Fiveleaper Graph

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 board 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 vertices connected by allowable fiveleaper moves are considered edges. The fiveleaper gets its name from the fact that all its move have a length of 5 squares.

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 graphs corresponding to antelope 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 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 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].

REFERENCES

Jelliss, G. Knight's Tour Notes. 2019. http://www.mayhematics.com/p/KTN10_Leapers.pdf

Marlow, T. W. and Jelliss, G. P. "Fiveleaper Tours." May 2002. https://www.mayhematics.com/t/pf.htm.