A zebra graph is a graph formed by all possible moves of a hypothetical chess piece called a "zebra" which moves analogously to a knight except that it is
restricted to moves that change by two squares along one axis of the board and three
squares along the other. To form the graph, each chessboard square is considered
a vertex, and vertices connected by allowable zebra moves are considered edges. The
graphs above gives the positions on square chessboards that are reachable by zebra
moves. Zebra graphs are therefore a 
-leaper graphs, as well
as the Euclidean distance-
graph.
Zebra graphs are bicolorable, bipartite, class 1, perfect, triangle-free, and weakly perfect.
The square (
)
zebra graph is connected for 
and 
.
It is traceable for 
, 10, 14, 15, 16, 17, 18, 19, and 20, with the status of
13 open.
The smallest nontrivial square board where a tour exists (i.e., for which the underlying zebra graph is Hamiltonian) is the 
board, first solved in 1886 by Frost (Jelliss). There
are a total of 
Hamiltonian cycles on this board. For 
, the square board is Hamiltonian
for exactly 
,
10, 14, 16, 18, and 20.
Precomputed properties of zebra graphs are implemented in the Wolfram Language as GraphData[
"Zebra", 
m, n
].
See also
Antelope Graph, Camel Graph, Fairy Chess, Fiveleaper Graph, Giraffe Graph, Knight Graph, Leaper Graph
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References
Cross, H. H. Problem 4709 in Fairy Chess Review. Feb. 1941.Frost, A. H. Plate VII in M. Frolow. Les
Carrés Magiques. Paris, 1886.Jelliss, G. "The Big Beasts:
Zebra 
2,
3
." §10.31 in Knight's Tour
Notes. 2019. http://www.mayhematics.com/p/KTN10_Leapers.pdfJelliss,
G. Chessics.Jelliss, G. P. "Generalized Knights and
Hamiltonian Tours." J. Recr. Math. 27, 191-200, 1995.Jelliss,
G. P. "Longer Leaper Tours with Quaternary Symmetry." The Games
and Puzzles Journal 2, No. 2, p. 290, 1999.Kraitchik,
M. 'Mathematical
Recreations. New York: W. W. Norton, pp. 70-73, 1942.Willcocks,
T. H. Chessics. 1978.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Zebra Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ZebraGraph.html