The 
-triangular
honeycomb king graph, called the hex king graph by Wagon (2014), is a graph consisting
of vertices in triangular honeycomb board
with 
vertices along each side, where vertices are connected by an edge if they are adjacent
along a horizontal, 
, or 
line of the chessboard (DeMaio and Tran 2013, Wagon
2014). It is denoted 
by DeMaio and Tran (2013) and 
by Wagon (2014). The graphs for 
and 4 are illustrated above.
As is clear from the diagrams, the 
-triangular king graph is isomorphic to the triangular
grid graph 
of Wagon (2014) and the 
-triangular grid graph
using the indexing convention of West (2000).
Triangular honeycomb king graphs are apex, bridgeless, connected, Eulerian, Hamiltonian, linklessly embeddable, map, matchstick, planar, projective planar, quadratically embeddable, rigid, traceable, triangular grids, uniquely colorable, unit-distance, and weakly perfect.
Triangular honeycomb king graphs are implemented in the Wolfram Language as GraphData[
"TriangularHoneycombKing",
n
].
See also
King Graph, Triangular Grid Graph, Triangular Honeycomb Board
Explore with Wolfram|Alpha
References
DeMaio, H. and Tran, L. "Domination and Independence on a Triangular Honeycomb Chessboard." College Math. J. 44, 307-314, 2013.Konhauser, J. D. E.; Vellman, D.; and Wagon, S. Which Way Did the Bicycle Go and Other Intriguing Mathematical Mysteries. Washington, DC: Amer. Math. Soc., 1996.Wagon, S. "Graph Theory Problems from Hexagonal and Traditional Chess." College Math. J. 45, 278-287, 2014.West, D. B. Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 390-392, 2000.
Referenced on Wolfram|Alpha
Triangular Honeycomb King Graph
Cite this as:
Weisstein, Eric W. "Triangular Honeycomb King Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TriangularHoneycombKingGraph.html