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Zebra Graph


ZebrasTour

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 (2,3)-leaper graphs.

Zebra graphs are bicolorable, bipartite, class 1, perfect, triangle-free, and weakly perfect.

The square (n×n) zebra graph is connected for n=1 and n>=6.

It is traceable for n=1, 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 10×10 board, first solved in 1886 by Frost (Jelliss). There are a total of 80320 Hamiltonian cycles on this board. For n<=20, the square board is Hamiltonian for exactly n=1, 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.

Cite this as:

Weisstein, Eric W. "Zebra Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ZebraGraph.html

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