1 | singleton graph π K_1 |
| 2 | path
graph π P_3 |
| 3 | cis-square with two triangles |
The π n
-triangular
honeycomb bishop graph has vertex count and edge
count given by
| π V_n | π = | π (n+1; 2) |
(1)
|
| π Image | π = | π 1/2n(n+1) |
(2)
|
| π E_n | π = | π 1/3(n-1)n(n+1), |
(3)
|
where π (n; k)
is a binomial coefficient.
Triangular honeycomb bishop graphs are black bishops, class 1, claw-free, connected, line, nongeometric, perfect, quadratically embeddable, traceable, and weakly perfect.
Triangular honeycomb bishop graphs are implemented in the Wolfram Language as [π {
,
nπ }
].
See also
Bishop Graph, Black Bishop Graph, Triangular Grid Graph, Triangular Honeycomb BoardExplore with Wolfram|Alpha
More things to try:
References
DeMaio, H. and Tran, L. "Domination and Independence on a Triangular Honeycomb Chessboard." College Math. J. 44, 307-314, 2013.GliΕski, W. Rules of Hexagonal Chess With Examples of First Openings. London: Hexagonal Chess Publications, 1973.Wagon, S. "Graph Theory Problems from Hexagonal and Traditional Chess." College Math. J. 45, 278-287, 2014.Referenced on Wolfram|Alpha
Triangular Honeycomb Bishop GraphCite this as:
Weisstein, Eric W. "Triangular Honeycomb Bishop Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TriangularHoneycombBishopGraph.html