A fiveleaper graph is a graph formed by all possible moves of a hypothetical chess piece called a "fiveleaper" which moves analogously to a knight except that it is restricted to moves that change by three squares along one axis of the chessboard and four squares along the other or by five squares along one axis. To form the graph, each chessboard square is considered a vertex, and edges join vertices related by allowable fiveleaper moves.
The 
fiveleaper graph is therefore a Euclidean
distance graph with Euclidean distance
5 on the vertex set of the 
grid graph, meaning that
all its moves have a Euclidean distance of
5 squares, which is the reason for the graph's
name.
The fiveleaper is similar to the hypothetical chess piece called an "antelope," but it can make an antelope's move or a rook's move of exactly 5 squares.
The plots above show the fiveleaper graphs on 
chessboards for 
to 7.
The 
fiveleaper graph is connected for 
(trivially) and 
, Hamiltonian for
(trivially) and 8, 10, 12, 14, ...(and
all other even values of 
but for no odd 
up to at least 
), and traceable for
up to at least 
(and likely for all larger values as well).
The 
fiveleaper graph is connected for 
, Hamiltonian for
(trivially) and even 
(up to at least 
and likely all larger values), and traceable
for 
(up to at least 
and likely for all larger values as well).
Precomputed properties of fiveleaper graphs are implemented in the Wolfram Language as GraphData[
"Fiveleaper", 
m, n
].
