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
joined by edges. The graphs
above give the positions on square chessboards that
are reachable by zebra moves. Zebra graphs are therefore 
-leaper graphs, as well
as Euclidean distance graphs with distance
.
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 Hamiltonian cycle exists (i.e., for which the underlying zebra graph is Hamiltonian)
is the 
board, first solved by Frost (1886, cited by Jelliss 2019). 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
].
2,
3
." §10.31 in Knight's Tour
Notes. 2019.